Abstract Integration with Respect to Measures and Applications to Modular Convergence in Vector Lattice Setting

Abstract

A “Bochner-type” integral for vector lattice-valued functions with respect to (possibly infinite) vector lattice-valued measures is presented with respect to abstract convergences, satisfying suitable axioms, and some fundamental properties are studied. Moreover, by means of this integral, some convergence results on operators in vector lattice-valued modulars are proved. Some applications are given to moment kernels and to the Brownian motion.

1 Introduction

In the literature there are many studies concerning the problem of approximating a real-valued function f by Urysohn-type integral operators or discrete sampling operators. These topics together with some other kind of operators, have several applications in several branches, for instance neural networks and reconstruction of signals and images (see, e.g., [1, 2, 5,6,7,8,9, 22, 25,26,27,28,29]).

In the classical setting, a signal is viewed as a function f, defined on a suitable finite or infinite time interval, which is reconstructed by starting of sampled values of the type f(k/w), where k is a suitable (finite or not) subset of ${\mathbb {Z}}$, and w is the theoretical minimum sampling rate necessary to reconstruct the signal entirely (see, e.g., [8] and the references therein). Thus, the original unknown signal can be considered as a function with values in a space consisting of random variables, that is (classes of equivalence of) measurable functions. Among them, the spaces $L^0$ or $L^2$ are widely used in stochastic processes. In these kinds of spaces, it is often advisable to deal with almost everywhere convergence, which does not have a topological nature, but corresponds to unbounded order convergence (see [34, 41] and the references therein), if we endow the involved space with the “component-wise” order and supremum. Thus, it is natural to investigate vector lattice-valued functions, defined on a set of (possibly infinite) measure, like for example the Lebesgue measure on a halfline or the whole of ${\mathbb {R}}$ (see, e.g., [13]).

In this paper we extend to the vector lattice setting the problem of approximating a function f by means of Urysohn-type integral operators in the setting of modular convergence, extending some earlier results proved in [6, 7, 13, 15]. We consider three vector lattices ${{\textbf{X}}}_1$, ${{\textbf{X}}}_2$, ${{\textbf{X}}}$, “linked” by a suitable “product” structure, we deal with ${{\textbf{X}}}_1$-valued functions defined on a metric space, (possibly infinite) ${{\textbf{X}}}_2$-valued measures defined on an algebra of subsets of the set and construct an ${{\textbf{X}}}$-valued integral, which will be an extension of the integrals investigated in [11], where only finite measures are considered, and in [13], where it is supposed that ${{\textbf{X}}}$ and ${{\textbf{X}}}_1$ are endowed with stronger order units e and $e_1$, respectively. We drop this hypothesis in our context. We endow ${{\textbf{X}}}_1$, ${{\textbf{X}}}_2$ and ${{\textbf{X}}}$ with abstract convergences and structures of “limit superior”, satisfying suitable axioms, in order to include usual relative uniform, order convergence and the related almost and Cesàro convergences in the general case, the order filter convergence at least in the $L^p$ case (with $0 \le p \le \infty$), and the relative uniform filter convergence when ${{\textbf{X}}}$ and ${{\textbf{X}}}_1$ have order units. Note that this type of convergence is equivalent to the norm filter convergence in ${{\mathbb {R}}}$, where we have the norm generated by the order unit.

The paper is structured as follows. In Sect. 2 we present the abstract axioms on convergence and limit superior and give some examples. In Sect. 3 we develop integration theory in our setting, giving Vitali-type theorems and a version of the Lebesgue dominated convergence theorem, we compare our integral with the classical Lebesgue integral, and state some main propertiers, among which some versions of Jensen’s inequality, in connection with uniformly continuous and convex functions. The proof of these results will be given in the Sect. 7. In Sect. 4 we recall the theory of vector lattice-valued modulars introduced in [13]. In Sect. 5 we give the structural hypotheses on the involved operators, and in Sect. 6 we present our main results on modular convergence of operators in Orlicz spaces in the vector lattice setting. In Sect. 6.1, as examples and applications, we deal with moment kernels in the vector lattice context, and with Itô-type integrals with respect to Brownian motion.

2 Preliminaries

Let $({\textbf{X}}, \le _X)$ be a vector lattice and let the symbols $\vee$ and $\wedge$ denote the lattice suprema and infima in ${\textbf{X}}$. We say that ${\textbf{X}}$ is Dedekind complete iff every nonempty subset $A \subset {\textbf{X}}$, order bounded from above, admits a lattice supremum in ${\textbf{X}}$, denoted by $\bigvee A$. From now on, ${\textbf{X}}$ is a Dedekind complete vector lattice, ${\textbf{X}}^{+}:=\{ x \in {\textbf{X}}: x \ge _X 0\}$ is its positive cone, and we denote by the symbol ${\mathbb {R}}_0^+$ (resp., ${\mathbb {R}}^+$), as usual, the set of non-negative (resp., strictly positive) real numbers. For every $x \in {\textbf{X}}$, set $\vert x\vert = x \vee (-x)$.

We add to ${\textbf{X}}$ an extra element $+\infty$, extending the order and the operations on ${\textbf{X}}$ in a natural way; see for example [32], where the author introduced the use of an embedding cone that has a largest element (=$\infty$) into which a Riesz space can be embedded.

Let $\overline{{\textbf{X}}}={\textbf{X}} \cup \{+ \infty \}$, $\overline{{\textbf{X}}}^{+}={\textbf{X}}^{+} \cup \{+ \infty \}$, and assume, by convention, $0 \cdot (+ \infty ) = 0$.

An (o)-sequence $(\sigma _l)_{l \in {\mathbb {N}}}$ in ${\textbf{X}}^{+}$ is a decreasing sequence with $\wedge _l \sigma _l=0$ (see, e.g., [19]). For unexplained notions we refer, for example, to [11, 16].

A strong order unit of a vector lattice ${{\textbf{X}}}$ is an element $e\in {{\textbf{X}}}^+ \setminus \{0\}$ with the property that, for every $x \in {{\textbf{X}}}$, there exists $\delta \in {{\mathbb {R}}}^+$ such that $\vert x\vert \le _X \delta \, e$. For example, if $e\in {{\textbf{X}}}^+ \setminus \{0\}$, then e is a strong order unit of $V[e]:=\{x \in {{\textbf{X}}} : \text { there is } \delta \in {\mathbb {R}}^+$ with $\vert x\vert \le _X \delta \, e \}.$ Moreover we observe that V[e] is solid in ${{\textbf{X}}}$ and Dedekind complete. Furthermore, by virtue of the Kakutani-Krein theorem (see, e.g., [45, Theorem II.7.4]), V[e] is algebraically and lattice isomorphic to the space of all real-valued continuous functions defined on a suitable compact and extremely disconnected topological space $\Omega$ (see, e.g., [38]). A similar, more general theorem (Maeda-Ogasawara-Vulikh-type representation theorem) holds also for any Dedekind complete vector lattice ${\textbf{X}} \hookrightarrow {\mathcal {C}}_{\infty }(\Omega ) = \{f \in \widetilde{{\mathbb {R}}}^{\Omega } : f \text { is continuous, and } \{\omega : \vert f(\omega ) \vert =+\infty \} \text { is nowhere dense in } \Omega \}$ (see, e.g., [33, Theorem 2.1], [42, 43, 46]). We will use the symbols $\sup$ and $\inf$ to denote both the pointwise suprema and infima in ${{\mathcal {C}}}_{\infty }(\Omega )$ and those in the (extended) real line.

Let ${{\mathcal {T}}}$ be the set of all sequences $(x_n)_n$ in ${\textbf{X}}$ and ${{\mathcal {T}}}^{+}=\{ (x_n)_n \in {{\mathcal {T}}}$: $x_n \ge _X 0$ for each $n \in {\mathbb {N}}\}$. Now we give an axiomatic approach to convergence in vector lattices (see, e.g., [11, Definition 2.1] and [3]).

Axioms 2.1

A convergence is a pair $({\mathcal {S}},\ell )$, where ${\mathcal {S}}$ is a linear subspace of ${{\mathcal {T}}}$ and $\ell$ is a function $\ell :{\mathcal {S}} \rightarrow {\textbf{X}}$, which satisfies the following conditions:

• 2.1.a) $\ell ((\zeta _1 \, x_n + \zeta _2 \, y_n)_n)= \zeta _1 \, \ell ( (x_n)_n)+ \zeta _2 \, \ell ( (y_n)_n)$ for every pair of sequences $(x_n)_n$, $(y_n)_n \in {\mathcal {S}}$ and for each $\zeta _1$, $\zeta _2 \in {\mathbb {R}}$ (linearity).

• 2.1.b) If $(x_n)_n$, $(y_n)_n \in {\mathcal {S}}$ and $x_n \le _X y_n$ definitely, namely for all $n \in {\mathbb {N}}$ greater than a suitable integer m, then ${\ell }((x_n)_n) \le _X \ell ((y_n)_n)$ (monotonicity).

• 2.1.c) If $(x_n)_n$ is such that $x_n=l$ definitely, then $(x_n)_n \in {\mathcal {S}}$ and $\ell ((x_n)_n)=l$; if $(x_n)_n$, $(y_n)_n$ are such that the set $\{n \in {\mathbb {N}}: x_n \ne y_n \}$ is finite and $(x_n)_n \in {\mathcal {S}}$, then $(y_n)_n \in {\mathcal {S}}$ and $\ell ((y_n)_n)=\ell ((x_n)_n)$.

• 2.1.d) If $(x_n)_n \in {\mathcal {S}}$, then $(\vert x_n\vert )_n \in {\mathcal {S}}$ and $\ell ((\vert x_n\vert )_n)=\vert \ell ((x_n)_n)\vert$.

• 2.1.e) Given three sequences $(x_n)_n$, $(y_n)_n$, $(z_n)_n$, with $(x_n)_n$, $(z_n)_n \in {\mathcal {S}}$, $\ell ((x_n)_n)=\ell ((z_n)_n)$, and $x_n \le _X y_n \le _X z_n$ definitely, then $(y_n)_n \in {\mathcal {S}}$ (and hence from 2.1.b) it follows that $\ell ((x_n)_n)= \ell ((y_n)_n)=\ell ((z_n)_n))$.

• 2.1.f) If $u \in {\textbf{X}}^{+}$, then the sequence $\Bigl (\dfrac{1}{n} u\Bigr )_n$ belongs to ${\mathcal {S}}$ and $\ell \Bigl (\Bigl (\dfrac{1}{n} u\Bigr )_n\Bigr )=0$.

The next property is a consequence of 2.1.c) and 2.1.e).

• 2.1.g) If $x \in {\textbf{X}}$, $(y_n)_n \in {\mathcal {S}}$ and $x \le _X y_n$ definitely, then $x \le _X \ell ((y_n)_n)$.

Analogously, we now present an axiomatic approach of a “limit superior”-type vector lattice-valued operator related to a convergence $({\mathcal {S}}, \ell )$, satisfying Axioms 2.1 (see, e.g., [3, Section 2]).

Axioms 2.2

Let ${{\mathcal {T}}}$, ${\mathcal {S}}$ be as in Axioms 2.1 and define a function ${\overline{\ell }}: {{\mathcal {T}}}^{+} \rightarrow \overline{{\textbf{X}}}^{+}$ satisfying the following conditions:

• 2.2.a) If $(x_n)_n, (y_n)_n \in {{\mathcal {T}}}^{+}$ are such that $x_n=y_n$ definitely, then ${\overline{\ell }}((x_n)_n)={\overline{\ell }}((y_n)_n)$.

• 2.2.b) If $(x_n)_n, (y_n)_n \in {{\mathcal {T}}}^{+}$, then ${\overline{\ell }}(( x_n + y_n)_n) \le _X {\overline{\ell }}((x_n)_n) + {\overline{\ell }}((y_n)_n)$ (subadditivity).

• 2.2.c) If $(x_n)_n$, $(y_n)_n \in {{\mathcal {T}}}^{+}$ and $x_n \le _X y_n$ definitely, then ${\overline{\ell }} ((x_n)_n) \le _X {\overline{\ell }}((y_n)_n)$ (monotonicity).

• 2.2.d) If a sequence $(x_n)_n \in {{\mathcal {T}}}^{+} \cap {\mathcal {S}}$, then ${{\overline{\ell }}}((x_n)_n) =\ell ((x_n)_n)$.

• 2.2.e) If a sequence $(x_n)_n \in {{\mathcal {T}}}^{+}$ is such that ${{\overline{\ell }}}((x_n)_n) =0$, then $(x_n)_n \in {\mathcal {S}}$ and $\ell ((x_n)_n)=0$.

A filter on ${\mathbb {N}}$ is a family ${\mathcal {F}}$ of subsets of ${{\mathbb {N}}}$ such that $\emptyset \not \in {{\mathcal {F}}}$, $A \cap B \in {{\mathcal {F}}}$ whenever A, $B \in {\mathcal {F}}$, and for every $A \in {\mathcal {F}}$ and $B \subset {\mathbb {N}}$ with $B \supset A$, we have that $B\in {\mathcal {F}}$. We denote by ${{\mathcal {F}}}_{\text {cofin}}$ the filter of all cofinite subsets of ${{\mathbb {N}}}$. We say that a filter on ${{\mathbb {N}}}$ is free iff it contains ${{\mathcal {F}}}_{\text {cofin}}$. An example of free filter is the filter of all subsets of ${\mathbb {N}}$ whose asymptotic density is equal to one (see, e.g., [16]).

Remark 2.3

Some examples of convergences satisfying Axioms 2.1 are: if ${{\mathcal {F}}}$ is a free filter, the relative uniform filter $((r{{\mathcal {F}}})$-)convergence and the order filter $((o{{\mathcal {F}}})$-)convergence (see [13, Definition 1.1]); the almost convergence and the Cesàro convergence (see, e.g., [16, 21]).

We recall that a sequence $(x_n)_n$ in ${\textbf{X}}$ is relatively uniformly filter $((r{{\mathcal {F}}})-)$ convergent to $x \in {\textbf{X}}$, iff there exists $u \in {\textbf{X}}^+ \setminus \{0 \}$ such that $\{ n \in {\mathbb {N}}: \vert x_n - x\vert \le _X \varepsilon \, u \} \in {{\mathcal {F}}}$ for each $\varepsilon \in {\mathbb {R}}^+$; order filter $((o{{\mathcal {F}}})$-)convergent to $x \in {\textbf{X}}$, iff there exists an (o)-sequence $(\sigma _l)_l$ with the property that, for all $l \in {\mathbb {N}}$, $\{ n \in {\mathbb {N}}: \vert x_n - x\vert \le _X \sigma _l \} \in {{\mathcal {F}}}$. If ${{\mathcal {F}}}$ is the filter ${{\mathcal {F}}}_\mathrm{{cofin}}$ of all subsets of ${{\mathbb {N}}}$ whose complement is finite, then the $(r{{\mathcal {F}}}_\mathrm{{cofin}})$- and $(o{{\mathcal {F}}}_\mathrm{{cofin}})$-convergence coincide with the usual (r)- and (o)-convergence, respectively (see, e.g., [16]).

A sequence $(x_n)_n$ in ${\textbf{X}}$ is (r)- ( resp., (o)-) almost convergent to $x \in {\textbf{X}}$, iff

$$\begin{aligned} (r)\text {-} ( \text {resp.,} (o)\text {-}) \lim _n \left( \bigvee _{m\in {\mathbb {N}}} \left| \frac{x_{m+1} + x_{m+2} +\ldots + x_{m+n}}{n}- x \right| \right) =0; \end{aligned}$$

(r)- ( resp., order or (o)-) Cesàro convergent to $x \in {\textbf{X}}$, iff

$$\begin{aligned} (r)\text {-} ( \text {resp.,} (o)\text {-}) \lim _n \frac{x_1 + x_2 +\ldots + x_n}{n}= x. \end{aligned}$$

Observe that $(r{{\mathcal {F}}})$-convergence implies $(o{{\mathcal {F}}})$-convergence, and in general they do not coincide. However, when ${\textbf{X}}={\mathbb {R}}$, these convergences are equivalent. In this case, we will denote both convergences by $({{\mathcal {F}}})$-convergence.

Note that, in general, the usual convergence is strictly stronger than almost convergence, and almost convergence is strictly stronger than the Cesàro one. Moreover, observe that almost and Cesàro convergences are not equivalent to $({{\mathcal {F}}})$-convergence for any free filter. Indeed, if a function $\phi :{\mathbb {R}} \rightarrow {\mathbb {R}}$ is sequentially continuous at 0 with respect to almost (resp., Cesàro) convergence (that is, $\phi$ maps sequences almost (resp., Cesàro) convergent to 0 into sequences almost (resp., Cesàro) convergent to 0), then $\phi$ is linear, but sequential continuity with respect to $({{\mathcal {F}}})$-convergence coincides with usual continuity (see, e.g., [3, 21, 39]).

Example 2.4

Now we give some examples of “limsup”-type operators satisfying Axioms 2.2 and some related properties.

• 2.4.a) The order limit superior of a sequence $(x_n)_n$ in ${{\textbf{X}}}^{+}$ is the element of $\overline{{\textbf{X}}}^{+}$ defined by ${ (o)\limsup _n x_n=\wedge _{m=1}^{\infty } (\vee _{n \ge m} \, x_n)}$ (see, e.g., [46]).

• 2.4.b) Let ${\textbf{X}}={\mathbb {R}}$ and ${{\mathcal {F}}}$ be any fixed free filter on ${\mathbb {N}}$. Given any sequence $(x_n)_n$ in ${\mathbb {R}}^{+}_0$, we call $({{\mathcal {F}}})$-limit superior of $(x_n)_n$ (shortly, $({\mathcal {F}})\limsup _n x_n$) the element of $\overline{{\mathbb {R}}}^{+}_0$ defined by ${ ({\mathcal {F}})\limsup \nolimits _n x_n= \inf \nolimits _{F \in {\mathcal {F}}} (\sup \nolimits _{n \in F}\, x_n)}$ (see, e.g., [12, 30]).

• 2.4.c) Given any sequence $(x_n)_n$ in ${{\textbf{X}}}^{+}$, we call order filter limit superior the element of $\overline{{\textbf{X}}}^{+}$ defined by ${(o{\mathcal {F}})\limsup _n x_n= \bigwedge \nolimits _{F \in {\mathcal {F}}} \Bigl (\bigvee \nolimits _{n \in F} x_n \Bigr )}$. The order filter limit superior satisfies Axioms 2.2.a) - 2.2.d) (see, e.g., [12]). Observe that, in the setting of vector lattices, when ${\textbf{X}} \ne {\mathbb {R}}$, the problem of finding suitable “limsup”-type operators satisfying Axiom 2.2.e) is still open, though some positive partial answers are found in [12]. For example, if T is any nonempty set and ${{\textbf{X}}}={{\mathbb {R}}}^T$, the space of all real-valued functions defined on T, or if R is a Dedekind complete vector lattice and ${{\textbf{X}}}= R^{\sim }$ (resp., $R^{\times }$) is the space of all linear order bounded (resp., order continuous) functionals, then the lattice and “component-wise” suprema and infima coincide (see, e.g., [10]). From this, since Axiom 2.2.e) is satisfied when ${{\textbf{X}}}={{\mathbb {R}}}$, then it holds also when ${{\textbf{X}}}={{\mathbb {R}}}^T$, $R^{\sim }$ or $R^{\times }$. Moreover, observe that the above argument can be applied even if ${{\textbf{X}}}$ is a vector lattice whose elements are equivalence classes of functions, and in which the lattice suprema/infima coincide with the respective classes of equivalence to which the pointwise suprema/infima belong. This is the case of the so-called “Dedekind complete $\rho$-spaces”, whose examples are the spaces $L^p(\Omega , \Sigma , \nu )$, where $0 \le p \le \infty$, $(\Omega , \Sigma , \nu )$ is a measure space and $\nu : \Sigma \rightarrow {\mathbb {R}}^{+}_0$ is a $\sigma$-additive and $\sigma$-finite measure, which do not have a strong order unit and in which the order convergence does not have a topological nature (see, e.g., [10, 46]). These spaces can be viewed also “directly” as spaces of classes of equivalences of real-valued functions defined on $\Omega$ up to $\nu$-null sets, in which the suprema and infima have the aforementioned property.

• 2.4.d) Suppose that ${{\textbf{X}}}$ has a strong order unit e, fix any free filter ${{\mathcal {F}}}$ on ${\mathbb {N}}$ and let us endow ${{\textbf{X}}}$ with $(r{{\mathcal {F}}})$-convergence (see, e.g., [13]). Then, it is not difficult to see that a sequence $(x_n)_n$ in ${\textbf{X}}$ is $(r{{\mathcal {F}}})$-convergent to $x \in {\textbf{X}}$ if and only if $\{ n \in {\mathbb {N}}: \vert x_n - x\vert \le _X \varepsilon \, e\} \in {{\mathcal {F}}}$ for every $\varepsilon \in {\mathbb {R}}^+$.

  Now we endow ${{\textbf{X}}}$ with the norm $\Vert \cdot \Vert _e$, defined by

  $$\begin{aligned} \Vert x\Vert _e=\inf \{\varepsilon \in {\mathbb {R}}^+: \vert x\vert \le _X \varepsilon \, e\} \end{aligned}$$

  (1)

  (see, e.g., [17, 23, 24, 44, 45]). Observe that $\Vert \cdot \Vert _e$ is monotone, that is $\Vert x\Vert _e \le \Vert y\Vert _e$ whenever $x,y \in {\textbf{X}}$ and $\vert x\vert \le \vert y\vert$. Moreover, thanks to the properties of the infimum, it is not difficult to see that

  $$\begin{aligned}{} & {} \vert x \vert \le \Vert x \Vert _e \, e , \end{aligned}$$

  (2)
  $$\begin{aligned}{} & {} \vert x\vert \le _X \alpha \, e \quad \Longleftrightarrow \quad \Vert x\Vert _e \le \alpha , \end{aligned}$$

  (3)

  for all $\alpha \in {\mathbb {R}}^+$ and $x \in {{\textbf{X}}}$ (see, e.g., [46, §7.4 (2)], [44, Proposition 1.2.13]), and

  $$\begin{aligned} \Vert \alpha \, e \Vert _e = \vert \alpha \vert \quad \text {for all } \alpha \in {\mathbb {R}}. \end{aligned}$$

  (4)

  Then, a sequence in ${{\textbf{X}}}$ $(r{{\mathcal {F}}})$-converges if and only if it $({{\mathcal {F}}})$-converges with respect to the norm $\Vert \cdot \Vert _e$. Thus, analogously as in [12, 30], the operator ${{\overline{\ell }}}((x_n)_n)= {({{\mathcal {F}}})\limsup \nolimits _n \Vert x_n\Vert _{e}}$ satisfies Axioms 2.2, because they are fulfilled by the usual filter limit superior in ${{\mathbb {R}}}$, and taking into account the properties of the norm.

• 2.4.e) Analogously as in Remark 2.3, it is possible to associate in a natural way an “order limsup”-type operator to (o)-almost and (o)-Cesàro convergences defined in Remark 2.3, satisfying Axioms 2.2.

• 2.4.f) Note that, if ${\mathbb {R}}$ is endowed with any convergence $\ell _{{\mathbb {R}}}$ satisfying Axioms 2.1 and whose corresponding “limit superior” operator ${\bar{\ell }}_{{\mathbb {R}}}$ fulfils Axioms 2.2, and $(x_n)_n$ is any sequence of non-negative real numbers such that ${\lim _n x_n=0}$ in the usual sense, then $\ell _{{\mathbb {R}}}(x_n)_n=0$. Indeed, fixed arbitrarily $\varepsilon \in {{\mathbb {R}}}^+$, there is a positive integer $n_0$ with $x_n \le \varepsilon$ whenever $n \ge n_0$. From this and Axioms 2.1.c), 2.2.c) and 2.2.d) we deduce that ${\bar{\ell }}_{{\mathbb {R}}}(x_n)_n \le \varepsilon$. By the arbitrariness of $\varepsilon$ and Axiom 2.2.e) it follows that $\ell _{{\mathbb {R}}}(x_n)_n=0$.

3 The Integral with Respect to Abstract Convergences

In this section we give the construction of an abstract integral for vector lattice-valued functions with respect to (possibly infinite) vector lattice-valued measures, extending the integrals presented in [11, 13]. The abstract integral in the vector space setting was first studied by R. G. Bartle in [4] and also, in a series of papers, by I. Dobrakov (see for example [31]). Bartle’s approach is similar to our approach. Dobrakov consider a bilinear form defined by an operator-valued measure, where the operator maps the values of the vector valued variables into a vector space. The technique we have used is to define an algebra structure in which the vector-valued measure can be multiplied by the values of the variables. It fits into the framework of Dobrakov by considering multiplication from the left (or from the right) as the operator used by Dobrakov. In [35] J. J. Grobler and C. C. A. Labuschagne showed how the Dobrakov integral can be defined in vector lattices, and dealt with a stochastic integral where the measure is related to a martingale. In this area we focus only on Brownian motion (see the end of Sect. 6.1).

Let ${\textbf{X}}_1$, ${\textbf{X}}_2$, ${\textbf{X}}$ be three Dedekind complete vector lattices.

Assumption 3.1

We say that $({\textbf{X}}_1,{\textbf{X}}_2,{\textbf{X}})$ is a product triple iff a “product operation” $\cdot :{\textbf{X}}_1 \times {\textbf{X}}_2 \rightarrow {\textbf{X}}$ is defined, satisfying the following conditions [11, Assumption 2.2]:

• 3.1.1) $(x_1+y_1) \cdot x_2= x_1 \cdot x_2 +y_1 \cdot x_{2}$,

• 3.1.2) $x_1 \cdot (x_2+y_2) = x_1 \cdot x_2+x_1 \cdot y_2$,

• 3.1.3) $[x_1 \ge _{X_1} y_1$, $x_2 \ge _{X_2} 0] \Rightarrow [x_{1} \cdot x_{2} \ge _{X} y_{1} \cdot x_{2}]$,

• 3.1.4) $[x_1 \ge _{X_1} 0$, $x_2 \ge _{X_2} y_2] \Rightarrow [x_{1} \cdot x_{2} \ge _{X} x_{1} \cdot y_{2}]$ for every $x_j$, $y_j \in {{\textbf{X}}}_j$, $j=1,2$;

• 3.1.5) if $(x_n)_n$ is an (o)-sequence in ${{\textbf{X}}}_1$ and $y \in {{\textbf{X}}}_2^{+}$, then $(x_n \cdot y )_n$ is an (o)-sequence in ${{\textbf{X}}}$;

• 3.1.6) if $x \in {{\textbf{X}}}_1^{+}$ and $(y_n)_n$ is an (o)-sequence in ${{\textbf{X}}}_2$, then $(x \cdot y_n)_n$ is an (o)-sequence in ${{\textbf{X}}}$ ( see, e.g., [37, 38]),

and if ${\textbf{X}}_1$, ${\textbf{X}}_2$, ${\textbf{X}}$ are endowed with three convergences, $\ell _1$, $\ell _2$, $\ell$, respectively, each of which satisfying Axioms 2.1 and the following “compatibility” conditions:

• 3.1.7) if $(x_n)_n$ is a sequence in ${{\textbf{X}}}_1$ with $\ell _1((x_n)_n) = 0$ and $y \in {{\textbf{X}}}_2$, then ${\ell }((x_n \cdot y )_n)=0$;

• 3.1.8) if $x \in {{\textbf{X}}}_1$ and $(y_n)_n$ is a sequence in ${{\textbf{X}}}_2$ with $\ell _2 ((y_n)_n) = 0$, then ${\ell }((x \cdot y_n )_n)=0$;

• 3.1.9) if $(x_n)_n$ is a sequence in ${{\textbf{X}}}_1^+$ with ${{\overline{\ell }}}_1((x_n)_n) =x$ and $y \in {{\textbf{X}}}_2^+$, then ${{\overline{\ell }}}((x_n \cdot y )_n)=x \cdot y$;

• 3.1.10) if $x \in {{\textbf{X}}}_1^+$ and $(y_n)_n$ is a sequence in ${{\textbf{X}}}_2^+$ with ${{\overline{\ell }}}_2 ((y_n)_n) = y$, then ${{\overline{\ell }}}((x \cdot y_n )_n)=x \cdot y$.

A Dedekind complete vector lattice ${{\textbf{X}}}$ is said to be a product algebra iff $({\textbf{X}},{\textbf{X}},{\textbf{X}})$ is a product triple. We endow the real line ${\mathbb {R}}$ with a convergence $\ell _{{\mathbb {R}}}$ satisfying Axioms 2.1 and assume that ${\textbf{X}}_1$, ${\textbf{X}}_2$, ${\textbf{X}}$ are product algebras and $({\mathbb {R}},{\textbf{X}},{\textbf{X}})$, $({\mathbb {R}},{\textbf{X}}_j,{\textbf{X}}_j)$, $j=1,2$, are product triples. Moreover, suppose that there exist two product algebras ${\textbf{X}}_1^{\prime }$ and ${\textbf{X}}_1^{\prime \prime }$, endowed with respective convergences $\ell _1^{\prime }$ and $\ell _1^{\prime \prime }$ satisfying Axioms 2.1, such that $({\textbf{X}}_1^{\prime }, {\textbf{X}}_1^{\prime \prime }, {\textbf{X}}_1)$ is a product triple.

Let G be any (possibly infinite) nonempty set, ${{\mathcal {P}}}(G)$ be the class of all subsets of G, ${{\mathcal {A}}} \subset {{\mathcal {P}}}(G)$ be an algebra, $\mu :{{\mathcal {A}}} \rightarrow \overline{{\textbf{X}}}^{+}_2$ be a finitely additive measure. We say that $\mu$ is $\sigma$-finite iff there is a sequence $(B_n)_n$ in ${{\mathcal {A}} }$, with $\mu (B_n) \in {{\textbf{X}}}^{+}_2$, $n \in {\mathbb {N}}$, and ${\bigcup \nolimits _{n\in {\mathbb {N}}} B_n=G}$. Note that, without loss of generality, the $B_n$’s can be taken increasing or disjoint.

Now we introduce an integral for ${\textbf{X}}_1$-valued functions with respect to a finitely additive and $\sigma$-finite $\overline{{\textbf{X}}}^+_2$-valued measure $\mu$, related to convergences $\ell _1$, $\ell _2$, $\ell$, satisfying Axioms 2.1. The integral will be an element of ${\textbf{X}}$. Similar constructions in the vector lattice setting were given in [11], where $\mu$ is finite, and in [13], where $\mu$ is possibly infinite and ${\textbf{X}}_1$, ${\textbf{X}}$ are endowed with two strong order units $e_1$, e, respectively.

We consider a sequence of functions of finite range and vanishing outside of a set of finite measure $\mu$, as follows. A function $f:G \rightarrow {\textbf{X}}_1$ is said to be simple iff it can be expressed as $f=\sum _{j=1}^r c_j \, \chi _{A_j},$ where $r \in {\mathbb {N}}$, $c_j \in {{\textbf{X}}}_1$, $A_j \in {{\mathcal {A}}}$, $\mu (A_j) \in {{\textbf{X}}}^{+}_2$ and $\chi _{A_j}$ is the characteristic function of the set $A_j$. We denote by ${\mathscr {S}}$ the set of all simple functions. For each $A \in {{\mathcal {A}}}$ and $f \in {\mathscr {S}}$, where f is as before, set

$$\begin{aligned} \int _A f(g) \, d\mu (g)= \sum _{j=1}^r c_j \, \mu (A \cap A_j). \end{aligned}$$

The integral of a simple function is a linear and monotone functional, and does not depend on the choice of the representation of f. Now we present the concepts of uniform convergence and convergence in measure.

Definition 3.2

Let $A \in {{\mathcal {A}}}$ and $(f_n)_n$ be a sequence of functions in ${\textbf{X}}_1^G$ :

• $(f_n)_n$ is said to be uniformly convergent to $f\in {\textbf{X}}_1^G$ on A iff

  $$\begin{aligned} {\ell _1 \Bigl (\Bigl ( \bigvee _{g \in A} \, \vert f_n(g)-f(g) \vert \Bigr )_n\Bigr )=0}; \end{aligned}$$

• $(f_n)_n$ converges in $\mu$-measure to $f\in {\textbf{X}}_1^G$ on A iff there is a sequence $(A_n)_n$ in ${\mathcal {A}}$, such that $\ell _2((\mu (A \cap A_n))_n)=0$ and $\ell _1 \Bigl (\Bigl (\bigvee \nolimits _{g \in A \setminus A_n} \vert f_n(g)-f(g) \vert \Bigr )_n\Bigr )=0$.

Observe that uniform convergence implies convergence in measure, and that, when ${{\textbf{X}}}_1={{\textbf{X}}}_2={\mathbb {R}}$ and the involved convergence is the usual one, the convergences in Definition 3.2 are equivalent to the classical ones (see, e.g., [11, Remark 3.3]). The following result extends [11, Proposition 3.4] when $\mu$ is not necessarily finite.

Proposition 3.3

Let $A \in {{\mathcal {A}}}$. If $(f_n)_n$, $(h_n)_n \in {{\textbf{X}}}_1^G$ converge in measure to f, h on A, respectively, and $c \in {{\mathbb {R}}}$, then $(f_n+h_n)_n$, $(f_n\vee h_n)_n$, $(f_n\wedge h_n)_n$, $(\vert f_n \vert )_n$ converge in measure on A to $f+h$, $f \vee h$, $f \wedge h$, $\vert f \vert$, respectively, and $(c \, f_n)_n$ converges in $\mu$-measure to $c \, f$ on A. The same results hold for uniform convergence.

Proof

We consider only convergence in measure, since the case of uniform convergence is analogous. We begin with proving the convergence in measure of the sequence $(f_n+h_n)_n$ to $f+h$ on A. Let $(A_n)_n$ and $(D_n)_n$ be two sequences in ${{\mathcal {A}}}$, related with the convergence in measure of $(f_n)_n$ and $(h_n)_n$ to f and h on A, respectively, and set $E_n=A_n \cup D_n$, $n \in {\mathbb {N}}$. Note that $0 \le _{X_2} \mu (E_n) \le _{X_2} \mu (A_n)+ \mu (D_n)$, and thus from Axioms 2.1.a) and 2.1.e) we deduce

$$\begin{aligned} \ell _2((\mu (E_n))_n)=0. \end{aligned}$$

(5)

Moreover, for each $n \in {\mathbb {N}}$ we have

$$\begin{aligned} 0{} & {} \le _{X_1} \Bigl ( \bigvee _{g \in A \setminus E_n} \, \vert f_n(g)+ h_n(g)-f(g)- h(g) \vert \Bigr )\\{} & {} \le _{X_1} \Bigl ( \bigvee _{g \in A \setminus E_n} \, \vert f_n(g)-f(g) \vert \Bigr ) + \Bigl ( \bigvee _{g \in A \setminus E_n} \, \vert h_n(g)-h(g)\vert \Bigr )\\{} & {} \le _{X_1} \Bigl ( \bigvee _{g \in A \setminus A_n} \, \vert f_n(g)-f(g)\vert \Bigr ) + \Bigl ( \bigvee _{g \in A \setminus D_n} \, \vert h_n(g)-h(g)\vert \Bigr ) . \end{aligned}$$

From the previous inequality and Axioms 2.1 it follows that the sequence $(f_n+h_n)_n$ converges in measure to $f+h$ on A.

Now we show that the sequence $(f_n\vee h_n)_n$ converges in measure to $f \vee h$ on A. Taking into account the Birkhoff inequalities (see [42, Theorem 12.4 (ii)]), we get

$$\begin{aligned}{} & {} 0 \le _{X_1} \Bigl ( \bigvee _{g \in A \setminus E_n} \, \vert f_n(g) \vee h_n(g)- f(g) \vee h(g) \vert \Bigr ) \nonumber \\{} & {} \quad \le _{X_1} \Bigl ( \bigvee _{g \in A \setminus E_n} \, \vert f_n(g) \vee h_n(g)- f_n(g) \vee h(g) \vert \Bigr ) \nonumber \\{} & {} \qquad + \Bigl ( \bigvee _{g \in A \setminus E_n} \, \vert f_n(g) \vee h(g)- f(g) \vee h(g) \vert \Bigr ) \nonumber \\{} & {} \quad \le _{X_1} \Bigl ( \bigvee _{g \in A \setminus A_n} \,\vert f_n(g)-f(g) \vert \Bigr ) + \Bigl ( \bigvee _{g \in A \setminus D_n} \,\vert h_n(g)-h(g)\vert \Bigr ). \end{aligned}$$

(6)

From (6) and Axioms 2.1 again it follows that the sequence $(f_n \vee h_n)_n$ converges in measure to $f \vee h$ on A. The proof of the convergence in measure of the sequence $(f_n \wedge h_n)_n$ to $f \wedge h$ is analogous.

The convergence in $\mu$-measure of the sequence $(\vert f_n\vert )_n$ to $\vert f \vert$ follows from the previous results, taking into account the definition of $\vert x \vert$. The proof of the convergence in $\mu$-measure of the sequence $(c \, f_n)_n$ to $c \, f$ is analogous to the previous ones. $\square$

Definition 3.4

We say that a sequence $(f_n)_n$ in ${\mathscr {S}}$ converges in $L^1$ to $f \in {\mathscr {S}}$ iff

$$\begin{aligned} \ell \left( \left( \int _G \vert f_n(g)-f(g)\vert \, d\mu (g) \right) _n \, \right) =0. \end{aligned}$$

Definition 3.5

We say that the integrals of a sequence of functions $(f_n)_n$ in ${\mathscr {S}}$ are $\mu$-equiabsolutely continuous iff the following two properties hold:

• 3.5.1) ${\ell \left( \left( \int _{A_n}\,\vert f_n(g)\vert \, d\mu (g)\right) _n \right) =0}$   if    $\ell _2((\mu ( A_n))_n)=0$;

• 3.5.2) there is an increasing sequence $(B_m)_m$ in ${{\mathcal {A}}}$ with $\mu (B_m)\in {{\textbf{X}}}_2$ for all $m \in {\mathbb {N}}$, and

  $$\begin{aligned} \ell \Bigl (\Bigl ({\overline{\ell }}\Bigl ( \Bigl (\int _{G \setminus B_m} \vert f_n(g)\vert \, d\mu (g) \Bigr )_n \Bigr ) \Bigr )_m \Bigr )=0. \end{aligned}$$

The next result extends [11, Theorem 3.7] when $\mu$ is not necessarily finite.

Theorem 3.6

Let $f_n \in {\mathscr {S}}$, $n \in {\mathbb {N}}$. If $(f_n)_n$ converges in measure to $f \equiv 0$ on every set $A \in {{\mathcal {A}}}$ with $\mu (A) \in {{\textbf{X}}}_2$ and their integrals are $\mu$-equiabsolutely continuous, then $(f_n)_n$ converges in $L^1$ to $f \equiv 0$.

Proof

Let $(B_m)_m$ be a sequence related to property 3.5.2) of the $\mu$-equiabsolute continuity of the integrals of the $f_n$’s. By the convergence in measure of $(f_n)_n$ to 0 on each $B_m$, there is a double sequence $(A_{m,n})_n$ in ${{\mathcal {A}}}$ with $\ell _2((\mu (B_m \cap A_{m,n}))_n)=0$ and ${\ell \Bigl (\Bigl (\bigvee \nolimits _{g \in B_m \setminus A_{m,n}} \vert f_n(g)\vert \Bigr )_n\Bigr )=0}$ for each $m \in {\mathbb {N}}$. For every fixed m, $n \in {\mathbb {N}}$ we have

$$\begin{aligned} 0&\le _{X} \int _G\vert f_n(g)\vert \, d\mu (g) \le _{X} \int _{G\setminus B_m}\vert f_n(g)\vert \, d\mu (g) + \int _{B_m \cap A_{m,n}}\vert f_n(g)\vert \, d\mu (g) \\&\quad + \int _{B_m \setminus A_{m,n}}\vert f_n(g)\vert \, d\mu (g). \end{aligned}$$

By 3.5.2), we get

$$\begin{aligned} \ell \Bigl ( \Bigl ( {\overline{\ell }} \Bigl (\Bigl (\int _{G\setminus B_m}\vert f_n(g)\vert \, d\mu (g) \Bigr )_n\Bigr )\Bigr )_m\Bigr )=0. \end{aligned}$$

(7)

From 3.5.1) we obtain

$$\begin{aligned} \ell \Bigl (\Bigl (\int _{B_m \cap A_{m,n}}\vert f_n(g)\vert \, d\mu (g)\Bigr )_n\Bigr )=0\quad \text { for every } m \in {\mathbb {N}}. \end{aligned}$$

Taking into account Axioms 2.1 and 2.2.d), then we deduce

$$\begin{aligned} \ell \Bigl ( \Bigl ( {\overline{\ell }} \Bigl (\Bigl (\int _{B_m \cap A_{m,n}}\vert f_n(g)\vert \, d\mu (g) \Bigr )_n\Bigr )\Bigr )_m\Bigr )=0. \end{aligned}$$

(8)

Furthermore, for each $m \in {\mathbb {N}}$ we have

$$\begin{aligned}{} & {} 0 \le _{X} \int _{B_m \setminus A_{m,n}}\vert f_n(g)\vert \, d\mu (g) \le _{X} \Bigl (\bigvee _{g \in B_m \setminus A_{n,m}} \vert f_n(g)\vert \Bigr ) \mu (B_m \setminus A_{n,m})\\{} & {} \quad \le _{X} \Bigl (\bigvee _{g \in B_m \setminus A_{n,m}} \vert f_n(g)\vert \Bigr ) \mu (B_m). \end{aligned}$$

Since $\mu (B_m) \in {\textbf{X}}_2$, from the previous inequality, convergence in measure on $B_m$ and Axioms 2.1 we get

$$\begin{aligned} \ell \Bigl (\Bigl (\int _{B_m \setminus A_{m,n}}\vert f_n(g)\vert \, d\mu (g)\Bigr )_n\Bigr )=0\quad \text { for each } m \in {\mathbb {N}}, \end{aligned}$$

(9)

and hence, analogously as in (8), from (9) we obtain

$$\begin{aligned} \ell \Bigl ( \Bigl ( {\overline{\ell }} \Bigl (\Bigl (\int _{B_m \setminus A_{m,n}}\vert f_n(g)\vert \, d\mu (g) \Bigr )_n\Bigr )\Bigr )_m\Bigr )=0. \end{aligned}$$

(10)

From (7), (8), (10) and Axioms 2.1, 2.2 it follows that

$$\begin{aligned} 0 \le _{X}&{\overline{\ell }}\Bigl (\Bigl (\int _G\vert f_n(g)\vert \, d\mu (g) \Bigr )_n\Bigr ) = \ell \Bigl ( \Bigl ( {\overline{\ell }} \Bigl (\Bigl (\int _G \vert f_n(g)\vert \, d\mu (g) \Bigr )_n\Bigr )\Bigr )_m \Bigr )\\ \le _{X}&\ell \Bigl ( \Bigl ( {\overline{\ell }} \Bigl (\Bigl (\int _{G\setminus B_m} \vert f_n(g)\vert d\mu (g) \Bigr )_n\Bigr )\Bigr )_m\Bigr ) + \ell \Bigl ( \Bigl ( \ell \Bigl (\Bigl (\int _{B_m \cap A_{m,n}} \vert f_n(g)\vert d\mu (g) \Bigr )_n\Bigr )\Bigr )_m\Bigr ) \\&+ \ell \Bigl ( \Bigl ( {\overline{\ell }} \Bigl (\Bigl (\int _{B_m \setminus A_{m,n}}\vert f_n(g)\vert \, d\mu (g) \Bigr )_n \Bigr )\Bigr )_m\Bigr )=0. \end{aligned}$$

From this and Axiom 2.2.e) we obtain ${ {\ell } \Bigl (\Bigl (\int _G\vert f_n(g)\vert \, d\mu (g)\Bigr )_n \Bigr )=0}$, that is the convergence in $L^1$ to 0 of the sequence $(f_n)_n$. $\square$

Now we turn to the construction of the integral.

Definition 3.7

Let $f \in {{\textbf{X}}}_1^G$. A sequence $(f_n)_n$ in ${\mathscr {S}}$ is said to be defining for f iff it converges in $\mu$-measure to f on every set $A \in {{\mathcal {A}}}$ with $\mu (A) \in {\textbf{X}}_2$ and their integrals are $\mu$-equiabsolutely continuous.

Definition 3.8

A positive function $f \in {{\textbf{X}}}_1^G$ is said to be integrable on G iff there exist a defining sequence $(f_n)_n$ for f and a map $l:{{\mathcal {A}}} \rightarrow {\textbf{X}}$, with

$$\begin{aligned} \ell \Bigl ( \Bigl (\bigvee _{A\in {{\mathcal {A}}}}\Bigl \vert \int _A f_n(g) \, d\mu (g) - l(A)\Bigr \vert \Bigr )_n \Bigr )=0, \end{aligned}$$

(11)

and in this case we set

$$\begin{aligned} \int _A f(g) \, d\mu (g):=l(A) \quad \text { for every } A \in {{\mathcal {A}}}. \end{aligned}$$

(12)

Now we prove that the integral in (12) is well-defined, extending [11, Proposition 3.11] and [13, Theorem 3.5].

Proposition 3.9

For every $A \in {{\mathcal {A}}}$, we have that $l(A):= \ell \left( \left( {\int _A} \, f_n(g)\,d\mu (g) \right) _n \right)$, and l(A) does not depend on the choice of the defining sequence.

Proof

We prove that the assertion is a consequence of Definition 3.8 (formula (11)) and Axioms 2.1. For $j=1,2$, let $(f_n^{(j)})_n$ be two defining sequences for f, put

$$\begin{aligned} l^{(j)} (A) :=\ell \left( \left( \int _A \, f_n^{(j)} (g)\,d\mu (g)\right) _n \right) \quad \text {for each } \,\, A \in {{\mathcal {A}}}, \end{aligned}$$

and set $q_n(g)=\vert f_n^{(1)}(g)-f_n^{(2)}(g) \vert$, $g \in G$, $n \in {\mathbb {N}}$. The sequence $(q_n)_n$ converges in $\mu$-measure to 0 on every set $A\in {{\mathcal {A}}}$ with $\mu (A) \in {{\textbf{X}}}_2$. Moreover, the integrals of $(f_n^{(j)})_n$, $j=1,2$, and thus also those of the $q_n$’s, are $\mu$-equiabsolutely continuous. Thus, by Theorem 3.6, the sequence $(q_n)_n$ converges to 0 in $L^1$. Moreover we get, for every $n \in {\mathbb {N}}$,

$$\begin{aligned} 0 \le _{X}&\vert l^{(1)}(A)-l^{(2)}(A)\vert \le _{X} \left| \int _A f_n^{(1)}(g) d\mu (g)-l^{(1)}(A)\right| \nonumber \\&+\left| l^{(2)}(A)- \int _A f_n^{(2)}(g) d\mu (g)\right| + \left| \int _A f_n^{(1)}(g) \, d\mu (g) - \int _A f_n^{(2)}(g) \, d\mu (g)\right| \nonumber \\ \le _{X}&\left| \int _A f_n^{(1)}(g) \, d\mu (g)-l^{(1)}(A)\right| \nonumber \\&+ \left| l^{(2)}(A)- \int _A f_n^{(2)}(g) d\mu (g)\right| + \int _G q_n(g) d\mu (g). \end{aligned}$$

(13)

From the definition of $l^{(j)}$, formula (13), the convergence in $L^1$ to 0 of $(q_n)_n$ and Axioms 2.1.a), 2.1.b), 2.1.g) it follows that $\vert l^{(1)}(A)-l^{(2)}(A) \vert =0$, namely $l^{(1)}(A)=l^{(2)}(A)$, for all $A \in {{\mathcal {A}}}$. $\square$

Now we define our integral for not necessarily positive functions.

Definition 3.10

A function $f: G \rightarrow {{\textbf{X}}}_1$ is said to be integrable on G iff the functions $f^+(g)=f(g)\vee 0$, $f^-(g)=(-f(g))\vee 0$, $g \in G$, are integrable on G, and in this case we set

$$\begin{aligned} \int _A f(g) \, d\mu (g) = \int _A f^+(g) \, d\mu (g) - \int _A f^-(g) \, d\mu (g)\quad \text {for every } \,\,A \in {{\mathcal {A}}}. \end{aligned}$$

Remark 3.11

• 3.11.a) Since $\vert f(g) \vert =f^+(g)+f^-(g)$ for every $g \in G$, if $(f_n)_n$ and $(h_n)_n$ are two sequences of functions defining for $f^+$ and $f^-$, respectively, then $(f_n+h_n)_n$ is a sequence of functions defining for $\vert f \vert$. From this it follows that, if f is integrable, then $\vert f \vert$ is integrable too, and

  $$\begin{aligned} \int _A \vert f(g) \vert \, d\mu (g) = \int _A f^+(g) \, d\mu (g) + \int _A f^-(g) \, d\mu (g) \quad \text {for each } \, A \in {{\mathcal {A}}}. \end{aligned}$$

  Moreover, note that the integral defined in Definition 3.10 is a linear positive ${{\textbf{X}}}$-valued functional.

• 3.11.b) Observe that, if $v_{1} \in {\textbf{X}}_1$ and $h:G \rightarrow {\mathbb {R}}$ is integrable, then the function $f:G \rightarrow {\textbf{X}}_1$ defined by $f(g)=(h(g)) \, v_{1}, \,\, g \in G$, is integrable, and

  $$\begin{aligned} \int _A f(g) \, d\mu (g)= \Bigl ( \int _A h(g) \, d\mu (g) \Bigr ) \, v_{1}\quad \text {for every } \, A \in {{\mathcal {A}}}. \end{aligned}$$

  Indeed, taking into account 3.1.7), this follows from the fact that, is $(h_n)_n$ is a defining sequence for h, then $(h_n \, v_{1})_n$ is a defining sequence for f.

• 3.11.c) If $\mu$ is finite, then the integrals given in [11] and in Definition 3.10 coincide (indeed, it will be enough to take $B_m=G$ for each $m \in {\mathbb {N}})$.

• 3.11.d) The notions of convergence in $L^1$ and $\mu$-equiabsolute continuity can be given also for sequences of integrable functions, analogously as in Definitions 3.4 and 3.5, respectively.

• 3.11.e) Moreover, we observe that, if f and h integrable and $\alpha \in {\mathbb {R}}$, then $f+h$ and $\alpha \, f$ are integrable too, thanks to the properties of the defining sequences and the linearity of $\ell$ (Axiom 2.1.a)).

• 3.11.f) If f is non negative and integrable, then, by Proposition 3.3, its integral is positive too. In fact, if $(f_n)_n$ $\mu$-converges to f, then $(f_n^+)_n=(f_n \vee 0)_n$ $\mu$-converges to $f \vee 0=f$. In addition, $\vert f_n^+\vert =f_n^+ \le _{X_1} \vert f_n \vert$, for every $n \in {\mathbb {N}}$, and then we obtain the $\mu$-equiabsolute continuity of the $f_n^+$ integrals, which ensures that $(f_n^+)_n$ is also defining. Let $l: {\mathcal {A}} \rightarrow {\textbf{X}}$ be related to the integrability of f. Since l does not depend on the defining sequence, we obtain that l is also the limit of the $f_n^+$’s integrals, which are non negative. Hence, by Axiom 2.1.b), we obtain that $l \ge _{X} 0$. These properties imply also that the integral is monotone (that is for every integrable functions f, g with $f \le _{X_1} g$

  we have that ${ \int _G f d\mu \le _{X} \int _G g d\mu }$ ).

The next step is to prove the $\mu$-absolute continuity of the integral. The following result extends [11, Theorem 3.14].

Theorem 3.12

Let $f : G \rightarrow {{\textbf{X}}}_1$ be an integrable function. Then, the integral ${\int _{(\cdot )} f(g) \, d\mu (g)}$ is $\mu$-absolutely continuous, that is

• 3.12.1) ${ \ell \left( \left( \int _{A_n}\,\vert f(g) \vert \, d\mu (g)\right) _n \right) =0}$   whenever    $\ell _2((\mu ( A_n))_n)=0$;

• 3.12.2) there exists an increasing sequence $(B_m)_m$ in ${{\mathcal {A}}}$, such that $\mu (B_m)\in {{\textbf{X}}}_2$ for all $m \in {\mathbb {N}}$, and

  $$\begin{aligned} \ell \Bigl (\Bigl (\int _{G \setminus B_m} \vert f(g) \vert \, d\mu (g) \Bigr )_m \Bigr )=0. \end{aligned}$$

Proof

Without loss of generality, we can suppose that $f \ge _{X_1} 0$. Let $(f_n)_n$ be a defining sequence for f. We begin with proving 3.12.1). Let $(A_n)_n$ be a sequence of elements of ${{\mathcal {A}}}$ with $\ell _2((\mu (A_n))_n)=0$. For each $n \in {\mathbb {N}}$, we have

$$\begin{aligned} 0 \le _{X}&\int _{A_n} \, f(g)\, d\mu (g) \le _{X} \Bigl \vert \int _{A_n} f(g) \, d\mu (g) -\int _{A_n} f_n(g) \,d\mu (g) \Bigr \vert + \Bigl \vert \int _{A_n} f_n(g) \, d\mu (g)\Bigr \vert \nonumber \\ \le _{X}&\bigvee _{A \in {{\mathcal {A}}}} \, \left| \int _A f(g) \, d\mu (g) -\int _A f_n(g) \,d\mu (g) \right| + \int _{A_n} \vert f_n(g)\vert \, d\mu (g). \end{aligned}$$

(14)

By (11), we get

$$\begin{aligned} \ell \Bigl ( \Bigl (\bigvee _{A \in {{\mathcal {A}}}} \, \Bigl \vert \int _A f(g) \, d\mu (g) -\int _A f_n(g) \,d\mu (g) \Bigr \vert \Bigr )_n\Bigr )=0. \end{aligned}$$

(15)

From condition 3.5.1) of $\mu$-equiabsolute continuity of the integrals of the $f_n$’s, we get

$$\begin{aligned} \ell \Bigl (\Bigl (\int _{A_n} \vert f_n(g)\vert \, d\mu (g)\Bigr )_n\Bigr )=0. \end{aligned}$$

(16)

From (14), (15), (16) and Axioms 2.1 we obtain ${\ell \Bigl (\Bigl ( \int _{A_n} \, f(g)\, d\mu (g) \Bigr )_n\Bigr )=0}$,   that is condition 3.12.1). Now we turn to 3.12.2). Let $(f_n)_n$ be a defining sequence for f and $(B_m)_m$ be a sequence from ${{\mathcal {A}}}$, according to condition 3.5.2) of the $\mu$-equiabsolute continuity of the integrals of the $f_n$’s. For every n, $m \in {\mathbb {N}}$, we have

$$\begin{aligned} 0 \le _{X}&\int _{G \setminus B_m}f(g)\, d\mu (g) \le _{X} \Bigl \vert \int _{G \setminus B_m} f(g) \, d\mu (g) -\int _{G \setminus B_m} f_n(g) \,d\mu (g) \Bigr \vert \\&+ \Bigl \vert \int _{G \setminus B_m}f_n(g) \, d\mu (g)\Bigr \vert \le _{X} \bigvee _{A \in {{\mathcal {A}}}} \, \left| \int _Af(g) \, d\mu (g) -\int _A f_n(g) \,d\mu (g) \right| \\ {}&+ \int _{G \setminus B_m} \vert f_n(g)\vert \, d\mu (g). \end{aligned}$$

By the definition of integrability (11), we get

$$\begin{aligned} \ell \Bigl ( \Bigl ( \bigvee _{A \in {{\mathcal {A}}}} \, \Bigl \vert \int _Af(g) \, d\mu (g) -\int _A f_n(g) \,d\mu (g) \Bigr \vert \Bigr )_n\Bigr )=0. \end{aligned}$$

Thus, taking in the previous inequality the limit $\ell$ and the limit superior ${\overline{\ell }}$ as n tends to $+\infty$, by virtue of Axioms 2.1 and 2.2 we obtain

$$\begin{aligned} 0 \le _{X}&\ell \Bigl (\Bigl ( \int _{G \setminus B_m}f(g)\, d\mu (g)\Bigr )_n\Bigr ) \le _{X} \ell \Bigl (\Bigl ( \bigvee _{A \in {{\mathcal {A}}}} \, \Bigl \vert \int _Af(g) \, d\mu (g) -\int _A f_n(g) \,d\mu (g) \Bigr \vert \Bigr )_n\Bigr ) \\ {}&+ {\overline{\ell }}\Bigl (\Bigl ( \int _{G \setminus B_m} \vert f_n(g)\vert \, d\mu (g)\Bigr )_n\Bigr )= {\overline{\ell }} \Bigl (\Bigl (\int _{G \setminus B_m} \vert f_n(g)\vert \, d\mu (g)\Bigr )_n\Bigr ) \quad \text {for each }m \in {\mathbb {N}}. \end{aligned}$$

Taking the limit $\ell$ as m tends to $+\infty$, by condition 3.5.2) we have

$$\begin{aligned} 0 \le _{X} \ell \Bigl (\Bigl ( \int _{G \setminus B_m}f(g)\, d\mu (g) \Bigr )_m\Bigr ) \le _{X} \ell \Bigl ( \Bigl ( {\overline{\ell }} \Bigl (\Bigl (\int _{G \setminus B_m} \vert f_n(g)\vert \, d\mu (g) \Bigr )_n \Bigr )\Bigr )_m\Bigr )=0, \end{aligned}$$

that is ${\ell \Bigl (\Bigl ( \int _{G \setminus B_m}f(g)\, d\mu (g) \Bigr )_m\Bigr )=0}$. This ends the proof. $\square$

We now give two convergence results for integrable functions.

Theorem 3.13

(Vitali). Let $(f_n)_n$ be a sequence of integrable functions in ${{\textbf{X}}}_1^G$, convergent in measure to 0 on every set $A \in {{\mathcal {A}}}$ with $\mu (A) \in {{\textbf{X}}}_2$ and with $\mu$-equiabsolutely continuous integrals. Then $(f_n)_n$ converges in $L^1$ to 0.

Proof

The result follows by arguing analogously as in the proof of Theorem 3.6. $\square$

Corollary 3.14

(Lebesgue). Let $(f_n)_n$ be a sequence of integrable functions in ${{\textbf{X}}}_1^G$, convergent in measure to 0 on any set $A \in {{\mathcal {A}}}$ with $\mu (A) \in {{\textbf{X}}}_2$, and let there be an integrable map $h \in {{\textbf{X}}}_1^G$ with $\vert f_n(g)\vert \le _{X_1} h(g)$ for all $n \in {\mathbb {N}}$ and $g \in G$. Then $(f_n)_n$ converges in $L^1$ to 0.

Proof

Since h is integrable, then its integral is absolutely continuous. From this and the boundedness of the $f_n$’s we deduce that the integrals of the $f_n$’s are $\mu$-equiabsolute continuous. Therefore, Corollary 3.14 is a consequence of Theorem 3.13. $\square$

We now prove that, in the classical case, the above defined integral and the Lebesgue’s one are equivalent (for a related overview see, e.g., [36]).

A measure $\mu :{{\mathcal {A}}}\rightarrow \overline{{\mathbb {R}}}^{+}_0$ is said to be regular iff $\mu (C) < + \infty$ for each compact set $C \subset G$, and for every $\varepsilon \in {\mathbb {R}}^+$ and $E \in {{\mathcal {A}}}$ there are a compact set $C \subset E$ and an open set $U \supset E$ with $\mu (U \setminus C)$ $\le$ $\varepsilon$. Note that every regular finitely additive measure is $\sigma$-additive (see, e.g., [18]).

Proposition 3.15

Let $G=(G,d)$ be a metric space, ${{\textbf{X}}}={{\textbf{X}}}_1 ={{\textbf{X}}}_2= {{\mathbb {R}}}$ be endowed with the usual convergence, ${{\mathcal {A}}}$ be the $\sigma$-algebra of all measurable subsets of G, and $\mu :{{\mathcal {A}}}\rightarrow {\mathbb {R}}$ be a $\sigma$-finite regular measure. Then, a function $f \in {{\mathbb {R}}}^G$ is integrable if and only if it is Lebesgue integrable, and in this case we have

$$\begin{aligned} \int _A f(g) \, d\mu (g)=(L) \int _A f(g) \, d\mu (g) \quad \text { for all } \, A \in {{\mathcal {A}}}, \end{aligned}$$

where ${(L) \int _{(\cdot )}}$ denotes the Lebesgue integral.

Proof

Let $f:G \rightarrow {{\mathbb {R}}}$ be Lebesgue integrable on G. Without loss of generality, we can assume $f \ge 0$. Then there is an increasing sequence $(f_n)_n$ in ${{\mathbb {R}}}^G$, $\mu$-convergent to f on G, and hence $\mu$-convergent to f on every set $A \in {{\mathcal {A}}}$ with $\mu (A) < + \infty$. Condition 3.5.1) on the $\mu$-equiabsolute continuity of the integrals of the $f_n$’s follows from the absolute continuity of the Lebesgue integral, while condition 3.5.2) is a consequence of the $\sigma$-additivity of regular measures and of the Lebesgue integral. Thus, $(f_n)_n$ is a defining sequence for f. From this and Proposition 3.9 it follows that every Lebesgue integrable function is integrable according to Definitions 3.8 and 3.10.

Now we prove the converse implication. Let $f \in {{\mathbb {R}}}^G$ be integrable and let $(f_n)_n$ be an associated defining sequence of f. Since $f_n \in {\mathscr {S}}$ for each $n \in {\mathbb {N}}$, then the $f_n$’s are Lebesgue integrable. Without loss of generality, we may suppose that $0 \le f_n(g) \le f_{n+1}(g) \le f(g)$ for every $g \in G$. By the absolute continuity of our integral, $(f-f_n)_n$ is a defining sequence for the identically null function. By Theorem 3.13, $(f-f_n)_n$ converges in $L^1$ to 0, and hence the sequence ${\Bigl (\int _G f_n(g) \, d\mu (g)\Bigr )_n}$ is bounded. Thus, by the classical monotone convergence theorem (see, e.g., [36, Theorem 27.B]), we obtain that f is Lebesgue integrable.

The equality of the integrals follows from the fact that the values of both integrals do not depend on the chosen respective defining sequences. $\square$

3.1 Some Properties of the Integral

We now give some properties of the integral and present the concepts of uniform continuity and convexity in the vector lattice context. We give some versions of the Jensen inequality, which will be useful to prove some modular convergence theorems. The proofs of the results of this subsection are given in the Appendix. Let $({\textbf{X}}_1^{\prime }, {\textbf{X}}_1^{\prime \prime }, {\textbf{X}}_1)$, $({\textbf{X}}_1,{\textbf{X}}_2, {\textbf{X}})$ be two product triples.

Proposition 3.16

Let $h:G \rightarrow {\textbf{X}}_1^{\prime }$, $q:G \rightarrow {\textbf{X}}_1^{\prime \prime }$ be bounded and integrable on G according to Definitions 3.8 and 3.10. Then, the function $h \cdot q:G \rightarrow {\textbf{X}}_1$ is bounded and integrable on G too.

Corollary 3.17

Let $h:G \rightarrow {\textbf{X}}_1^{\prime }$, and $q:G \rightarrow {\textbf{X}}_1^{\prime \prime }$. Assume that h is integrable on G and bounded on every set $B \in {{\mathcal {A}}}$ with $\mu (B) \in {{\textbf{X}}}_2$. Moreover, suppose that q is bounded and integrable on G, and that there is ${\overline{B}} \in {{\mathcal {A}}}$ with $\mu ({\overline{B}}) \in {{\textbf{X}}}_2$ and such that q vanishes on $G \setminus {\overline{B}}$. Then, the function $h \cdot q:G \rightarrow {\textbf{X}}_1$ is bounded and integrable on G.

Now we give the concept of uniform continuity for vector lattice-valued functions (see, e.g., [13, Definition 3.9]). Let $G=(G,d)$ be a metric space. We say that

• $f:G \rightarrow {\textbf{X}}_1$ is $uniformly continuous on$ G iff there exists an element $u \in {{\textbf{X}}}_1^+$ such that for each $\varepsilon \in {\mathbb {R}}^+$ there is $\delta \in {\mathbb {R}}^+$ with $\vert f(g_1) -f(g_2) \vert \le _{X_1} \varepsilon \, u$ whenever $g_1$, $g_2\in G$, $d(g_1,g_2) \le \delta$.

• A function $\psi :{\textbf{X}}_1\rightarrow {\textbf{X}}_1$ is said to be uniformly continuous on ${\textbf{X}}_1$ iff for every $u \in {{\textbf{X}}}_1^+$ and $\varepsilon \in {\mathbb {R}}^+$ there exist $w \in {{\textbf{X}}}_1^+$ and $\delta \in {\mathbb {R}}^+$ with $\vert \psi (x_1)-\psi (x_2)\vert \le _{X_1} \, \varepsilon \, w$ whenever $\vert x_1-x_2 \vert \le _{X_1} \delta \, u$.

Observe that, when $G={{\textbf{X}}}_1={\mathbb {R}}$ endowed with the usual topology, the two presented concepts of uniform continuity coincide with the classical one. Indeed, it is enough to consider $u=w={\textbf{1}}$, the constant function which associates the real number 1 to every $g \in G$.

Now we present some properties on uniformly continuous functions defined in a metric space (G, d).

Proposition 3.18

Let $f:G \rightarrow {{\textbf{X}}}_1$, $\psi :{\textbf{X}}_1\rightarrow {\textbf{X}}_1$ be uniformly continuous on G and ${\textbf{X}}_1$, respectively. Then the composite function $\psi \circ f: G \rightarrow {{\textbf{X}}}_1$ is uniformly continuous on G.

The next result extends the first part of [13, Proposition 3.12.4] to this setting.

Proposition 3.19

Let $\mu$ be such that $\mu (K) \in X_2$  for every compact subset $K \in {{\mathcal {A}}}$. Let $C \in {{\mathcal {A}}}$ be a compact set, and $f: G \rightarrow {\textbf{X}}_1$ be a uniformly continuous function on G, such that $f(g)=0$ whenever $g \in G \setminus C$. Then, f is bounded and integrable on G.

Now, to deal with Orlicz-type spaces in the setting of vector lattice-valued modulars, we investigate some properties of convex functions.

Definition 3.20

Let ${\textbf{X}}$ be any Dedekind complete product algebra. A function $\varphi : {\textbf{X}} \rightarrow {\textbf{X}}$ is said to be convex iff for each $v \in {\textbf{X}}$ there is an element $\beta _v \in {\textbf{X}}$ with

$$\begin{aligned} \varphi (s) \ge _{X} \varphi (v) + \beta _v (s-v) \quad \text { for all } s \in {\textbf{X}} \end{aligned}$$

and such that the set $E_u :=\{ \beta _v : v \in [-u,u] \}$ is order bounded in ${{\textbf{X}}}$ for every $u \in {{\textbf{X}}}^+ \setminus \{0\}$.

Remark 3.21

Note that, when ${\textbf{X}} = {\mathbb {R}}$, $\varphi$ is convex if and only if

$$\begin{aligned} \varphi (t) \le \varphi (t_1) + \dfrac{\varphi (t_2) - \varphi (t_1)}{t_2 - t_1} \, (t-t_1) \text { for all } t, t_1, t_2 \in {\mathbb {R}} \, \, \text { with }\, t_1< t < t_2 \end{aligned}$$

(see [14, Definition 4.3]), and the set $E_u$ is order bounded for all $u \in {\mathbb {R}}^+$, since every convex function $\varphi : {{\mathbb {R}}} \rightarrow {{\mathbb {R}}}$ is Lipschitz on every bounded subinterval of the real line.

Example 3.22

Now we give some examples of convex functions in the vector lattice setting.

• 3.22.a) Define $\varphi : {\textbf{X}} \rightarrow {\textbf{X}}$ by $\varphi (x)= x^2, \,\, x \in {\textbf{X}}$, which makes sense, since ${\textbf{X}}$ is a product algebra. It is

  $$\begin{aligned} s^2 \ge _{X} v^2 + 2 \, v (s-v) \quad \text {for all } s,v \in {\textbf{X}}. \end{aligned}$$

  (17)

  From this it follows that $\varphi$ is convex, since the set $E_u =\{2 v : v \in [-u,u]\}$ is evidently order bounded for each $u \in {\textbf{X}}^+$.

• 3.22.b) More generally observe that, at least in certain cases, it is possibile to construct convex ${\textbf{X}}$-valued functions by starting with convex real-valued functions. For example, let ${\widehat{\varphi }}: {\mathbb {R}} \rightarrow {\mathbb {R}}$ be a function of class $C^1({{\mathbb {R}}})$, and ${\textbf{X}}={{\mathcal {C}}}_{\infty }(\Omega )$ be as in [33, Theorem 2.1]. For each $x\in {\textbf{X}}$ and $\omega \in \Omega$, set

  $$\begin{aligned} \varphi (x)(\omega )= {\widehat{\varphi }}(x(\omega )). \end{aligned}$$

  (18)

  Note that ${\widehat{\varphi }} \circ x \in {\textbf{X}}$ and ${\widehat{\varphi }}^{\prime } \circ x \in {\textbf{X}}$ whenever $x \in {\textbf{X}}$, and hence the function $\varphi$ in (18) is well-defined. The convexity of ${\widehat{\varphi }}$ implies that

  $$\begin{aligned} {\widehat{\varphi }}(s(\omega )) \ge {\widehat{\varphi }}(v(\omega )) +{\widehat{\varphi }}^{\prime } (v(\omega )) \, (s(\omega ) - v(\omega )) \end{aligned}$$

  (19)

  for every $\omega \in \Omega \setminus N$, where $N \subset \Omega$ is a suitable nowhere dense set (in general, depending on s and v). Since, by the Baire category theorem, the complement of a meager subset of $\Omega$ is dense in $\Omega$, and since ${\widehat{\varphi }}^{\prime } \circ v$ is continuous, the inequality in (19) holds for every $\omega \in \Omega$. Setting $\beta _v (\omega )={\widehat{\varphi }}^{\prime } (v(\omega ))$, $\omega \in \Omega$, we obtain that $\beta _v \in {\textbf{X}}$. Moreover observe that, if $v \in [-u,u]$, then by the monotonicity of ${\widehat{\varphi }}^{\prime }$ it follows that ${\widehat{\varphi }}^{\prime } (-u(\omega )) \le {\widehat{\varphi }}^{\prime } (v(\omega )) \le {\widehat{\varphi }}^{\prime } (u(\omega ) )$. Thus, the set $E_u$ is bounded in ${{\mathcal {C}}}_{\infty }(\Omega )$. From this and (19) we deduce the convexity of $\varphi$.

  An analogous property holds when ${{\textbf{X}}}= L^0(\Omega , \Sigma , \nu )$, where $\nu : \Sigma \rightarrow {\mathbb {R}}^{+}_0$ is a $\sigma$-additive and $\sigma$-finite measure. Indeed, in this case it is enough to argue as above, and we get the inequality in (19) for all $\omega \in \Omega$, directly from the convexity of ${\widehat{\varphi }}$.

• 3.22.c) Let ${\textbf{X}} \subset {{\mathcal {C}}}_{\infty }(\Omega )$ be as in [33, Theorem 2.1], and set

  $$\begin{aligned} \varphi (x)=\vert x \vert ^p, \,\, {\widehat{\varphi }}(t)=\vert t \vert ^p, \quad x \in {\textbf{X}}, \,\, t \in {\mathbb {R}}, \, \, p \in {\mathbb {N}}, \, \, p \ge 3. \end{aligned}$$

  Since ${\widehat{\varphi }}^{\prime }(t)=p \, \vert t \vert ^{p-2} \, t$, $t \in {\mathbb {R}}$, then $\varphi \circ x$ and $\varphi ^{\prime } \circ x$ belong to ${\textbf{X}}$ for all $x \in {\textbf{X}}$. From the convexity of ${\widehat{\varphi }}$ on ${\mathbb {R}}$, we obtain

  $$\begin{aligned} \vert s(\omega )\vert ^p \ge \vert v(\omega )\vert ^p + p \,\vert v(\omega )\vert ^{p-2} \, v(\omega ) \, \, (s(\omega )-v(\omega )), \quad \omega \in \Omega \setminus N, \end{aligned}$$

  (20)

  where $N \subset \Omega$ is a suitable nowhere dense set. Taking into account that, by the Baire category theorem, the complement of a meager subset of $\Omega$ is dense in $\Omega$, we get that the inequality in (20) holds for any $\omega \in \Omega$. From (20) it follows that $\varphi$ satisfies the inequality in (17), taking $\beta _v=p \,\vert v \vert ^{p-2} \, v$, $\omega \in \Omega$. Moreover, since $\beta _v(\omega )$ is well-defined and $\vert \beta _v(\omega )\vert \le p \, \vert u(\omega )\vert ^{p-1} \text { for each }\omega \in \Omega ,$ then the set $E_u= \{ \beta _v : v \in [-u,u] \}$ is order bounded in ${{\mathcal {C}}}_{\infty }(\Omega )$. Since, by the Maeda-Ogasawara-Vulikh representation theorem [33, Theorem 2.1], ${\textbf{X}}$ can be viewed as a solid subspace in ${{\mathcal {C}}}_{\infty }(\Omega )$, it follows that the set $E_u$ is order bounded also in ${{\textbf{X}}}$. Thus, $\varphi$ is convex.

Now we prove some fundamental properties of convex functions, extending [13, Theorem 4.5].

Proposition 3.23

Let $\varphi : {\textbf{X}}_1 \rightarrow {\textbf{X}}_1$ be convex and such that $\varphi (0)=0$. Then $\varphi$ satisfies the following property:

$$\begin{aligned} \text {if } x \in {{\textbf{X}}}_1 \text { and } \xi \in {\mathbb {R}}, \,\, 0 \le \xi \le 1, \text { then } \varphi (\xi x )\le _{X_1} \xi \, \varphi (x). \end{aligned}$$

(21)

Proposition 3.24

Let $\varphi :{\textbf{X}}_1 \rightarrow {\textbf{X}}_1$ be a convex function. Then for every $u \in {\textbf{X}}_1^+ \setminus \{0\}$ there is an element $\beta ^*_u \in {\textbf{X}}_1^+ \setminus \{0\}$ such that

$$\begin{aligned} \vert \varphi (x_1)-\varphi (x_2)\vert \le _{X_1} \beta ^*_u \, \vert x_1 - x_2\vert \text { for all }x_1, x_2 \in [-u,u], \end{aligned}$$

(22)

where $[-u,u]=\{v \in {\textbf{X}}_1$: $-u \le _{X_1} v \le _{X_1} u \}$.

Proposition 3.25

Let $f:G \rightarrow {\textbf{X}}_1$ be uniformly continuous on G and vanishing outside of a compact set $C \subset G$, $\mu$ be regular and $\varphi : {\textbf{X}}_1 \rightarrow {\textbf{X}}_1$ be convex and with $\varphi (0)=0$. Then the composite function $\varphi \circ f: G \rightarrow {\textbf{X}}_1$ is uniformly continuous on G, integrable on G, and vanishes outside C.

Now we give the following versions of the Jensen inequality, which extends [14, Theorem 4.6].

Theorem 3.26

Let $C \subset G$ be a compact set, $({{\textbf{X}}},{\mathbb {R}}, {{\textbf{X}}})$ be a product triple, $\mu :{{\mathcal {A}}} \rightarrow \overline{{\mathbb {R}}}^{+}_0$ be a $\sigma$-finite regular measure, $\varphi : {{\textbf{X}}} \rightarrow {{\textbf{X}}}$ be a convex function with $\varphi (0)=0$, $f: G \rightarrow {{\textbf{X}}}$ be uniformly continuous on G and such that $f(g)=0$ for all $g \in G \setminus C$. Let $h: G \rightarrow {\mathbb {R}}_0^+$ be a Lebesgue integrable function such that $\int _G h(g) \, d\mu (g)=1,$ and bounded on each subset $B \subset G$ with $\mu (B) < + \infty$. Then, we get

$$\begin{aligned} \varphi \Bigl (\int _G h(g) \, f(g) \, d\mu (g) \Bigr ) \le _{X} \int _G h(g) \, \varphi (f(g)) \, \, d\mu (g) . \end{aligned}$$

Corollary 3.27

Under the same hypotheses as in Theorem 3.26, assume that

$$\begin{aligned} 0 < \int _G \, h(g) \, d\mu (g) \le 1. \end{aligned}$$

Then,

$$\begin{aligned} \varphi \Bigl (\int _G h(g) \, f(g) \, d\mu (g) \Bigr ) \le _{X} \int _G h(g) \, \varphi (f(g)) \, \, d\mu (g) . \end{aligned}$$

4 Vector Lattice-Valued Modulars

In this section we define the modulars in the setting of vector lattices (for a related literature, see, e.g., [8, 13, 14, 40] and the references therein). Let T be a linear subspace of ${{\textbf{X}}}_1^G$, such that $\vert f \vert \in T$ whenever $f \in T$, and such that, if $f \in T$ and $A \in {{\mathcal {A}}}$, then $f \cdot \chi _A \in T$. We say that a functional $\rho :T \rightarrow \overline{{\textbf{X}}}^{+}$ is a modular on T iff

($\rho _0$):

$\rho (0)=0$;

($\rho _1$):

$\rho (-f)=\rho (f)$ for all $f \in T$;

($\rho _2$):

$\rho (\alpha _1 f + \alpha _2 h) \le _{X} \rho (f) + \rho (h)$ for any f, $h \in T$ and $\alpha _1$, $\alpha _2 \in {\mathbb {R}}_0^{+}$ with $\alpha _1 + \alpha _2 =1$.

A modular $\rho$ is said to be monotone iff $\rho (f) \le _{X} \rho (h)$ for each f, $h \in T$ with $\vert f \vert \le _{X_1} \vert h \vert$. Observe that, if $f \in T$, then $\vert f \vert \in T$ and $\rho (f) = \rho (\vert f \vert )$ (see, e.g., [8, 13]).

We say that a modular $\rho$ is convex iff $\rho (\alpha _1 f + \alpha _2 h) \le _{X} \alpha _1 \, \rho (f) + \alpha _2 \, \rho (h)$ whenever f, $h \in T$, $\alpha _1$, $\alpha _2 \in {\mathbb {R}}_0^{+}$, $\alpha _1 + \alpha _2 =1$.

We now give some examples of modulars. First, we formulate the following condition on functions $\varphi : {{\textbf{X}}}_1 \rightarrow {{\textbf{X}}}_1$ which will be useful for proving monotonicity of modulars:

$$\begin{aligned} \varphi (x_1 \vee x_2) \le _{X_1} \varphi (x_1) + \varphi (x_2) \text { for each } x_1, x_2 \in {{\textbf{X}}}_1^{+}. \end{aligned}$$

(23)

Let $\varphi$, ${\widehat{\varphi }}$ and $\Omega$ be as in (18), where ${\widehat{\varphi }}$ is not necessarily convex or differentiable. Observe that, in this case, the definition of $\varphi$ makes sense if and only if ${\widehat{\varphi }} \circ x \in {{\textbf{X}}}_1$ whenever $x \in {{\textbf{X}}}_1$. If ${\widehat{\varphi }} \in {{\mathbb {R}}}^{{\mathbb {R}}}$ is increasing, then for every $x_1$, $x_2 \in { {\textbf{X}}}_1^+$ and $\omega \in \Omega \setminus N$, where N is a suitable nowhere dense subset of $\Omega$, we have

$$\begin{aligned} {\widehat{\varphi }}(\max \{x_1(\omega ),x_2(\omega )\}) \le {\widehat{\varphi }}(x_1(\omega ))+{\widehat{\varphi }}(x_2(\omega )). \end{aligned}$$

(24)

Thanks to the Baire category theorem, $\Omega \setminus N$ is dense in $\Omega$, and hence the inequality in (24) holds for all $\omega \in \Omega$. Thus, from (18) and (24) we deduce (23). Some other examples of functions satisfying (23) can be found in [13]. Observe that, proceeding analogously as in [13, Proposition 3.1], it is possible to see that, if $\varphi : {{\textbf{X}}}_1 \rightarrow {{\textbf{X}}}_1$ is increasing on ${{\textbf{X}}}_1^{+}$, $\varphi (0)=0$, $\varphi$ satisfies (23), and

$$\begin{aligned} { {\mathcal {L}}}^{\varphi }= \Bigl \{ f \in {{{\textbf{X}}}_1}^G: \, \, \int _G \varphi (\vert f(g) \vert ) \, d\mu (g) \text { exists in } {{\textbf{X}}} \Bigr \} , \end{aligned}$$

(25)

then the operator $\rho ^{\varphi }$ defined by

$$\begin{aligned} \rho ^{\varphi }(f)=\int _G \varphi (\vert f(g) \vert ) \, d\mu (g), \quad f \in {{\mathcal {L}}}^{\varphi }, \end{aligned}$$

(26)

is a monotone modular and, if $\varphi$ is convex, then $\rho ^{\varphi }$ is convex on the set of the positive functions of ${{\mathcal {L}}}^{\varphi }$. The set

$$\begin{aligned} {L^{\varphi }(G)= \{f \in {{\textbf{X}}}_1^G: (o)\text {-}\lim _{\alpha \rightarrow 0^+} \rho ^{\varphi }(\alpha \, f) = \bigwedge _{\alpha \in {\mathbb {R}}^+} \rho ^{\varphi }(\alpha \, f) =0 \}} \end{aligned}$$

(27)

is the Orlicz space generated by $\varphi$ (here, ${(o)\text {-}\lim \nolimits _{\alpha \rightarrow 0^+} \rho ^{\varphi }(\alpha \, f)=0}$ means that there exists an (o)-sequence $(\sigma _l)_l$ in X such that for every $l \in {\mathbb {N}}$ there is ${\overline{\alpha }} \in {\mathbb {R}}^+$ with $\rho ^{\varphi }(\alpha \, f)\le _{X} \sigma _l$ whenever $0 <\alpha \le {\overline{\alpha }}$).

Thanks to the properties of the modulars, it is not difficult to check that $L^{\varphi }(G)$ is actually a linear space and that, if $\alpha \in {\mathbb {R}}^+$ and $\alpha \, f \in L^{\varphi }(G)$, then $\beta \, f \in L^{\varphi }(G)$ for each $\beta \in {\mathbb {R}}^+$, $\beta < \alpha$.

The subspace of $L^{\varphi }$ defined by

$$\begin{aligned} E^{\varphi }(G)=\{ f \in L^{\varphi }(G): \rho ^{\varphi } (\alpha \, f) \in {\textbf{X}} \text { for every }\alpha \in {\mathbb {R}}^+ \} \end{aligned}$$

(28)

is called the space of the finite elements of $L^{\varphi }(G)$.

A sequence $(f_n)_n$ in $L^{\varphi }(G)$ is modularly convergent to $f \in L^{\varphi }(G)$ iff

$$\begin{aligned} {{\ell } (( \rho ^{\varphi }(\alpha (f_n-f)))_n)=0} \quad \text {for at least one } \alpha \in {\mathbb {R}}^+ . \end{aligned}$$

(29)

Note that, since $\rho ^{\varphi }$ is a monotone modular and $\alpha \in {\mathbb {R}}^+$ satisfies the condition in (29), thanks to Axioms 2.1 we get

$$\begin{aligned} {{\ell } ((\rho ^{\varphi }(\beta (f_n-f)))_n)=0} \quad \text {for all } \beta \in {{\mathbb {R}}}^+, \, \, \beta < \alpha . \end{aligned}$$

(30)

5 The Structural Assumptions on the Operators

To prove our results about modular convergence with respect to the convergences introduced axiomatically in Axioms 2.1 in the vector lattice setting, we give some structural hypotheses.

We begin with the next technical assumption.

• H*) If $e \in {{\textbf{X}}}_1^+\setminus \{0\}$, V[e] is as in Sect. 2 and $(x_n)_n$ is a sequence in V[e] such that $\ell _{{\mathbb {R}}}((\Vert x_n\Vert )_n )=0$, then $\ell _1((\vert x_n \vert )_n )=0$.

Remark 5.1

Taking into account (2) and Axioms 2.1, it is not difficult to see that condition H*) is satisfied, for instance, when $\ell _{{\mathbb {R}}}$ is the usual (resp., filter, almost, Cesàro) convergence on ${\mathbb {R}}$, and $\ell _1$ is the usual (resp., filter, almost, Cesàro) order (or (r)-) convergence on ${{{\textbf{X}}}}_1$.

Now, similarly as in [6], we give the following

Assumption 5.2

Let G be a metric space, ${{\mathcal {A}}}$ be the $\sigma$-algebra of all Borel subsets of G, ${\textbf{X}}_2= {\mathbb {R}}$, and $\mu :{{\mathcal {A}}} \rightarrow \overline{{\mathbb {R}}}_0^{+}$ be $\sigma$-finite and regular. We denote by ${{\mathcal {C}}}_c(G)$ the space of all uniformly continuous functions $f \in {\textbf{X}}_1^G$ with compact support on G.

• 5.2.a) Let ${{\mathcal {M}}}$ be the class of all measurable functions $L:G \times G \rightarrow {\mathbb {R}}_0^{+}$ with respect to the product $\sigma$-algebra, such that the sections $L(\cdot ,t)$ and $L(s, \cdot )$ are integrable (with respect to $\mu$) and bounded on every set of finite measure $\mu$ for every t, $s \in G$, respectively.

• 5.2.b) Let $\Psi$ be the family of all functions $\psi :{{\textbf{X}}}_1^{+} \rightarrow {{\textbf{X}}}_1^{+}$ such that

• 5.2.b.1) $\psi$ is uniformly continuous and increasing on ${{\textbf{X}}}_1^{+}$, $\psi (0)=0$ and $\psi (v) \in {{\textbf{X}}}_1^+\setminus \{0\}$ whenever $v \in {{\textbf{X}}}_1^+\setminus \{0\}$.

Let $\Xi = (\psi _n)_n \subset \Psi$ be a sequence of functions such that:

• 5.2.b.2) $(\psi _n)_n$ is equicontinuous at 0, that is for every $u \in {{\textbf{X}}}_1^+\setminus \{0\}$ and $\varepsilon \in {\mathbb {R}}^+$ there are $w \in {{\textbf{X}}}_1^+\setminus \{0\}$ and $\delta \in {\mathbb {R}}^+$ with $\psi _n(x) \le _{X_1} \varepsilon \, w$ whenever $x \le _{X_1} \delta \, u$ and $n \in {\mathbb {N}}$;

• 5.2.b.3) for every $v \in {{\textbf{X}}}_1^{+}$ the sequence $(\psi _n(v))_n$ is order equibounded, that is there exists $A_v \in {{\textbf{X}}}_1^+\setminus \{0\}$ with $\psi _n(v) \le _{X_1} A_v$ for all $n \in {\mathbb {N}}$.

Let ${{\mathcal {K}}}_{\Xi }$ be the class of all sequences of functions $K_n: G \times G \times {{\textbf{X}}}_1\rightarrow {{\textbf{X}}}_1$, $n \in {\mathbb {N}}$, such that:

• 5.2.b.4) $K_n(\cdot ,t,u)$ and $K_n(s,\cdot ,u)$ are integrable on G with respect to the measure $\mu$ for any $u \in {{\textbf{X}}}_1$ and $n \in {\mathbb {N}}$;

• 5.2.b.5) $K_n(s,t,0) = 0$ for each $n \in {\mathbb {N}}$ and s, $t \in G$;

• 5.2.b.6) there are two positive sequences $(L_n)_n \subset {{\mathcal {N}}}$ and $(\psi _n)_n \subset \Psi$, with

  $$\begin{aligned} \vert K_n(s,t,u)-K_n(s,t,v) \vert \le _{X_1} L_n(s,t) \, \psi _n(\vert u-v \vert ) \end{aligned}$$

  for each $n \in {\mathbb {N}}$, s, $t \in G$ and u, $v \in {{\textbf{X}}}_1$.

• 5.2.c) Let $\varphi : {{\textbf{X}}}_1 \rightarrow {{\textbf{X}}}_1$ be a function, convex on ${{\textbf{X}}}_1$, increasing on ${{\textbf{X}}}_1^{+}$ and such that

• 5.2.c.1) $\varphi (0)=0$;

• 5.2.c.2) $\varphi (x) \in {{\textbf{X}}}_1^+\setminus \{0\}$ whenever $x \in {{\textbf{X}}}_1^+\setminus \{0\}$.

• 5.2.d) Let ${\mathbb {K}}=(K_n)_n \in {{\mathcal {K}}}_{\Xi }$, and ${{\textbf {T}}}=(T_n)_n$ be a sequence of Urysohn-type operators defined by

$$\begin{aligned} (T_n f)(s)=\int _G K_n(s,t,f(t)) \, d\mu (t), \quad s \in G, \end{aligned}$$

(31)

where $f \in$ Dom ${{\textbf {T}}}=$ ${\bigcap \nolimits _{n=1}^{\infty }}$ Dom $T_n$, and for every $n \in {\mathbb {N}}$, Dom $T_n$ is the set on which $T_n f$ is well-defined.

Now we extend the concept of singularity given in [6] to the setting of the convergences introduced axiomatically in 2.1.

Definition 5.3

A family ${\mathbb {K}} \in {{\mathcal {K}}}_{\Xi }$ is said to be singular iff there are: two sequences $(L_n)_n$ and $(\psi _n)_n$, satisfying 5.2.b.6), an infinite set $H \subset {\mathbb {N}}$, and an element $D^{(1)} \in {{\mathbb {R}}}^+$, such that

• 5.3.a.1)

  $$\begin{aligned} {\int _G L_n(s,t) \, d\mu (t) \le D^{(1)}} \text { for every } s \in G \text { and } n \in H, \end{aligned}$$

• 5.3.a.2)

  $$\begin{aligned} {\int _G L_n(s,t) \, d\mu (s) \le D^{(1)}} \text { for any } t \in G \text { and } n \in H, \end{aligned}$$

• 5.3.a.3) ${{\overline{\ell }}}_{{\mathbb {R}}}((x_n)_{n\in {\mathbb {N}}})= {{\overline{\ell }}}_{{\mathbb {R}}}((x_n)_{n\in H})$ for every sequence $(x_n)_n$ in ${\mathbb {R}}^+_0$,

• 5.3.a.4) ${{\overline{\ell }}}((x_n)_{n\in {\mathbb {N}}})= {{\overline{\ell }}}((x_n)_{n\in H})$ for any sequence $(x_n)_n$ in ${{\textbf{X}}}_1^+$.

A singular family ${{\mathcal {K}}}$ is (M)-singular (resp., (U)-singular) iff

• 5.3.b.1)

  $$\begin{aligned} \int _G L_n(s,t) \, d\mu (t) >0 \quad \text {for any } s \in G \text { and } n \in H; \end{aligned}$$

• 5.3.b.2) for each $A\in {{\mathcal {A}}}$ with $\mu (A) < + \infty$ there is a sequence $(A_n)_n$ in ${{\mathcal {A}}}$ with $\ell _2((\mu (A_n))_n) =0$ and such that

  $$\begin{aligned} { {\ell }_{{\mathbb {R}}} \Bigl ( \bigvee _{s \in A \setminus A_n} \int _{A \setminus B(s,\delta )} L_n(s,t) \, d\mu (t) }\Bigr )=0 \end{aligned}$$

  (resp., for every $A \in {{\mathcal {A}}}$ with $\mu (A) < + \infty$, we have

  $$\begin{aligned} { {\ell }_{{\mathbb {R}}} \Bigl ( \bigvee _{s \in A} \int _{A \setminus B(s,\delta )} L_n(s,t) \, d\mu (t) }\Bigr )=0 ) \end{aligned}$$

  for each $\delta \in {\mathbb {R}}^+$, where $B(s,\delta )=\{t \in G$: $d(s,t)\le \delta \}$;

• 5.3.b.3) for any $A \in {{\mathcal {A}}}$ with $\mu (A) < +\infty$ there are a sequence $(D_n)_n$ in ${{\mathcal {A}}}$ with $\ell _2((\mu (D_n))_n)=0$, an element $z \in {{\textbf{X}}}^+ \setminus \{0\}$ and an (o)-sequence $(\varepsilon _n)_n$ in ${\mathbb {R}}^+$, such that

  $$\begin{aligned} { \bigvee _{u \in {\textbf{X}}_1 \setminus \{0\} } \Bigl ( \bigvee _{s \in A \setminus D_n} \Bigl \vert \int _G K_n(s,t,u) \, d\mu (t) - u \Bigr \vert \Bigr ) \le _X \varepsilon _n \, z} \end{aligned}$$

  for all $n \in H$ (resp., for every $A \in {{\mathcal {A}}}$ with $\mu (A) < +\infty$ there exist $z \in {{\textbf{X}}}^+ \setminus \{0\}$ and an (o)-sequence $(\varepsilon _n)_n$ in ${{\mathbb {R}}}^+$, with

  $$\begin{aligned} { \bigvee _{u \in {\textbf{X}}_1 \setminus \{0\} } \Bigl ( \bigvee _{s \in A} \Bigl \vert \int _G K_n(s,t,u) \, d\mu (t) - u \Bigr \vert \Bigr ) \le _X \varepsilon _n \, z } \end{aligned}$$

  for each $n \in H$).

Remark 5.4

• 5.4.a) Note that, if $H={\mathbb {N}}$, condition 5.3.a.1) (resp., 5.3.a.2)) is equivalent to the order equiboundedness of the integrals

  $$\begin{aligned} \int _G L_n(s,t) \, d\mu (t) \, \text { (resp., }\, \int _G L_n(s,t) \, d\mu (s)\, ), \, n \in {\mathbb {N}}, \, s \in G \, \text { (resp., } t \in G). \end{aligned}$$

• 5.4.b) As an example, let ${{\mathcal {F}}}$ be any fixed free filter on ${{\mathbb {N}}}$. A sequence $(x_n)_n$ in ${\mathbb {R}}$ is said to be (order) ${{\mathcal {F}}}$-bounded iff there are $M_0 \in {{\mathbb {R}}}$ and $H \in {{\mathcal {F}}}$ with

  $$\begin{aligned} \vert x_n \vert \le M_0 \text { for all }n \in H \end{aligned}$$

  (32)

  (see, e.g., [15]). It is not difficult to see that, if the sequences

  $$\begin{aligned} \sup _{s \in G} \, \int _G L_n(s,t) \, d\mu (t), \, \, \, \sup _{t \in G} \, \int _G L_n(s,t) \, d\mu (s), \, \, \, n \in {\mathbb {N}}, \end{aligned}$$

  are ${{\mathcal {F}}}$-bounded and H is as in (32), then H satisfies the conditions in 5.3.a.j), $j=1,2$, with respect to the ${{\mathcal {F}}}$-limit superior, Now we prove 5.3.a.3). To this aim, it is enough to see that, for any sequence $(x_n)_n$ in ${{\mathbb {R}}}$, ${ \inf \nolimits _{F \in {\mathcal {F}}} (\sup \nolimits _{n \in F} x_n)= \inf \nolimits _{F \in {\mathcal {F}}} (\sup \nolimits _{n \in F\cap H} x_n) }.$ Indeed, since $F \cap H \in {{\mathcal {F}}}$ whenever F, $H \in {{\mathcal {F}}}$, the inequality $\le$ follows from the properties of the infimum. Moreover, as ${\sup \nolimits _{n \in F} x_n \ge \sup _{n \in F\cap H} x_n}$ for each $F \in {{\mathcal {F}}}$, taking the infimum as F varies in ${{\mathcal {F}}}$ we obtain the converse inequality. Analogously as above, it is possible to check that condition 5.3.a.4) is fulfilled if we endow ${{\textbf{X}}}_1$ with $(o{{\mathcal {F}}})$-convergence. Moreover, when we deal with usual, almost and Cesàro (order) convergence, we will take $H={\mathbb {N}}$ in the definition of (M)- and (U)-singularity.

6 The Main Results

In this section we present general modular convergence results for the involved operators, in connection with the convergences introduced in Axioms 2.1, extending [6, Theorem 3]. We begin with the following result on equiabsolute continuity.

Theorem 6.1

Under Assumptions 5.2, assume that ${{\textbf{X}}}_1={{\textbf{X}}}$, ${\mathbb {K}}$ is singular and that the following condition is satisfied:

• 6.1.1) for every compact set $C \subset G$ there exists an increasing sequence $(B_m)_m$ of subsets of G having finite measure $\mu$ and with

  $$\begin{aligned} \ell _{{\mathbb {R}}} \Bigl ( \Bigl ( {\bar{\ell }}_{{\mathbb {R}}} \Bigl ( \Bigl ( \sup _{t \in C} \int _{G \setminus B_m} L_n(s,t) \, d\mu (s) \Bigr )_n \Bigr ) \Bigr )_m \Bigr )=0. \end{aligned}$$

  (33)

Then there is a constant $\lambda \in {\mathbb {R}}^+$ such that, for every $f \in {{\mathcal {C}}}_c(G)$, the sequence $\varphi (\lambda \vert T_n f \vert )$, $n \in H$, has $\mu$-equiabsolutely continuous integrals.

Proof

Let H and $D^{(1)}$ be as in 5.3.a.1) and 5.3.a.2), $\lambda \in {\mathbb {R}}^+$ be such that $\lambda \, D^{(1)} \le 1$, $f \in {{\mathcal {C}}}_c(G)$, C be a compact set such that $f(s)=0$ for all $s \in G \setminus C$, ${u^*=\bigvee \nolimits _{s \in G} \vert f(s) \vert }$, and $(B_m)_m$ be a sequence from ${{\mathcal {A}}}$, according to (33).

Set $\Theta ^*= {\varphi \Bigl (\bigvee \nolimits _{n \in {\mathbb {N}}} \psi _n (u^*)\Bigr )}$. Note that $\Theta ^* \in {\textbf{X}}$, thanks to the order equiboundedness of the sequence $((\psi _n(u^*))_n)$ (see 5.2.b.3)). Pick arbitrarily $s \in G$ and $n\in H$. Taking into account the monotonicity of $\varphi$ and using Corollary 3.27, where the roles of f and h are played by the functions $\lambda \, D^{(1)} \, \psi _n(\vert f(\cdot )\vert )$, $\dfrac{L_n(s,\cdot )}{D^{(1)}},$ respectively, we have

$$\begin{aligned}{} & {} \varphi \Bigl (\lambda \int _G \vert K_n(s,t,f(t)) \vert \, d\mu (t) \Bigr ) \le _{X} \varphi \Bigl (\lambda \int _G L_n(s,t) \, \psi _n(\vert f(t) \vert ) \,d\mu (t) \Bigr ) \nonumber \\{} & {} \quad \le _{X} \varphi \Bigl (\lambda \, D^{(1)} \int _C \dfrac{L_n(s,t)}{D^{(1)}} \, \psi _n(\vert f(t) \vert ) \, d\mu (t) \Bigr )\nonumber \\{} & {} \quad \le _{X} \frac{1}{D^{(1)}} \int _C L_n(s,t) \, \varphi (\lambda \, D^{(1)} \, \psi _n(\vert f(t) \vert ) \,) \, \, d\mu (t)\nonumber \\{} & {} \quad \le _{X} \frac{1}{D^{(1)}} \int _C L_n(s,t) \cdot \Bigl ( \varphi \Bigl (\bigvee _{n \in {\mathbb {N}}} \psi _n (u^*) \Bigr ) \Bigr ) \, d\mu (t) \le _{X} \frac{1}{D^{(1)}} \, \Bigl ( \int _C L_n(s,t) \, d\mu (t) \Bigr ) \cdot \Theta ^*.\nonumber \\ \end{aligned}$$

(34)

By applying the Fubini-Tonelli theorem to $L_n$ and integrating with respect to $\mu (s)$ on $G \setminus B_m$, $m \in {\mathbb {N}}$, we get, for each $m \in {\mathbb {N}}$ and $n \in H$,

$$\begin{aligned} \int _{G \setminus B_m} \Bigl (\int _C L_n(s,t) \, d\mu (t) \Bigr ) \, d\mu (s) = \int _C \Bigl (\int _{G \setminus B_m} L_n(s,t) \, d\mu (s) \Bigr ) \, d\mu (t), \end{aligned}$$

(35)

and hence

$$\begin{aligned}{} & {} \int _{G \setminus B_m} \Bigl (\dfrac{1}{D^{(1)}} \int _C L_n(s,t) \, d\mu (t) \Bigr ) \cdot \Theta ^*\, d\mu (s) \nonumber \\{} & {} \quad = \dfrac{1}{D^{(1)}} \Bigl (\int _C \Bigl (\int _{G \setminus B_m} L_n(s,t) \, d\mu (s) \Bigr ) \, d\mu (t) \Bigr ) \cdot \Theta ^*\nonumber \\{} & {} \quad \le _{X} \dfrac{1}{D^{(1)}} \Bigl (\int _C \Bigl ( \sup _{t \in C} \int _{G \setminus B_m} L_n(s,t) \, d\mu (s) \Bigr ) \, d\mu (t) \Bigr ) \cdot \Theta ^* \le _{X} \dfrac{1}{D^{(1)}} \, \iota _{m,n} \, \mu (C)\, \cdot \Theta ^* ,\nonumber \\ \end{aligned}$$

(36)

where ${ \iota _{m,n}= \sup \nolimits _{t \in C} \int _{G \setminus B_m} L_n(s,t) \, d\mu (s)}.$

From (31), (34), (35) and (36) we obtain

$$\begin{aligned} 0 \le _{X}&\int _{G\setminus B_m} \varphi (\lambda \vert T_n(f)(s) \vert )\, d\mu (s)\nonumber \\ \le _{X}&\int _{G \setminus B_m} \varphi \Bigl (\lambda \int _G \vert K_n(s,t,f(t)) \vert \, d\mu (t) \Bigr ) \, d\mu (s) \le _{X} \dfrac{\iota _{m,n} \, \mu (C)}{D^{(1)}} \, \Theta ^* . \end{aligned}$$

(37)

By 6.1.1), we know that $\ell _{{\mathbb {R}}} ( ( {\bar{\ell }}_{{\mathbb {R}}} ( ( \iota _{m,n})_{n\in {\mathbb {N}}} ) )_{m\in {\mathbb {N}}} )=0$. From this and 5.3.a.3) it follows that $\ell _{{\mathbb {R}}} ( ( {\bar{\ell }}_{{\mathbb {R}}} ( ( \iota _{m,n})_{n\in H} ) )_{m\in {\mathbb {N}}} )=0$. Thus, taking into account that $\mu (C) < + \infty$, from (37), Axioms 2.1, 2.2 and conditions 3.1.7), 3.1.9) we deduce

$$\begin{aligned} \ell \Bigl ( \Bigl ( {\bar{\ell }} \Bigl ( \Bigl ( \int _{G\setminus B_m} \varphi (\lambda \vert T_n(f)(s) \vert ) \, d\mu (s) \Bigr )_{n \in H} \Bigr ) \Bigr )_{m \in {\mathbb {N}}}\Bigr )=0. \end{aligned}$$

From this and 5.3.a.4) we obtain

$$\begin{aligned} \ell \Bigl ( \Bigl ( {\bar{\ell }} \Bigl ( \Bigl ( \int _{G\setminus B_m} \varphi (\lambda \vert T_n(f)(s) \vert ) \, d\mu (s) \Bigr )_{n \in {\mathbb {N}}} \Bigr ) \Bigr )_{m \in {\mathbb {N}}} \Bigr )=0. \end{aligned}$$

Thus, 3.5.2) follows.

Now we prove 3.5.1). Let $(A_n)_n$ be any sequence in ${{\mathcal {A}}}$, such that $\ell _2((\mu (A_n))_n)=0$. By arguing analogously as in (34) and (36), we obtain

$$\begin{aligned} 0 \le \int _{A_n} \varphi (\lambda \vert T_n(f)(s) \vert )\, d\mu (s) \le _{X} \mu (A_n) \, \mu (C) \, \, \Theta ^* . \end{aligned}$$

From the previous inequality and Axioms 2.1 we get

$$\begin{aligned} {\ell \Bigl ( \Bigl ( \int _{A_n} \varphi (\lambda \vert T_n(f)(s) \vert )\, d\mu (s) \Bigr )_n \Bigr )=0}, \end{aligned}$$

that is 3.5.1). This ends the proof. $\square$

A consequence of Theorem 6.1 is the following

Corollary 6.2

Under the same hypotheses as in Theorem 6.1, if ${\mathbb {K}}$ is singular, then there is a constant $\beta \in {\mathbb {R}}^+$ such that, for every $f \in {{\mathcal {C}}}_c(G)$, the sequence $\varphi (\beta \vert T_n f-f \vert )$, $n \in H$, has $\mu$-equiabsolutely continuous integrals.

Proof

Let $\lambda$ be as in Theorem 6.1, and choose arbitrarily $f \in {{\mathcal {C}}}_c(G)$. Note that $\lambda \, \vert f \vert \in {{\mathcal {C}}}_c(G)$. By Proposition 3.25, the function $s \mapsto \varphi (\lambda \, \vert f(s) \vert )$ is integrable on G. By Theorem 3.12, the integral

$$\begin{aligned} \int _{(\cdot )} \varphi (\lambda \, \vert f(s) \vert ) \, d\mu (s) \end{aligned}$$

(38)

is absolutely continuous. Moreover, since $\rho ^{\varphi }$ is a modular, we get

$$\begin{aligned} \int _A \varphi \Bigl (\dfrac{\lambda }{2} \, \vert T_n f (s) - f(s) \vert \Bigr ) \, d\mu (s) \le _{X} \int _A \varphi (\lambda \, \vert T_n f(s)\vert ) \, d\mu (s) + \int _A \varphi (\lambda \, \vert f(s)\vert ) \, d\mu (s) \end{aligned}$$

for every $A \in {{\mathcal {A}}}$. The assertion follows from the absolute continuity of the integral in (38) and Theorem 6.1, taking $\beta = \lambda /2$. $\square$

The following result on convergence in measure (resp., uniform convergence) extends [6, Theorem 2] to our setting.

Theorem 6.3

Under Assumptions 5.2, if ${{\textbf{X}}}_1={{\textbf{X}}}$ and ${\mathbb {K}} \in {{\mathcal {K}}}_{\Xi }$ is (M)- (resp., (U)-) singular, then there exists $\alpha \in {\mathbb {R}}^+$ such that, for every $f \in {{\mathcal {C}}}_c(G)$, the sequence

$$\begin{aligned} \varphi (\alpha \vert T_n f - f \vert ), \, \, n \in {\mathbb {N}}, \end{aligned}$$

converges to 0 in measure (resp., uniformly) on every set $A \in {{\mathcal {A}}}$ with $\mu (A) < + \infty$.

Moreover, the sequence $(T_n f)_n$ converges to f in measure (resp., uniformly) on each set $A \in {{\mathcal {A}}}$ such that $\mu (A) < + \infty$.

Proof

First, we prove the results concerning convergence in measure. Let $D^{(1)}$ be as in 5.3.1) and 5.3.2), $\alpha \in {\mathbb {R}}^+$ be such that $2\, \alpha \le 1$ and $4 \, \alpha \, D^{(1)} \le 1$, and $u \in {{\textbf{X}}}^+ \setminus \{0\}$ be related to the uniform continuity of f. Fix arbitrarily $\varepsilon \in {\mathbb {R}}^+$. Without loss of generality, we can suppose $\varepsilon \le 1$. Since the sequence $(\psi _n)_n$ is equicontinuous at 0 (see 5.2.b.2)), then there exist $w \in {{\textbf{X}}}^+ \setminus \{0\}$ and $\sigma \in {\mathbb {R}}^+$ with

$$\begin{aligned} \psi _n(\vert x_1-x_2 \vert ) \le _{X} \varepsilon \, w \text { whenever } \vert x_1-x_2\vert \le _{X} \sigma \, u \text { and } n\in {\mathbb {N}}. \end{aligned}$$

(39)

By the uniform continuity of f, in correspondence with $\sigma$ there is $\delta \in {\mathbb {R}}^+$ such that

$$\begin{aligned} \vert f(t)-f(s)\vert \le _{X} \sigma \,u \text { whenever } t,s \in G, \, \, d(t,s)\le \delta . \end{aligned}$$

(40)

Since $\varphi$ is convex, increasing on ${{\textbf{X}}}^+ \setminus \{0\}$ and $\varphi (0)=0$, from (39), (40) and Proposition 3.23 it follows that

$$\begin{aligned} \varphi (\psi _n(\vert f(t)-f(s) \vert )) \le _{X} \varphi (\varepsilon \, w) \le _{X} \varepsilon \, \varphi (w) \end{aligned}$$

(41)

whenever $t,s \in G$, $d(t,s)\le \delta$ and $n\in {\mathbb {N}}$. Fix $A \in {{\mathcal {A}}}$ with $\mu (A) < + \infty$. Let $z \in {{\textbf{X}}}^+ \setminus \{0\}$, $(D_n)_n$ be a sequence in ${\mathcal {A}}$ and $(\varepsilon _n)_n$ be an (o)-sequence in ${\mathbb {R}}^+$, related to 5.3.b.3). If $\varepsilon _1 >1$, we can replace z with $\varepsilon _1 \, z$ and $(\varepsilon _n)_n$ with $(\varepsilon _n/\varepsilon _1)_n$ in 5.3.b.3), so that without loss of generality we can assume that $\varepsilon _1\le 1$. Let $(A_n)_n$ be a sequence in ${\mathcal {A}}$ associated with 5.3.b.2), and put $E_n= A_n \cup D_n$, $n \in {\mathbb {N}}$. As in (5), since $\ell _2((\mu (A_n))_n)= \ell _2((\mu (D_n))_n)=0,$ we get $\ell _2((\mu (E_n))_n)=0.$ Now, set

$$\begin{aligned} {v^*=\varphi (w) + \varphi (z) + \varphi \Bigl (4 \, \alpha \, D^{(1)} \bigvee _{n \in {\mathbb {N}}} \psi _n (2 \, u^*) \Bigr )}, \end{aligned}$$

(42)

and let $\delta$ be as in (40). By 5.3.b.2), we have

$$\begin{aligned} \ell _{{\mathbb {R}}} \Bigl ( \Bigl ( \sup _{s \in A \setminus E_n} { \int _{G \setminus B(s,\delta )} L_n(s,t) \, d\mu (t) }\Bigr )_n \Bigr ) =0. \end{aligned}$$

(43)

By 5.2.b.6), (41), (43), thanks to the convexity of the modular $\rho ^{\varphi }$, and using Corollary 3.27, where the roles of f and h are played by the functions $4 \, \alpha \, D^{(1)} \, \psi _n(\vert f(\cdot )-f(s)\vert )$ and $\dfrac{L_n(s,\cdot )}{D^{(1)}}$ respectively (with n and s fixed), for each $n \in H$ and $s \in A \setminus E_n$ we get

$$\begin{aligned}{} & {} 0 \le _{X} \varphi (\alpha \vert (T_n f)(s)-f(s) \vert ) =\varphi \Bigr (\alpha \Bigl \vert \int _G K_n(s,t,f(t) )\, d\mu (t) -f(s)\Bigr \vert \Bigr )\nonumber \\{} & {} \le _{X} \dfrac{1}{2} \, \varphi \Bigr (2 \, \alpha \Bigl \vert \int _G K_n(s,t,f(s)) \, d\mu (t) -f(s)\Bigr \vert \Bigr ) \nonumber \\{} & {} \quad + \dfrac{1}{2} \, \varphi \Bigr (2 \, \alpha \int _G \vert K_n(s,t,f(t))-K_n(s,t,f(s)) \vert \, d\mu (t) \Bigr )\nonumber \\ {}{} & {} \le _{X} \dfrac{1}{2} \, \bigvee _{s \in A \setminus E_n} \,\varphi \Bigr (2 \, \alpha \Bigl \vert \int _G K_n(s,t,f(s)) \, d\mu (t) -f(s)\Bigr \vert \Bigr ) \nonumber \\{} & {} \quad + \dfrac{1}{2} \, \varphi \Bigr (2 \, \alpha \int _G L_n(s,t) \, \psi _n(\vert f(t)-f(s) \vert )\, d\mu (t) \Bigr ) \nonumber \\{} & {} \le _{X} \dfrac{1}{2} \, \varphi \Bigr (2 \, \alpha \bigvee _{s \in A \setminus E_n, f(s)\ne 0} \Bigl \vert \int _G K_n(s,t,f(s)) \, d\mu (t) -f(s)\Bigr \vert \Bigr ) \nonumber \\{} & {} + \dfrac{1}{4} \, \varphi \Bigr (4 \, \alpha \int _{G \cap B(s,\delta )} L_n(s,t) \,\psi _n(\vert f(t)-f(s) \vert ) \, d\mu (t) \Bigr ) \end{aligned}$$

(44)

$$\begin{aligned}{} & {} \quad + \dfrac{1}{4} \, \varphi \Bigr (4 \, \alpha \int _{G \setminus B(s,\delta )} L_n(s,t) \, \psi _n(\vert f(t)-f(s) \vert ) \, d\mu (t)\Bigr ) \nonumber \\ {}{} & {} \le _{X} \dfrac{1}{2} \, \varphi \Bigr (2 \, \alpha \bigvee _{u \in {{\textbf{X}}} \setminus \{0\}} \Bigl ( \bigvee _{s \in A \setminus E_n} \Bigl \vert \int _G K_n(s,t,u) \, d\mu (t) -u\Bigr \vert \Bigr ) \Bigr )\nonumber \\{} & {} \quad + \dfrac{1}{4 \,D^{(1)}} \int _{C \cap B(s, \delta )} L_n(s,t) \, \varphi (4 \, \alpha \,D^{(1)} \psi _n(\vert f(t)-f(s) \vert ) \, d\mu (t)\Bigr ) \nonumber \\{} & {} \quad + \dfrac{1}{4 \,D^{(1)}} \int _{C \setminus B(s, \delta )} L_n(s,t) \, \varphi (4 \, \alpha \,D^{(1)} \psi _n(\vert f(t)-f(s) \vert ) \, d\mu (t)\Bigr ) \nonumber \\{} & {} \le _{X} \dfrac{1}{2} \varphi ( \varepsilon _n \, z) + \dfrac{1}{4} \varphi ( \varepsilon \, w)\nonumber \\{} & {} \quad +\frac{1}{4} \Bigl ( \int _{G\setminus B(s,\delta )} L_n(s,t) d\mu (t) \Bigr ) \cdot \varphi \Bigl ( 4 \alpha D^{(1)} \bigvee _{n \in {\mathbb {N}}} \psi _n (2 u^*) \Bigr ) \nonumber \\{} & {} \le _{X} \dfrac{1}{2} \, \varepsilon _n \, \varphi (z)+ \Bigl ( \varepsilon + \sup _{s \in A \setminus E_n} \int _{G\setminus B(s,\delta )} L_n(s,t) \, d\mu (t) \Bigr ) \cdot v^*\nonumber \\ {}{} & {} \le _{X} \Bigl ( \varepsilon + \varepsilon _n + \sup _{s \in A \setminus E_n} \int _{G\setminus B(s,\delta )} L_n(s,t) \, d\mu (t) \Bigr ) \cdot v^* , \end{aligned}$$

(45)

taking into account that $\varepsilon _1 \le 1$ and by virtue of formula (21) in Proposition 3.23. Thus, all terms of (44) belong to the vector lattice $V[v^*]$, defined by

$$\begin{aligned} V[v^*]=\{x \in {{\textbf{X}}} : \text { there exists } \sigma \in {\mathbb {R}}^+ \text { with } \vert x \vert \le \sigma \, v^* \}. \end{aligned}$$

(46)

As seen in Sect. 2, $v^*$ is a strong order unit in $V[v^*]$. Thus, taking in (44) the norm $\Vert \cdot \Vert _{v^*}$, defined analogously as in (1), from monotonicity and subadditivity of $\Vert \cdot \Vert _{v^*}$, 5.3.a.3), (4) and (43), and taking into account the monotonicity of $(\varepsilon _n)$ and 2.4.f), we obtain

$$\begin{aligned} 0\le & {} \, {\bar{\ell }}_{{\mathbb {R}}} \Bigl ( \Bigl ( \Bigl \Vert \bigvee _{s \in A \setminus E_n} \varphi (\alpha \vert (T_n f)(s)-f(s) \vert ) \Bigr \Vert _{v^*}\Bigr )_{n \in {\mathbb {N}}} \Bigr ) \nonumber \\= & {} \, {\bar{\ell }}_{{\mathbb {R}}} \Bigl ( \Bigl ( \Bigl \Vert \bigvee _{s \in A \setminus E_n} \varphi (\alpha \vert (T_n f)(s)-f(s) \vert ) \Bigr \Vert _{v^*}\Bigr )_{n \in H} \Bigr ) \nonumber \\\le & {} {\bar{\ell }}_{{\mathbb {R}}} \, \Bigl ( \Bigl ( \Bigl \Vert \Bigl ( \varepsilon + \varepsilon _n \,+ \sup _{s \in A \setminus E_n} \int _{G\setminus B(s,\delta )} L_n(s,t) \, d\mu (t) \Bigr ) \cdot v^*\Bigr \Vert _{v^*}\Bigr )_{n \in H} \Bigr ) \nonumber \\ {}= & {} {\bar{\ell }}_{{\mathbb {R}}} \, \Bigl ( \Bigl ( \Bigl \Vert \Bigl ( \varepsilon + \varepsilon _n \, + \sup _{s \in A \setminus E_n} \int _{G\setminus B(s,\delta )} L_n(s,t) \, d\mu (t) \Bigr ) \cdot v^*\Bigr \Vert _{v^*}\Bigr )_{n \in {\mathbb {N}}} \Bigr ) \nonumber \\ {}\le & {} \varepsilon + {\bar{\ell }}_{{\mathbb {R}}} \Bigl ( \Bigl ( \sup _{s \in A \setminus E_n} \int _{G \setminus B(s,\delta )} L_n(s,t) \, d\mu (t) \Bigr )_{n \in {\mathbb {N}}} \Bigr ) + {\bar{\ell }}_{{\mathbb {R}}} ((\varepsilon _n)_n) \le \varepsilon . \end{aligned}$$

(47)

From (47) and the arbitrariness of $\varepsilon \in {\mathbb {R}}^+$ it follows that

$$\begin{aligned} {\bar{\ell }}_{{\mathbb {R}}} \Bigl ( \Bigl ( \Bigl \Vert \bigvee _{s \in A \setminus E_n} \varphi (\alpha \vert (T_n f)(s)-f(s) \vert ) \Bigr \Vert _{v^*}\Bigr )_{n \in {\mathbb {N}}} \Bigr )=0, \end{aligned}$$

and hence, by Axiom 2.2.e), we obtain

$$\begin{aligned} \ell _{{\mathbb {R}}} \Bigl ( \Bigl ( \Bigl \Vert \bigvee _{s \in A \setminus E_n} \varphi (\alpha \vert (T_n f)(s)-f(s) \vert ) \Bigr \Vert _{v^*}\Bigr )_{n \in {\mathbb {N}}} \Bigr )=0. \end{aligned}$$

From this and condition H*) we deduce

$$\begin{aligned} \ell \Bigl ( \Bigl ( \bigvee _{s \in A \setminus E_n} \varphi (\alpha \vert (T_n f)(s)-f(s) \vert ) \Bigr )_{n \in {\mathbb {N}}} \Bigr )=0, \end{aligned}$$

and hence the sequence $\varphi (\alpha \vert T_n f - f\vert ), \, \, n \in {\mathbb {N}},$ converges in $\mu$-measure to 0 on A. The proof of the results about uniform convergence is analogous, taking $A_n=D_n=\emptyset$. To prove the last statement it is enough to argue analogously as above, taking $\alpha =1$ and $\varphi$ equal to the identity map. $\square$

Now we define modular convergence in the context of abstract convergence in vector lattices.

Definition 6.4

A sequence $(f_n)_n$ of functions in $L^{\varphi }(G)$ is ${\ell }$-modularly convergent to $f \in L^{\varphi }(G)$ iff there exists a constant $\alpha \in {\mathbb {R}}^+$ such that

$$\begin{aligned} {\ell }((\rho ^{\varphi }(\alpha (f_n-f))_n))=0. \end{aligned}$$

We are ready to present the main result on modular convergence in our setting, with respect to the convergences introduced axiomatically in Axioms 2.1.

Theorem 6.5

Under Assumptions 5.2, if ${{\textbf{X}}}_1={{\textbf{X}}}$, ${\mathbb {K}}$ is (M)-singular and condition 6.1.1) holds, then there is a constant $\alpha \in {\mathbb {R}}^+$ with ${\ell }((\rho ^{\varphi }(\alpha (T_n f-f))_n))=0$ for every $f \in {{\mathcal {C}}}_c(G)$.

Proof

Theorem 6.5 is a consequence of Corollary 6.2, Theorem 6.3 and Theorem 3.6 applied to the sequence $\varphi (\alpha \vert T_n f - f \vert ),$ $n \in H$. $\square$

6.1 Applications to Mellin Kernel

In this setting, we formulate the following structural assumptions:

Assumption 6.6

• 6.6.a) Let $G=({\mathbb {R}}^+,d_{\ln })$, where $d_{\ln } (t_1, t_2) =\vert \ln t_1 - \ln t_2\vert$, $t_1, t_2 \in {\mathbb {R}}^+$, let ${{\textbf{X}}}_1={{\textbf{X}}}$, ${{\textbf{X}}}_2={{\mathbb {R}}}$, and for any measurable set $S \subset {\mathbb {R}}^+$ put ${\mu (S)= \int _S \frac{dt}{t}}.$ Let $\widetilde{{\mathcal {M}}}$ be the set of all sequences of non-negative functions ${\widetilde{L}}_n$ defined on ${\mathbb {R}}^+$, integrable with respect to $\mu$ and bounded on every subset $A \subset {\mathbb {R}}^+$ with $\mu (A) < +\infty$.

• 6.6.b) Let $\Xi = (\psi _n)_n \subset \Psi$ be as in 5.2 (c), and denote by ${\widetilde{{\mathcal {K}}}}_{\Xi }$ the set of all sequences of functions ${\widetilde{K}}_n: {\mathbb {R}}^+ \times {\mathbb {R}} \rightarrow {{\textbf{X}}}$, $n \in {\mathbb {N}}$, such that:

• 6.6.b.i) ${\widetilde{K}}_n(\cdot ,u)$ is integrable for each $u \in {{\textbf{X}}}$ and $n \in {\mathbb {N}}$, and ${\widetilde{K}}_n(t,0) = 0$ for all $n \in {\mathbb {N}}$ and $t \in {\mathbb {R}}^+$;

• 6.6.b.ii) there are sequences $({\widetilde{L}}_n)_n \subset \widetilde{{\mathcal {M}}}$ and $(\psi _n)_n \subset \Psi$ with

  $$\begin{aligned} \vert {\widetilde{K}}_n(t,u)-{\widetilde{K}}_n(t,v) \vert \le _{X} {\widetilde{L}}_n(t) \, \psi _n(\vert u-v\vert ) \end{aligned}$$

  for all $n \in {\mathbb {N}}$, $t \in {\mathbb {R}}^+$ and u, $v \in {\textbf{X}}$.

• 6.6.c) Let $\widetilde{{\mathbb {K}}}=({\widetilde{K}}_n)_n \in {\widetilde{{\mathcal {K}}}}_{\Xi }$, and consider a sequence $\widetilde{{{\textbf {T}}}}=(\widetilde{T_n})_n$ of nonlinear Mellin-type operators defined by

  $$\begin{aligned} (\widetilde{T_n} f)(s)=\int _0^{+ \infty } \, {\widetilde{K}}_n\Bigl (\frac{t}{s},f(t)\Bigr ) \frac{dt}{t}, \quad s \in \mathcal {{\mathbb {R}}}^+, \end{aligned}$$

  where $f \in$ Dom $\widetilde{{{\textbf {T}}}}=$ ${\bigcap \nolimits _{n=1}^{\infty }}$ Dom $\widetilde{T_n}$, and Dom $\widetilde{T_n}$ is the set on which $\widetilde{T_n} f$ is well-defined, for each $n \in {\mathbb {N}}$.

It is not difficult to check that the function $d_{\ln }$ is actually a distance.

Now we give the concept of singularity in the setting of Mellin operators.

Definition 6.7

We say that $\widetilde{{\mathbb {K}}}= ({\widetilde{K}}_n)_n$ is singular iff there are an infinite set $H \subset {\mathbb {N}}$ and $D^{(1)} \in {{\mathbb {R}}}^+$ with

• 6.7.1) ${ \int _0^{+ \infty } {\widetilde{L}}_n(t) \, \dfrac{dt}{t} \le D^{(1)} } \text { for every } n \in H$, and

• 6.7.2) ${\overline{\ell }}((a_n)_{n\in {\mathbb {N}}})= {\overline{\ell }}((a_n)_{n\in H})$ for each sequence $(a_n)_n$ in ${\mathbb {R}}$.

A singular family ${{\mathcal {K}}}$ is said to be (U)-singular iff:

• 6.7.3) ${ \int _0^{+ \infty } {\widetilde{L}}_n(t) \, \dfrac{dt}{t} >0} \quad \text {for all } n \in {\mathbb {N}};$

• 6.7.4) for any $\delta \in {\mathbb {R}}^+$, $\delta > 1$, one has

  $$\begin{aligned} { {\ell }_{{\mathbb {R}}} \Bigl ( \Bigl ( \int _{{\mathbb {R}}^+ \setminus [1/\delta , \delta ]} {\widetilde{L}}_n(t) \, \dfrac{dt}{t} }\Bigr )_n \Bigr )=0; \end{aligned}$$

• 6.7.5) there are $z \in {{\textbf{X}}}^+ \setminus \{0\}$ and an (o)-sequence $(\varepsilon _n)_n$ in ${{\mathbb {R}}}^+$, with

  $$\begin{aligned} \bigvee _{u \in {\textbf{X}}_1 \setminus \{0\} } \Bigl \vert \int _0^{+\infty } {\widetilde{K}}_n(t,u) \, \dfrac{dt}{t} - u \Bigr \vert \le _X \varepsilon _n \, z \end{aligned}$$

  whenever $n \in H$).

Remark 6.8

• 6.8.a) Note that, if we set ${L_n(s,t)={\widetilde{L}}_n\Bigl ( \frac{t}{s}\Bigr )}$, $K_n(s,t,u)={\widetilde{K}}_n\Bigl ( \frac{t}{s},u \Bigr )$ for all s, $t \in {\mathbb {R}}^+$, $u \in {\textbf{X}}$ and $n \in {\mathbb {N}}$, then it is not difficult to see that, if ${\widetilde{K}}_n$, ${\widetilde{L}}_n$, $n \in {\mathbb {N}}$, fulfil Assumptions 6.6, then $K_n$, $L_n$, $n \in {\mathbb {N}}$, satisfy Assumptions 5.3, and to check that

  $$\begin{aligned}{} & {} \int _0^{+\infty } {\widetilde{L}}_n\Bigl (\frac{t}{s}\Bigr ) \, \frac{dt}{t} = \int _0^{+\infty } {\widetilde{L}}_n(z) \, \frac{s \, dz}{sz}= \int _0^{+\infty } {\widetilde{L}}_n(z) \, \frac{dz}{z},\\{} & {} \int _0^{+\infty } {\widetilde{K}}_n\Bigl (\frac{t}{s},u\Bigr ) \, \frac{dt}{t} = \int _0^{+\infty } {\widetilde{K}}_n(z,u)\frac{s \, dz}{sz} = \int _0^{+\infty } {\widetilde{K}}_n(z,u)\frac{dz}{z} \end{aligned}$$

  for all $n \in {\mathbb {N}}$, $s \in {\mathbb {R}}^+$ and $u \in {\textbf{X}}$.

• 6.8.b) Analogously to [6, Example 1] we have that, if $({\widetilde{L}}_n)_n$ satisfies (6.7.4), namely

  $$\begin{aligned} { {\ell }_{{\mathbb {R}}} \Bigl ( \Bigl ( \int _{{\mathbb {R}}^+ \setminus [1/\delta , \delta ]} {\widetilde{L}}_n(t) \, \dfrac{dt}{t} }\Bigr )_n \Bigr )=0 \end{aligned}$$

  for each $\delta >1$, then for each compact subset $C \subset {\mathbb {R}}^+$ there exists a set $B \subset {\mathbb {R}}^+$ of finite $\mu$-measure such that

  $$\begin{aligned} {\bar{\ell }}_{{\mathbb {R}}} \Bigl ( \Bigl ( \sup _{t \in C}\int _{{{\mathbb {R}}}^+ \setminus B} {\widetilde{L}}_n \Bigl (\dfrac{s}{t}\Bigr ) \, \frac{ds}{s} \Bigr )_n \Bigr )=0. \end{aligned}$$

  In fact, pick arbitrarily a compact set $C \subset {\mathbb {R}}^+$. Without loss of generality, we may suppose that $C=[1/M,M]$, where $M>1$. Let $\delta = 2\, M.$ For every $t \in C$ it is $\frac{\delta }{t} \ge 2$, and $\frac{1}{\delta \, t} \le \frac{1}{2}$. From this and the non-negativity of the ${\widetilde{L}}_n$’s it follows that

  $$\begin{aligned} 0\le & {} \sup _{t \in C} \int _{{\mathbb {R}}^+ \setminus [1/\delta , \delta ]} {\widetilde{L}}_n \Bigl (\dfrac{s}{t}\Bigr ) \, \frac{ds}{s}\\\le & {} \sup _{t \in C} \int _{0}^{1/\delta } {\widetilde{L}}_n \Bigl (\dfrac{s}{t}\Bigr ) \, \frac{ds}{s} +\sup _{t \in C}\int _{\delta }^{+\infty }{\widetilde{L}}_n \Bigl (\dfrac{s}{t}\Bigr ) \, \frac{ds}{s} \\ {}= & {} \sup _{t \in C}\int _{0}^{1/(\delta \, t)}{\widetilde{L}}_n (z) \dfrac{dz}{z} + \sup _{t \in C}\int _{\delta /t}^{+\infty } {\widetilde{L}}_n (z) \dfrac{dz}{z} \\ {}\le & {} \int _{0}^{1/2}{\widetilde{L}}_n (z) \dfrac{dz}{z} + \int _{2}^{+\infty } {\widetilde{L}}_n (z) \dfrac{dz}{z} =J_n. \end{aligned}$$

  Since, by hypothesis, $\ell _{{\mathbb {R}}}((J_n)_n)=0$, then, taking $B=[1/\delta , \delta ]$, the assertion follows from previous inequalities and Axioms 2.1.

The next result follows directly from Theorem 6.5, adapting it to the setting of Mellin-type operators.

Theorem 6.9

Under Assumptions 6.6, assume that $\widetilde{{\mathbb {K}}}$ is (U)-singular, Then, for every $f \in {{\mathcal {C}}}_c({\mathbb {R}}^+)$, the sequence $(\widetilde{T_n} f )_n$ is uniformly convergent to f on ${{\mathbb {R}}}^+$, and modularly convergent to f with respect to the modular $\rho ^{\varphi }$, where the constant a in (29) can be chosen independently of f.

Indeed, observe that condition 6.1.1) follows from (6.7.4) and Remark 6.8 b).

In the linear case, a particular type of Mellin-type kernels is the moment kernel, defined by

$$\begin{aligned} {\widetilde{L}}_n(t)=n \, t^n \chi _{(0,1)}(t), \quad t \in {\mathbb {R}}^+. \end{aligned}$$

(48)

For any $n \in {\mathbb {N}}$, $t \in {\mathbb {R}}^+$ and $u \in {\textbf{X}}$, set

$$\begin{aligned} {\widetilde{K}}_n (t,u)={\widetilde{L}}_n (t) \cdot u. \end{aligned}$$

(49)

Observe that for any $\delta > 1$, $n \in {\mathbb {N}}$ and $u \in {\textbf{X}}$ we have

$$\begin{aligned} \int _0^{+\infty } {\widetilde{K}}_n (t,u) \, \frac{dt}{t}= & {} \Bigl ( \int _0^{+\infty } {\widetilde{L}}_n(t) \, \frac{dt}{t} \Bigr ) u= n \Bigl ( \int _0^1 t^{n-1} \, dt \Bigr ) u \, dt=u; \nonumber \\ \int _{{\mathbb {R}}^+ \setminus [1/\delta ,\delta ]} {\widetilde{L}}_n(t) \, \frac{dt}{t}= & {} n \int _0^{1/\delta } t^{n-1} \, dt= \Bigl ( \frac{1}{\delta } \Bigr )^n. \end{aligned}$$

(50)

From (48), (49) and (50) it follows that the conditions in 6.7 on (U)-singularity are fulfilled.

Note that it is possible to check that these conditions are satisfied also by other Mellin-type kernels, like for instance Mellin-Gauss-Weierstrass and Mellin-Poisson-Cauchy-type kernels (see, e.g., [15, 20]).

Furthermore, by proceeding and arguing similarly as in [13, Subsection 3.4], it is possible to see that our theory includes also stochastic integration, for example Itô-type integrals. Indeed, we can treat the standard Brownian motion ${{\textbf{B}}}=(B_t)_{0 \le t \le T}$ defined on a probability space $(\Omega , \Sigma , \nu )$ as a function defined on [0, T] and taking values in ${{\textbf{X}}}_2=L^2=L^2(\Omega , \Sigma , \nu )$ endowed with order convergence, or order filter convergence with respect to a fixed free filter ${{\mathcal {F}}}$ on ${{\mathbb {N}}}$. Moreover, we can consider a stochastic process with (uniformly) continuous trajectories $f:G \rightarrow {{\textbf{X}}}$, where $G=[0,T]$ is endowed with the usual distance and ${{\textbf{X}}}=L^0=L^0(\Omega , \Sigma , \nu )$ is equipped with order convergence, and to define the integral of f with respect to ${\textbf{B}}$ as an element of ${{\textbf{X}}}=L^0$, according to 3.8 and 3.10 (see, e.g., [13] and the related references therein). Note that, in this context, $({{\textbf{X}}}, {{\textbf{X}}}_2, {{\textbf{X}}})$ is a product triple, and Axioms 2.1, 2.2 are satisfied (see Examples 2.4).

7 Proofs of the results in Section 3

Proof of Proposition 3.16

It is readily seen that $h \cdot q$ is bounded of G. Now we prove its integrability. Set ${u= \bigvee \nolimits _{g \in G} \vert h(g) \vert }$. Note that $u \in {{\textbf{X}}}_1^{\prime }$, since h is bounded on G. Moreover, as h and q are integrable on G, there are two defining sequences $(h_n)_n$ and $(q_n)_n$ of simple functions for h and q, respectively. Pick arbitrarily $A \in {{\mathcal {A}}}$ with $\mu (A) \in {{\textbf{X}}}_2$. There are two sequences $(A_n)_n$ and $(D_n)_n$ in ${{\mathcal {A}}}$ with $\ell _2((\mu (A_n))_n)=$ $\ell _2((\mu (D_n))_n)=0$ and

$$\begin{aligned} \ell _1^{\prime }\Bigl (\Bigl ( \bigvee _{g \in A \setminus A_n} \, \vert h_n(g)-h(g) \vert \Bigr )_n\Bigr ) = \ell _1^{\prime \prime }\Bigl (\Bigl (\bigvee _{g \in A \setminus D_n} \, \vert q_n(g)-q(g) \vert \Bigr )_n\Bigr )=0.\qquad \end{aligned}$$

(51)

Put $E_n=A_n \cup D_n$, $n \in {\mathbb {N}}$. As in (5), we have $\ell _2( (\mu (E_n))_n)=0$. Since q is bounded on G, without loss of generality we can assume that $v= \bigvee \nolimits _{g \in G, n \in {\mathbb {N}}} \vert q_n(g) \vert \in {{\textbf{X}}}_1^{\prime \prime }$. Moreover, for all $g \in G$ and $n \in {\mathbb {N}}$ we get

$$\begin{aligned} 0 \le _{X_1}&\vert h(g) q(g) - h_n(g) q_n(g) \vert \nonumber \\ \le _{X_1}&\vert h(g) \vert \cdot \vert q_n(g) - q(g) \vert + \vert h_n(g) -h(g) \vert \cdot \vert q_n(g)\vert \nonumber \\ \le _{X_1}&u \, \vert q_n(g) - q(g) \vert + \vert h_n(g) -h(g)\vert \, v . \end{aligned}$$

(52)

From (51), (52), Axioms 2.1 and conditions 3.1.7), 3.1.8) we deduce

$$\begin{aligned} {\ell _1\Bigl (\Bigl ( \bigvee _{g \in A \setminus E_n} \, \vert h_n(g) \, q_n(g)-h(g) \, q(g)\vert \Bigr )_n\Bigr )=0}. \end{aligned}$$

Thus, the sequence $(h_n \cdot q_n)_n$ converges in measure to $h \cdot q$.

Now we claim the $\mu$-equiabsolute continuity of the integrals of the $h_n \cdot \, q_n$’s. For each $n \in {\mathbb {N}}$ and $g \in G$, we get $0 \le _{X_1} \vert h_n(g) \, q_n(g) \vert \le _{X_1} \vert h_n(g)\vert \, v,$ and hence

$$\begin{aligned} 0 \le _{X} \int _A \vert h_n(g) \, q_n(g) \vert \, d\mu (g) \le _{X} \Bigl ( \int _A \vert h_n(g)\vert \, d\mu (g) \Bigr ) \cdot v, \,\,\text { for all } n \in {\mathbb {N}} \text { and } A \in {{\mathcal {A}}}. \end{aligned}$$

The claim follows from this inequality, the $\mu$-equiabsolute continuity of the integrals of the $h_n$’s, Axioms 2.1 and condition 3.1.7).

Now we prove the existence of a map $l:{{\mathcal {A}}} \rightarrow {\textbf{X}}$, satisfying formula (11) in Definition 3.8. Without loss of generality, we can suppose that h, q, $h_n$ and $q_n$ are positive for every n, and that the sequences $(h_n)_n$, $(q_n )_n$ are increasing. Arguing analogously as in (52), for each n, $m \in {\mathbb {N}}$ with $m \ge n$ we get

$$\begin{aligned} 0 \le _{X}&\Bigl ( \bigvee _{A \in {{\mathcal {A}}}} \Bigl ( \int _A h_m(g) \, q_m(g) \, d\mu (g) -\int _A h_n(g) \, q_n(g) \, d\mu (g) \Bigr ) \Bigr )\\ \le _{X}&u \Bigl (\int _G (q_m(g) - q_n(g)) \, d\mu (g) \Bigr ) + \Bigl ( \int _G(h_m(g) -h_n(g)) \, d\mu (g) \Bigr ) v \\ \le _{X}&u \Bigl (\int _G (q(g) - q_n(g)) \, d\mu (g) \Bigr ) + \Bigl ( \int _G (h(g) -h_n(g)) \, d\mu (g) \Bigr ) v. \end{aligned}$$

Since q and h are integrable on G and their integrals are absolutely continuous, then the sequences $(q_n - q)_n$ and $(h_n - h)_n$ are defining sequences for the identically zero function, and so both of them converge in $L^1$ to 0, thanks to Theorem 3.13. From this last inequality, Axioms 2.1 and conditions 3.1.7), 3.1.8) it follows that

$$\begin{aligned} \ell \Bigl (\Bigl ( \bigvee _{A \in {{\mathcal {A}}}} \Bigl (\Bigl ( \bigvee _{m=1}^{\infty } \int _A h_m(g) \, q_m(g) \, d\mu (g)\Bigr ) -\int _A h_n(g) \, q_n(g) \, d\mu (g) \Bigr ) \Bigr )_n \Bigr )=0. \end{aligned}$$

Setting ${l(A) := \bigvee \nolimits _{m=1}^{\infty } \int _A h_m(g) \, q_m(g) \, d\mu (g)}$ for every $A \in {{\mathcal {A}}}$, from the previous equality we deduce that the map l satisfies (11). $\square$

Proof of Corollary 3.17

For each $g\in G$, set $h^*(g) = h(g) \chi _{{\overline{B}}}$. It is not difficult to check that $h^*$ is bounded and integrable on G. By Proposition 3.16, the function $h^* \cdot q$ is bounded and integrable on G too. By hypothesis, we have $h(g)\, q(g) = h^*(g) \, q(g) \chi _{{\overline{B}}}$. Thus, the boundedness and integrability of $h \cdot q$ follow from those of $h^* \cdot q$. $\square$

Proof of Proposition 3.18

Let $u \in {{\textbf{X}}}_1^+$ be related to the uniform continuity of f on G, and choose arbitrarily $\varepsilon \in {\mathbb {R}}^+$. By the uniform continuity of $\psi$ on ${\textbf{X}}_1$, there are $\sigma (\varepsilon )\in {\mathbb {R}}^+$ and $w \in {{\textbf{X}}}_1^+$ with $\vert \psi (x_1)-\psi (x_2)\vert \le _{X_1} \, \varepsilon \, w$ whenever $\vert x_1-x_2\vert \le _{X_1} \sigma (\varepsilon ) \, u$. By virtue of the uniform continuity of f on G, in correspondence with $\sigma (\varepsilon )$ there is $\delta (\varepsilon ) \in {\mathbb {R}}^+$ with $\vert f(g_1)-f(g_2) \vert \le _{X_1} \sigma (\varepsilon )\, u$ whenever $d(g_1,g_2) \le \delta (\varepsilon )$. So, for $x_i = f(g_i), \,i=1,2$, it follows that $\vert \psi (f(g_1))-\psi (f(g_2))\vert \le _{X_1} \varepsilon \, w$ if $d(g_1,g_2) \le \delta (\varepsilon ).$ $\square$

Proof of Proposition 3.19

We begin with proving the boundedness of f on G. Let $C \subset G$ be a compact set such that $f(g)=0$ whenever $g\in G \setminus C$. Observe that it is enough to prove that f is bounded on C. As f is uniformly continuous on G, there exists $u \in {{\textbf{X}}}^+_1 \setminus \{0\}$ such that for $\varepsilon =1$ there is $\delta _1 \in {\mathbb {R}}^+$ with

$$\begin{aligned} \bigvee _{g_1, g_2 \in G, \,\, d(g_1,g_2)\le \delta _1} \vert f(g_1) - f(g_2) \vert \le _{X_1} u. \end{aligned}$$

(53)

Since C is compact, then C is totally bounded too, and thus there is a finite number $t_1$, $t_2, \ldots , t_q$ of elements of G such that ${C \subset \bigcup \nolimits _{j=1}^q \{g \in G: d(g,t_j) \!\le \! \delta _1\} }.$ Let ${{\widehat{u}}= \bigvee \nolimits _{j=1}^q \, \vert f(t_j) \vert }$, and pick arbitrarily $g \in G$. There is ${\overline{j}} \in \{1,2,\ldots , q \}$ with $d(g,t_{{\overline{j}}}) \le \delta _1$. We have

$$\begin{aligned} \vert f(g) \vert \le _{X_1} \vert f(g) - f(t_{{\overline{j}}}) \vert + \vert f(t_{{\overline{j}}}) \vert \le _{X_1} u+ {\widehat{u}}. \end{aligned}$$

(54)

Thus, f is bounded on C. Now we prove the integrability of f on G. Without loss of generality, we can assume $f \ge 0$. Let $(\epsilon _n)_n$ be any (o)-sequence in ${\mathbb {R}}^+$. By the uniform continuity of f on G, there exists an (o)-sequence $(\delta _n)_n$ in ${\mathbb {R}}^+$ with

$$\begin{aligned} 0 \le _{X_1}\vert f(g_1) - f(g_2)\vert \le _{X_1} \varepsilon _n \, u \quad \text {whenever } d(g_1,g_2) \le \delta _n \end{aligned}$$

(55)

for every $n \in {\mathbb {N}}$. From 3.1.7) we deduce

$$\begin{aligned} {\ell _1 ((\varepsilon _n \, u )_n)=0}. \end{aligned}$$

(56)

Now we construct a defining sequence by induction. At the generic n-th step, $n \in {\mathbb {N}}$, we partition the set C into a finite number $l_n$ of pairwise disjoint sets of ${{\mathcal {A}}}$ of diameter less than or equal to $\delta _n$ (this is always possible, since G is a metric space and C is totally bounded) with $\mu (A) \in X_2$ for every $A \in {\mathcal {E}}_n$, thanks to the regularity of $\mu$.

Denote this family by ${{\mathcal {E}}}_n=\{E^{(j)}_n: j= 1,2, \ldots l_n \}$, and take ${{\mathcal {E}}}_n$ in such a way that ${{\mathcal {E}}}_n$ is a refinement of the family ${{\mathcal {E}}}_{n-1}$ constructed at the n-1-th step: that is, for each $E^{(j)}_{n} \in {{\mathcal {E}}}_n$, $j=1,2,\ldots , l_n$ there exists $E^{(i)}_{n-1} \in {{\mathcal {E}}}_{n-1}$ with $E^{(j)}_n \subset E^{(i)}_{n-1}$. For every $n \in {\mathbb {N}}$, set

$$\begin{aligned} f_n(g)= \bigwedge \{f(g):g \in E^{(j)}_n\} \quad \text { if } g \in E^{(j)}_n \quad (j=1,2, \ldots , l_n). \end{aligned}$$

(57)

Moreover, let $f_n(g)=0$ for each $n \in {\mathbb {N}}$ and $g \in G \setminus C$. It is not difficult to see that, for every $g \in G$ and $n \in {\mathbb {N}}$,

$$\begin{aligned} f_n \in {\mathscr {S}}\, \quad \text { and }\quad 0 \le _{X_1} f_n(g) \le _{X_1} f_{n+1}(g) \le _{X_1} f(g) \le _{X_1} u+ {\widehat{u}}. \end{aligned}$$

(58)

From (55) and (57) we obtain

$$\begin{aligned} 0 \le _{X_1} \bigvee _{g \in G} (f(g)-f_n(g)) \le _{X_1} \bigvee _{g_1, g_2 \in G, d(g_1,g_2)\le \delta _n} \vert f(g_1) - f(g_2) \vert \le _{X_1} \varepsilon _n \, u, \end{aligned}$$

and then by Axioms 2.1 we get

$$\begin{aligned} {\ell _1\Bigl (\Bigl ( \bigvee _{g \in G} \, \vert f_n(g)-f(g) \vert \Bigr )_n\Bigr )=0}. \end{aligned}$$

Thus, the sequence $(f_n)_n$ converges uniformly to f on G, and a fortiori it converges in measure to f on G.

Now we prove the $\mu$-equiabsolute continuity of the integrals of the $f_n$’s. We begin with 3.5.1). Choose arbitrarily a sequence $(A_n)_n$ from ${{\mathcal {A}}}$, with $\ell _2((\mu (A_n))_n)=0$. By (58) we get

$$\begin{aligned} 0 \le _{X} \int _{A_n} f_n(g) \, d\mu (g) \le _{X} \mu (A_n) ( u+{\widehat{u}}) \quad \text { for any } n \in {\mathbb {N}}. \end{aligned}$$

From the previous inequality, Axioms 2.1.b) and 2.1.c) and condition 3.1.7) it follows that

$$\begin{aligned} { \ell \left( \left( \int _{A_n}\,f_n(g) \, d\mu (g)\right) _n \right) =0}. \end{aligned}$$

Now we turn to 3.5.2). From (58), the positivity of $\mu$ and the monotonicity of the integral we deduce

$$\begin{aligned} 0 \le _{X} \int _{G \setminus C} f_n(g) \, d\mu (g) \le _{X} \int _{G \setminus C} f(g) \, d\mu (g)=0 \quad \text { for all } n \in {\mathbb {N}}. \end{aligned}$$

(59)

Condition 3.5.2) follows from (59) and Axioms 2.1, taking $B_m=C$ for each $m \in {\mathbb {N}}$.

Now we prove the existence of a map l, satisfying (11) in the definition of integrability. For each $n \ge m$ and $A \in {{\mathcal {A}}}$, we get

$$\begin{aligned} 0 \le _{X}&\int _A f_n(g) \, d\mu (g) -\int _A f_m(g) \, d\mu (g) = \int _A (f_n(g) -f_m(g)) \, d\mu (g) \\ \le _{X}&\Bigl ( \bigvee _{g \in C} (f_n(g) - f_m(g)) \Bigr ) \, \mu (C) \le _{X} \Bigl ( \bigvee _{g \in C} (f(g) - f_m(g)) \Bigr ) \, \mu (C) \le _{X} \, \varepsilon _n \,\mu (C) \, u. \end{aligned}$$

Taking in the previous formula the supremum as n varies in ${\mathbb {N}}$, and taking into account the monotonicity of the sequence $(f_n)_n$ and of the integral, we obtain

$$\begin{aligned} 0 \le _{X}&\bigvee _{A \in {{\mathcal {A}}}} \Bigl (\Bigl ( \bigvee _{n=1}^{\infty } \int _A f_n(g) \, d\mu (g) \Bigr ) -\int _A f_m(g) \, d\mu (g) \Bigr ) \le _{X}\varepsilon _n \, \mu (C) \, u . \end{aligned}$$

Put $l(A)= \bigvee _{n=1}^{\infty } \int _A f_n(g) \, d\mu (g)$, $A \in {{\mathcal {A}}}$. From (56), the previous inequality and Axioms 2.1 we obtain

$$\begin{aligned} \ell \Bigl ( \Bigl ( \bigvee _{A \in {\mathcal {A}}} \Bigl ( l(A) - \int _A \, f_m(g)\,d\mu (g) \Bigr ) \Bigr )_m \Bigr )=0. \end{aligned}$$

$\square$

Proof of Proposition 3.23

By the definition of convexity, if we consider $v= \xi x$, $s=0$, there is $\beta _{\xi x} \in {\textbf{X}}_1$ with $0 = \varphi (0) \ge _{X_1} \varphi (\xi x) - \beta _{\xi x} \, \xi x,$ while for $v= \xi x$ and $s=x$, we get $\varphi (x) \ge _{X_1} \varphi (\xi x) + \beta _{\xi x} (1-\xi ) x$. Multiplying by $1-\xi$ the first one and by $\xi$ the second one, we obtain

$$\begin{aligned} 0 \ge _{X_1} (1-\xi ) \varphi (\xi x) - \beta _{\xi x} \, \xi (1-\xi ) x , \quad \xi \, \varphi (x) \ge _{X_1} \xi \, \varphi (\xi x) + \beta _{\xi x} \, \xi (1-\xi ) x . \end{aligned}$$

Summing up, we get  $\xi \, \varphi (x) \ge _{X_1} \varphi (\xi x)$. $\square$

Proof of Proposition 3.24

Fix arbitrarily $u \in {\textbf{X}}_1$, pick $v \in [-u,u]$, let $\beta _v$ be as in Definition 3.20 and set ${\beta ^*_u =\bigvee \nolimits _{v \in [-u,u]} \vert \beta _v \vert }$. Note that $\beta ^*_u \in {\textbf{X}}_1$. Choose arbitrarily $x_1$, $x_2 \in {\textbf{X}}_1$. By the convexity of $\varphi$, if we set $v=x_1$ and $s=x_2$ (resp., $v=x_2$ and $s=x_1$), we get

$$\begin{aligned} \varphi (x_2) \ge _{X_1} \varphi (x_1) + \beta _{x_1} (x_2-x_1), \quad \text {(resp., } \varphi (x_1) \ge _{X_1} \varphi (x_2) + \beta _{x_2} (x_1-x_2) \, ). \end{aligned}$$

So, $\vert \varphi (x_2) - \varphi (x_1)\vert \le _{X_1} \beta ^*_u \, \vert x_1-x_2\vert$. $\square$

Proof of Proposition 3.25

Thanks to the uniform continuity of f on G, there is an element $u \in {\textbf{X}}_1^+ \setminus \{0\}$ with the property that for every $\varepsilon \in {\mathbb {R}}^+$ there exists $\delta \in {\mathbb {R}}^+$ with

$$\begin{aligned} \vert f(g_1) - f(g_2)\vert \le _{X_1} \varepsilon \, u \quad \text { whenever } g_1, g_2 \in G, \, \, d(g_1,g_2)\le \delta . \end{aligned}$$

Moreover, since $f(g)=0$ for all $g \in G \setminus C$ and C is compact, by virtue of Proposition 3.19 we get that f is bounded. Let ${u^*=\vee _{g \in G} \vert f(g) \vert }$. By Proposition 3.24, there is an element $\beta ^*_{u^*} \in {\textbf{X}}_1^+ \setminus \{0\}$ with $\vert \varphi (x_1)-\varphi (x_2) \vert \le _{X_1} \beta ^*_{u^*} \, \vert x_1 - x_2\vert$ for all $x_1, x_2 \in [-u^*,u^*]$. Then, it follows that

$$\begin{aligned} \vert \varphi (f(g_1)) - \varphi (f(g_2))\vert \le _{X_1} \beta ^*_{u^*} \, \vert (f(g_1)-f(g_2))\vert \le _{X_1} \varepsilon \, \beta ^*_{u^*} \, \, u \end{aligned}$$

whenever $g_1$, $g_2 \in G$, $d(g_1,g_2) \le \delta$. Therefore, $\varphi \circ f$ is uniformly continuous on G. Since $f(g)=0$ for every $g \in G \setminus C$ and $\varphi (0)=0$, we get that $\varphi \circ f$ vanishes outside C. Thus, $\varphi \circ f$ is integrable on C, thanks to Proposition 3.19. $\square$

Proof of Theorem 3.26

By Proposition 3.15 (resp., 3.19), h (resp., f) is integrable on G. Moreover, taking into account that $\varphi (0)=0$ and since $({{\textbf{X}}},{\mathbb {R}}, {{\textbf{X}}})$ is a product triple, thanks to Proposition 3.25, we have that $\varphi \circ f$ is uniformly continuous and integrable on G and vanishes outside C. If we apply twice Corollary 3.17 to the pairs (h, f) and $(h,\varphi \circ f)$ it follows that the functions $h \cdot f$ and $h \cdot (\varphi \circ f)$ are bounded and integrable on G. Let $\tau = \int _G h(g) \, f(g) \, d\mu (g) \in {\textbf{X}}.$ Since $\varphi$ is convex, in correspondence with $\tau$ there is an element $\beta _{\tau } \in {\textbf{X}}$, satisfying the definition of convexity with $s=f(g)$, $g \in G$,

$$\begin{aligned} \varphi (f(g)) \ge _{X} \varphi (\tau ) + \beta _{\tau } \, (f(g)-\tau ). \end{aligned}$$

By multiplying both members of by h(g) we get, for all $g \in G$,

$$\begin{aligned} h(g) \, \varphi (f(g)) \ge _{X} h(g) \, \varphi (\tau ) + h(g) \, \beta _{\tau } \, f(g) -\, h(g) \, \beta _{\tau } \, \tau . \end{aligned}$$

Observe that all members of the previous inequality are integrable on G. Since the integral is monotone, (see Remark 3.11.f) and ${\int _G h(g) \, d\mu (g)=1}$, we get

$$\begin{aligned}{} & {} \int _G h(g) \, \varphi (f(g)) \, d\mu (g) \\{} & {} \quad \ge _{X} \Bigl ( \int _G h(g) \, d\mu (g) \Bigr ) \, \cdot \, \Bigl ( \varphi \Bigl (\int _G h(g) \, f(g) \, d\mu (g) \Bigr ) \Bigr )+ \beta _{\tau } \int _G h(g) \, f(g) d\mu (g) \\{} & {} \qquad - \beta _{\tau } \Bigl ( \int _G h(g) \, d\mu (g) \Bigr ) \, \cdot \, \Bigl (\int _G h(g) \, f(g) \, d\mu (g) \Bigr ) = \varphi \Bigl (\int _G h(g) \, f(g) \, d\mu (g) \Bigr ). \end{aligned}$$

Therefore, the assertion follows. $\square$

Proof of Corollary 3.27

Let ${m_h=\int _G h(g) \, d\mu (g)}$, and for every $g \in G$, set $h^*(g)=\dfrac{h(g)}{m_h}$. By hypothesis, ${\int _G h^*(g) \, d\mu (g)=1}$. From Theorem 3.26 applied to $h^*$, we obtain

$$\begin{aligned} \dfrac{1}{m_h}\int _G h(g) \, \varphi (f(g)) \, d\mu (g){} & {} = \int _G h^*(g) \, \varphi (f(g)) \, d\mu (g) \\{} & {} \ge _{X} \varphi \Bigl (\int _G h^*(g) \, f(g) \, d\mu (g) \Bigr )\\{} & {} \ge _{X} \varphi \Bigl (\dfrac{1}{m_h} \int _G h(g) \, f(g) \, d\mu (g) \Bigr ). \end{aligned}$$

From Proposition 3.23 applied to $\varphi$, with ${x= \dfrac{1}{m_h}\int _Gh(g) \, f(g) \, d\mu (g)}$ and $\xi =m_h$, we get

$$\begin{aligned} \varphi \Bigl (\dfrac{1}{m_h} \int _G h(g) \, f(g) \, d\mu (g) \Bigr ) \ge _{X} \dfrac{1}{m_h} \, \varphi \Bigl (\int _G h(g) \, f(g) \, d\mu (g) \Bigr ). \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} \int _G h(g) \varphi (f(g)) \, d\mu (g) \ge _{X} \varphi \Bigl (\int _G h(g) f(g) \, d\mu (g) \Bigr ). \end{aligned}$$

$\square$