# Some monotonicity properties in F-normed Musielak–Orlicz spaces

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## Abstract

Strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity and their orthogonal counterparts are considered in the case of Musielak–Orlicz function spaces \(L^\Phi (\mu )\) endowed with the Mazur–Orlicz F-norm as well as in the case of their subspaces \(E^\Phi (\mu )\) with the F-norm induced from \(L^\Phi (\mu )\). The presented results generalize some of the results from Cui et al. (Aequ Math 93:311–343, 2019) and Hudzik et al. (J Nonlinear Convex Anal 17(10):1985–2011, 2016), obtained only for Orlicz spaces as well as their subspaces of order continuous elements equipped with the Mazur–Orlicz F-norm.

## Keywords

Musielak–Orlicz spaces Mazur–Orlicz F-norm F-normed Köthe spaces Strict monotonicity Orthogonal strict monotonicity Lower local uniform monotonicity Orthogonal lower local uniform monotonicity Upper local uniform monotonicity Orthogonal upper local uniform monotonicity## Mathematics Subject Classification

Primary 46E30 Secondary 46A80## 1 Introduction and preliminaries

Let us denote by \(\mathbb {N}\), \(\mathbb {R}\) and \(\mathbb {R}_+\) the sets of natural, real and nonnegative real numbers, respectively.

*X*, the functional \(X\ni x\rightarrow \Vert x\Vert \in \mathbb {R}_{+}:=[0,\infty )\), is called an

*F*-norm if the following conditions are satisfied:

- (i)
\(\Vert x\Vert =0\) if and only if \(x=0\),

- (ii)
\(\Vert -x\Vert =\Vert x\Vert \) for all \(x\in X\),

- (iii)
\(\Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert \) for all \(x,y\in X\),

- (iv)
\(\Vert \lambda _{n}x_{n}-\lambda x\Vert \rightarrow 0\) whenever \(\Vert x_{n}-x\Vert \rightarrow 0\) and \(\lambda _{n}\rightarrow \lambda \) for any \(x\in X\), \((x_{n})_{n=1}^{\infty }\) in

*X*, \(\lambda \in \mathbb {R}\) and \((\lambda _{n})_{n=1}^{\infty }\) in \(\mathbb {R}\).

*F*-normed space \(X=(X,\Vert \cdot \Vert )\) is said to be an

*F*-space if it is complete with respect to the

*F*-norm topology. If a lattice

*X*is endowed with a monotone

*F*-norm \(\Vert .\Vert \) (i.e., the condition \(|y| \le |x|\) implies \(\Vert y\Vert \le \Vert x\Vert \) for \(x, y \in X\)), under which

*X*is topologically complete, then \(X = (X, \Vert .\Vert )\) is said to be an

*F*-lattice.

The notation *S*(*X*) (resp. *B*(*X*)) stands for the unit sphere (resp. the closed unit ball) of a real *F*-normed space \((X,\left\| \cdot \right\| )\).

In the whole paper, we will assume that \((\Omega ,\Sigma ,\mu )\) is a complete \(\sigma \)-finite and non-atomic measure space. Let us denote by \(L^{0}(\mu )=L^{0}(\Omega ,\Sigma ,\mu )\) the space of all (equivalence classes of) real-valued and \(\Sigma -\)measurable functions on \(\Omega \).

### Definition 1.1

*F*-space \((E,\Vert \cdot \Vert _{E})\) is called an

*F*-normed Köthe space if it is a linear subspace of \(L^{0}\) satisfying the following condition:

Denote by \(E_{+}\) the positive cone of *E*, that is, \(E_{+}=\{x\in E:x\ge 0\}\).

Let us recall now the definitions of the necessary monotonicity properties.

### Definition 1.2

An *F*-normed Köthe space \((E,\Vert .\Vert _{E})\) is said to be strictly monotone \((E\in (SM)\) for short) if for any \(x,y\in E\) such that \(0\le y\le x\), we have \(\Vert y\Vert _{E}<\Vert x\Vert _{E}\) whenever \(y\ne x\) ( or equivalently \(\Vert x-y\Vert _{E}<\Vert x\Vert _{E}\) whenever \(0\le y\le x\) and \(y\ne 0) \) (see [1]).

### Definition 1.3

An *F*-normed Köthe space *E* is said to be orthogonally strictly monotone \((E\in (OSM)\) for short) if for any \(x\in E_{+}\backslash \{0\}\) and any \(A\in \Sigma \cap {\text {supp}}x\) with \(\mu (A)>0\), we have \(\Vert x\chi _{\Omega \backslash A}\Vert _{E}<\Vert x\Vert _{E}\).

### Definition 1.4

An *F*-normed Köthe space \((E,\Vert .\Vert _{E})\) is said to be lower locally uniformly monotone \((E\in (LLUM)\) for short) if for any \(x\in E\) and \((x_{n})_{n=1}^{\infty }\) in *E* such that \(0\le x_{n}\le x\) for all \(n\in \mathbb {N}\) and \(\Vert x_{n}\Vert _{E}\rightarrow \Vert x\Vert _{E}\) as \(n\rightarrow \infty \), the condition \(\Vert x-x_{n}\Vert _{E}\rightarrow 0\) as \(n\rightarrow \infty \) holds.

### Definition 1.5

*F*-normed Köthe space \((E,\Vert .\Vert _{E})\) is said to be orthogonally lower locally uniformly monotone \((E\in (OLLUM)\) for short) if for any \(x\in E_{+}\backslash \{0\}\) and any \((A_{n})_{n=1}^{\infty }\) in \(\Sigma \) the following implication is satisfied:

### Definition 1.6

An *F*-normed Köthe space \((E,\Vert .\Vert _{E})\) is said to be upper locally uniformly monotone \((E\in (ULUM)\) for short) if for any \(x\in E_{+} \) and \((x_{n})_{n=1}^{\infty }\) in \(E_{+}\) such that \(x\le x_{n}\) for all \(n\in \mathbb {N}\) and \(\Vert x_{n}\Vert _{E}\rightarrow \Vert x\Vert _{E}\) as \(n\rightarrow \infty \), the condition \(\Vert x_{n}-x\Vert _{E}\rightarrow 0\) as \(n\rightarrow \infty \) holds.

### Definition 1.7

An *F*-normed Köthe space \((E,\Vert .\Vert _{E})\) is said to be orthogonally upper locally uniformly monotone \((E\in (OULUM)\) for short) if for any \(x\in E_{+}\backslash \{0\}\) such that \(\mu ({\text {supp}}E\backslash {\text {supp}}x)>0\) and any \(x_{n}\in E_{+}\) with \({\text {supp}}x_{n}\subset {\text {supp}}E\backslash {\text {supp}}x\), if \(\Vert x+x_{n}\Vert _{E}\rightarrow \Vert x\Vert _{E}\) then \(\Vert x_{n}\Vert _{E}\rightarrow 0\) as \(n\rightarrow \infty \).

*u*was important (it is known that the convexity of such function implies its continuity on the interval, where its values are finite except for the right hand side point of the interval).

- 1.
\(\Phi \in \Delta _2\),

- 2.
there exist a set \(\Omega _1\in \Sigma \) with \(\mu (\Omega _1)=0\), a constant \(K\ge 1\) and a measurable function \(\delta \ge 0\) with \(\Phi (t,\delta (t))\in L^1(\Omega ,\Sigma ,\mu )\) such that \(\Phi (t,2u)\le K\Phi (t,u)\) for all \(t\in \Omega \backslash \Omega _1\) and any \(u\ge \delta (t)\).

*F*-norm (see [9, 10, 11]):

*F*-lattices when considered with the natural partial order.

In the whole paper we will write shortly \(L^\Phi (\mu )\) and \(E^\Phi (\mu )\) instead of \((L^\Phi (\mu ),\Vert .\Vert _\Phi )\) and \((E^\Phi (\mu ),\Vert .\Vert _\Phi )\), respectively, assuming that these function spaces are generated by a monotone Musielak–Orlicz function \(\Phi \).

Let us present now some useful results. The proof of Lemma 1.8 will be omitted because it is the same as the proof of implications \((i)\Rightarrow (ii)\Rightarrow (iii)\) of Theorem 5.5 from [2], where, in fact, the continuity of the function \(\Phi (t,.)\) for \(\mu -\)a.e. \(t\in \Omega \) was important.

### Lemma 1.8

### Theorem 1.9

(cf. [10], Theorem 8.13 (a)). If \(\mu \) is \(\sigma -\)finite and atomless and \(\Phi \) is a monotone Musielak–Orlicz function such that \(a_\Phi (t)=0\) and \(b_\Phi (t)=\infty \) for \(\mu -\)a.e. \(t\in \Omega \), then \(E^\Phi (\mu )=L^\Phi (\mu )\) if and only if \(\Phi \in \Delta _2\).

### Remark 1.10

Theorem 1.9 also holds true without the assumption that a generating monotone Musielak–Orlicz function \(\Phi \) is such that \(a_\Phi (t)=0\) for \(\mu -\)a.e. \(t\in \Omega \).

### Lemma 1.11

### Remark 1.12

The collection \(\{A_k\}_{k=1}^\infty \) of disjoint sets from Lemma 1.11 can be found in such a way that \(\mu (\Omega \backslash \underset{k=1}{\overset{\infty }{\bigcup }}A_k)>0\) (see the proof of Lemma 1.7.3 in [8]).

The results presented in the paper generalize some of the results from papers [3, 5], obtained there for Orlicz spaces and their subspaces of order continuous elements endowed with the Mazur–Orlicz F-norm. However, the proofs of the results presented in this paper require deeper techniques then the ones which concern only Orlicz spaces. The application of some methods or ideas of proofs from papers [2, 3, 4, 5, 6, 8] was useful.

It seems to be interesting to note that the case of Orlicz spaces equipped with the Mazur–Orlicz F-norm already shows that, for instance, the characterization of strict monotonicity and orthogonal strict monotonicity differ (see [3, 5]), which was not possible in the norm case.

## 2 Results

Let us start with presenting a generalization of Lemma 6.1 from [5].

### Lemma 2.1

*F*-norm \(\Vert .\Vert _\Phi \), we have:

- (1)
\(I_\Phi \left( \frac{x}{\Vert x\Vert _\Phi }\right) \le \Vert x\Vert _\Phi \).

- (2)
If \(I_\Phi \left( \lambda \frac{x}{\Vert x\Vert _\Phi }\right) <\infty \) for some \(\lambda >1\), then \(I_\Phi \left( \frac{x}{\Vert x\Vert _\Phi }\right) =\Vert x\Vert _\Phi \).

- (3)
If \(I_\Phi \left( \frac{x}{\lambda }\right) =\lambda \) for \(\lambda >0\), then \(\Vert x\Vert _\Phi =\lambda \).

### Proof

Although the proof of statements (1) and (2) goes in the same way as the proof of Lemma 6.1 from [5] and the proof of statement (3) is the same as that in [3, Lemma 2.16(iii)], we will present them just for the convenience of the reader.

*The proof of statement*(1). Note that function \(f(\lambda )=I_\Phi \left( \frac{x}{\lambda }\right) \) is non-increasing on \((0,\infty )\) for any \(x\in L^\Phi (\mu )\). From the definition of the Mazur–Orlicz

*F*-norm \(\Vert .\Vert _\Phi \), we obtain that if \(x\in L^\Phi (\mu )\), then \(I_\Phi \left( \frac{x}{\Vert x\Vert _\Phi +\varepsilon }\right) \le \Vert x\Vert _\Phi +\varepsilon \) for any \(\varepsilon >0\). Let us take a sequence \((\varepsilon _n)_{n=1}^\infty \) of positive numbers such that \(\varepsilon _n\searrow 0\) as \(n\rightarrow \infty \). Then

*The proof of statement*(2). Assume that \(I_\Phi \left( \lambda \frac{x}{\Vert x\Vert _\Phi }\right) <\infty \) for some \(\lambda >1\) and for any \(x\in L^\Phi (\mu )\). For any sequence \((\varepsilon _n)_{n=1}^\infty \) in \((0,\Vert x\Vert _\Phi )\) such that \(\varepsilon _n\searrow 0\) as \(n\rightarrow \infty \) and \(\varepsilon _1\in (0,\Vert x\Vert _\Phi )\) satisfying inequality the \(\frac{1}{\Vert x\Vert _\Phi -\varepsilon _1}\le \frac{\lambda }{\Vert x\Vert _\Phi }\), the condition

*The proof of statement*(3). By the definition of the F-norm \(\Vert .\Vert _\Phi \), we have that \(\Vert x\Vert _\Phi \le \lambda \). Assuming that \(\Vert x\Vert _\Phi <\lambda \), by statement (1) of this lemma, we get

### Corollary 2.2

- (1)
\(I_\Phi (x)\le \Vert x\Vert _\Phi \) whenever \(x\in B(L^\Phi (\mu ))\).

- (2)
If \(x\in S(L^\Phi (\mu ))\) and \(I_\Phi (\lambda x)<\infty \) for some \(\lambda >1\), then \(I_\Phi (x)=1\).

- (3)
The equality \(I_\Phi \left( \frac{x}{\Vert x\Vert _\Phi }\right) =\Vert x\Vert _\Phi \) holds for every \(x\in L^\Phi (\mu )\backslash \{0\}\) whenever \(\Phi \in \Delta _2\).

### Proof

Statement (1) follows directly from statement (1) of Lemma 2.1 and statements (2) and (3) come from statement (2) of Lemma 2.1 because the assumption that \(\Phi \in \Delta _2\) gives that \(I_\Phi (\lambda x)<\infty \) for any \(\lambda >0\). \(\square \)

### Remark 2.3

Note that the \(\Delta _2\)-condition for \(\Phi \) implies that \(b_\Phi (t)=\infty \) for \(\mu -\)a.e. \(t\in \Omega \), so in statement (3) of Corollary 2.2, the assumption that \(b_\Phi (t)=\infty \) for \(\mu -\)a.e. \(t\in \Omega \) is satisfied automatically.

### Theorem 2.4

If \(a_\Phi (t)>0\) for *t* from a measurable set of positive measure, then the space \(L^\Phi (\mu )\) is not orthogonally strictly monotone.

### Proof

Assume that \(a_\Phi (t)>0\) for \(t\in A\subset \Omega \) with \(\mu (A)>0\). Passing to a subset, if necessary, and denoting it again by *A*, we can assume that \(\mu (A)<\infty \) and \(\mu (\Omega \backslash A)>0\). We will consider two cases separately.

*Case 1.*Let \(b_\Phi (t)=\infty \) for \(\mu -\)a.e. \(t\in \Omega \). Then, there exists a measurable set \(B\subset \Omega \backslash A\) such that \(0<\int _B \Phi (t,a_\Phi (t)+1)d\mu =b<\infty \). Define

*Case 2.* Let \(b_\Phi (t)<\infty \) on a measurable set *A* of positive and finite measure. We will consider two subcases.

*B*of positive measure such that \(B\subset A\). Define the sets

### Remark 2.5

If we assume additionally in Theorem 2.4 that \(b_\Phi (t)=\infty \) for \(\mu -\)a.e. \(t\in \Omega \), then we can present the following simpler proof of this theorem.

If \(a_\Phi (t)>0\) for *t* belonging to a measurable set *B* of positive measure, then we can find (coming to a subset *A* of the set *B*, if necessary) \(a,b\in \mathbb {R}_+\) such that \(0< a<b\le a_\Phi (t)\) and \(\Phi (t,a)=\Phi (t,b)\) for all \(t\in A\subset B\). Indeed, let us define sets \(B_n:=\left\{ t\in B:\frac{1}{n}<a_\Phi (t) \right\} \). Then \(\underset{n=1}{\overset{\infty }{\cup }} B_n=B\), so there exists \(l\in \mathbb {N}\) such that \(\mu (B_l)>0\). Therefore, \(0<\frac{1}{l+1}<\frac{1}{l}<a_\Phi (t)\) for all \(t\in B_l\). Setting \(a=\frac{1}{l+1}\), \(b=\frac{1}{l}\) and \(A=B_l\), we have shown our thesis.

Next, there exists an increasing sequence \((A_n)_{n=1}^\infty \) of measureable sets such that \(\mu (A_n)<\infty \), \(\mu \left( \Omega \backslash \underset{n=1}{\overset{\infty }{\cup }} A_n\right) =0\) and \(\sup \limits _{t\in {A_n}}\Phi (t,u)<\infty \) for all \(u\in \mathbb {R}_+\) and \(n\in \mathbb {N}\) (see [6]), which gives that \(\chi _{A_n}\in E^\Phi (\mu )\) for all \(n\in \mathbb {N}\). There exists \(m\in \mathbb {N}\) such that \(\mu (A\cap A_m)>0\), whence \(\Phi (t,\lambda )\chi _{A\cap A_{m}}\in L^1(\mu )\) for any \(\lambda >0\).

*C*and

*D*of positive measures such that \(C\cup D=F\). Let us define

### Theorem 2.6

If \(b_\Phi (t)<\infty \) for *t* from a measurable set of positive measure, then the space \(L^\Phi (\mu )\) is not orthogonally strictly monotone.

### Proof

*t*from a set of positive measure, \(\mu (A_0)>0\). Let \(A\subset A_0\) be a measurable subset of the set \(A_0\) such that \(\mu (A)\in (0,\infty )\) and let \(u(t)=b_\Phi (t)\chi _A(t)\). Let \(\{A_k\}_{k=1}^\infty \) be a sequence of pairwise disjoint and measurable subsets of

*A*of positive measure and let \(\{\alpha _k\}_{k=1}^\infty \) be a sequence of real numbers such that \(0<\alpha _k<1\) for any \(k\in \mathbb {N}\) and \(\alpha _k \uparrow 1\) as \(k\rightarrow \infty \). Define

*n*(

*k*) such that \(\mu (A_k^{n(k)})>0\). Further, for every \(k\in \mathbb {N}\) there exists a set \(B_k\in \Sigma \), \(B_k\subset A_k^{n(k)}\) such that \(\mu (B_k)>0\) and \(\int _{B_k} \Phi (t,\alpha _k u(t))d\mu \le 2^{-k}\). Set \(x_k(t)=\alpha _k u(t)\chi _{B_k}(t)\) and define

### Remark 2.7

Another, simpler proof of Theorem 2.6 is possible and it looks as follow.

### Theorem 2.8

The space \(L^\Phi (\mu )\) is strictly monotone if and only if \(\Phi (t,.)\) is strictly increasing for \(\mu -\)a.e. \(t\in \Omega \) and \(\Phi \in \Delta _2\).

### Proof

*Sufficiency*. Assume that \(\Phi (t,.)\) is increasing for \(\mu -\)a.e. \(t\in \Omega \) and \(\Phi \in \Delta _2\). Then, taking any \(0\le y\le x\in L^\Phi (\mu )\), \(y\ne x\) and assuming that \(\Vert x\Vert _\Phi =\Vert y\Vert _\Phi \), by statement (2) of Lemma 2.1, we obtain

*Necessity*. By virtue of Theorem 2.6 we can assume that \(b_\Phi (t)=\infty \) for \(\mu -\)a.e. \(t\in \Omega \) because otherwise \(L^\Phi (\mu )\) is not even orthogonally strictly monotone. Under this assumption, first we will show that if \(\Phi \notin \Delta _2\), then \(L^\Phi (\mu )\) is not strictly monotone. In order to do so, assume that \(b_\Phi (t)=\infty \) for \(\mu -\)a.e. \(t\in \Omega \) but \(\Phi \notin \Delta _2\). Applying Lemma 1.8 with \(b_{n}=1+\frac{1}{n}\), \(p_{n}=2^n\) and \(q_n=\frac{1}{2^n}\), where \(n\in \mathbb {N}\), we can find a sequence \(\{\widetilde{x}_n(t)\}_{n=1}^\infty \) of \(\Sigma -\)measurable functions and mutually disjoint sets \(\{F_n\}_{n=1}^\infty \) in \(\Sigma \) such that \(\widetilde{x}_n(t)<\infty \) on the set \(F_n\) and we have

*A*of positive and finite measure and \(a,b\in \mathbb {R}_+\) such that \(0<a<b<\infty \) and \(\Phi (t,a)=\Phi (t,b)\) for all \(t\in A\). Indeed, since the set of positive rational numbers \(\mathbb {Q}_+\) is countable, we can write \(\mathbb {Q}_+=\{r_n\}_{n=1}^\infty \), where \(r_n\ne r_m\) for any \(m,n\in \mathbb {N}\) such that \(m\ne n\). For \(r_m,r_n\in \mathbb {Q}_+\) such that \(r_m<r_n\), let us define the sets

Next, there exists an increasing sequence \((A_n)_{n=1}^\infty \) of sets such that \(\mu (A_n)<\infty \), \(\mu \left( \Omega \backslash \underset{n=1}{\overset{\infty }{\cup }} A_n\right) =0\) and \(\sup \limits _{t\in {A_n}}\Phi (t,u)<\infty \) for all \(u\in \mathbb {R}_+\) and \(n\in \mathbb {N}\) (see [6]), which gives that \(\chi _{A_n}\in E^\Phi (\mu )\) for all \(n\in \mathbb {N}\). Then, there exists \(m\in \mathbb {N}\) such that \(\mu (A\cap A_m)>0\), whence \(\Phi (t,\lambda )\chi _{A\cap A_{m}}\in L^1(\mu )\) for any \(\lambda >0\). We will consider two cases separately.

*Case 1*. Let \(\Phi (t,a)>0\) for any \(t\in A\). Define \(y=a\chi _{A\cap A_m}\) and \(x=b\chi _{A\cap A_m}\) and let us denote \(d=I_\Phi (a\chi _{A\cap A_m})\). Then \(0\le y\le x\), \(y\ne x\) and \(I_\Phi (y)=\int \nolimits _{A\cap A_m}\Phi (t,a)d\mu (t)=\int \nolimits _{A\cap A_m}\Phi (t,b)d\mu (t)=I_\Phi (x)\), so \(I_\Phi (y)=I_\Phi (x)=d<\infty \), whence \(y,x\in L^\Phi (\mu )\). Writing the last equalities equivalently, we obtain that \(I_\Phi \left( \frac{dy}{d}\right) =I_\Phi \left( \frac{d x}{d}\right) =d,\) whence by statement (3) of Lemma 2.1, we get that \(\Vert d y\Vert _\Phi =\Vert d x\Vert _\Phi =d\). By virtue of the facts that \(0\le d y\le d x\) and \(d y\ne d x\), we conclude that \(L^\Phi (\mu )\) is not strictly monotone.

*Case 2*. Let us assume that \(\Phi (t,a)=0\) for any \(t\in A\). Then \(a_\Phi (t)>0\) for \(t\in A\). By virtue of Theorem 2.4, in this case \(L^\Phi (\mu )\) is not even orthogonally strictly monotone. \(\square \)

Let us present now a useful lemma.

### Lemma 2.9

If \((\Omega ,\Sigma ,\mu )\) is a non-atomic, complete and \(\sigma -\)finite measure space and \(\Phi \) is a monotone Musielak–Orlicz function, then the space \(E^\Phi (\mu )\) is nontrivial if and only if the set \(\Omega _{oc}\) has positive measure. Moreover, in this case \({\text {supp}}E^\Phi (\mu )=\Omega _{oc}\), that is, there exists a function \(x\in E^\Phi (\mu )\) such that \(x(t)>0\) for any \(t\in \Omega _{oc}\) as well as if \(x\in L^0(\mu )\) and \(\mu [(\Omega \backslash \Omega _{oc})\cap {\text {supp}}x]>0\) then \(x\notin E^\Phi (\mu )\).

### Proof

*Sufficiency*. Assume that \(\mu (\Omega _{oc})>0\). Then (see [6]), there exists a sequence of increasing sets \(\{A_n\}_{n=1}^\infty \) in \(\Omega _{oc}\cap \Sigma \) such that \(\chi _{A_n}\in E^\Phi (\mu )\), \(0<\mu (A_n)<\infty \) and \({\bigcup \nolimits _{n=1}^\infty } A_n=\Omega _{oc}\). This shows that \(E^\Phi (\mu )\ne \{0\}\).

*Necessity*. Assume that \(x\in L^0(\mu )\) and \(\mu [(\Omega \backslash \Omega _{oc})\cap {\text {supp}}x]>0\). Let us denote \(A_x=(\Omega \backslash \Omega _{oc})\cap {\text {supp}}x\). Then \(b_\Phi (t)<\infty \) for any \(t\in A_x\). Let us define

### Corollary 2.10

The space \(E^\Phi (\mu )\) is strictly monotone if and only if \(\mu (\Omega _{oc})=0\) or if \(\mu (\Omega _{oc})>0\), then \(\mu (\Omega _{oc}\cap \Omega _{nsi})=0\).

### Proof

*Sufficiency*. If \(\mu (\Omega _{oc})=0\), then \(E^\Phi (\mu )=\{0\}\), whence \(E^\Phi (\mu )\) is strictly monotone. Assume that \(\mu (\Omega _{oc})>0\). Then \(\mu (\Omega _{oc}\cap \Omega _{nsi})=0\). Consequently, \(\mu (\Omega _{oc}\cap \Omega _{si})=\mu (\Omega _{oc})>0\), where \(\Omega _{si}=\{t\in \Omega :\Phi (t,.)\text { is strictly increasing on }\mathbb {R}_+\}\). By virtue of Lemma 2.9, \(E^\Phi (\mu )\ne \{0\}\) and \({\text {supp}}E^\Phi (\mu )=\Omega _{oc}\). Therefore, we can (and we will) further consider below the space \(E^\Phi (\mu )\) on the measure space \((\Omega _{oc},\Sigma |_{\Omega _{oc}},\mu |_{\Omega _{oc}})\) only. Note that \(a_\Phi (t)=0\) for all \(t\in \Omega _{oc}\) (because \(\Phi (t,.)\) is strictly increasing on \(\mathbb {R}_+\) for \(\mu -\)a.e. \(t\in \Omega _{oc}\)). Next, the proof goes in a similar way as the proof of sufficiency for Theorem 2.8.

*Necessity*. Assume that \(E^\Phi (\mu )\) is strictly monotone but \(\mu (\Omega _{oc})>0\) and \(\mu (\Omega _{oc}\cap \Omega _{nsi})>0\). Then \({\text {supp}}E^\Phi (\mu )=\Omega _{oc}\), so \(b_\Phi (t)=\infty \) for all \(t\in \Omega _{oc}\). Since the proof goes in a similar way as the proof of the necessity of the fact that \(\Phi (t,.)\) must be strictly increasing on \(\mathbb {R}_+\) for \(\mu -\)a.e. \(t\in \Omega \) in Theorem 2.8, it is omitted (in the case when \(a_\Phi (t)>0\) on a set of positive measure we can proceed similarly as in the proof of Remark 2.5). \(\square \)

Let us present now Lemmas 2.11 and 2.12 which will be useful in the proof of Theorem 2.13.

### Lemma 2.11

Let \(\Phi (t,.)\) be a monotone Musielak–Orlicz function on \(\mathbb {R}_+\) for \(\mu -\)a.e. \(t\in \Omega \). Then, for any \(x\in E^\Phi (\mu )\) the function \(f_x(\lambda )=I_\Phi (\lambda x)\) is continuous on the interval \((0,\infty )\) and right continuous at 0.

Since the easy proof is similar to the proof of Lemma 4.1(i) from [3], it is omitted.

### Lemma 2.12

For any monotone Musielak–Orlicz function \(\Phi \), any measure space \((\Omega ,\Sigma ,\mu )\) and any sequence \((x_n)_{n=1}^\infty \) in \(L^\Phi (\mu )\), modular convergence and F-normed convergence are equivalent, i.e. \(\Vert x_n\Vert _\Phi \rightarrow 0\) if and only if \(I_\Phi (\lambda x_n)\rightarrow 0\) for any \(\lambda >0\), as \(n\rightarrow \infty \).

Since an easy proof is similar to the proof of Lemma 6.4, p. 2004, from [5], it is omitted.

### Theorem 2.13

- (a)
\(E^{\Phi }(\mu )\) is lower locally uniformly monotone,

- (b)
\(E^{\Phi }(\mu )\) is strictly monotone,

- (c)
\(\mu (\Omega _{oc})=0\) or if \(\mu (\Omega _{oc})>0\), then \(\mu (\Omega _{oc}\cap \Omega _{nsi})=0\).

### Proof

By virtue of Corollary 2.10, statements (b) and (c) are equivalent. Moreover, statement (a) implies statement (b). So, we need only to prove that statement (b) implies statement (a).

### Corollary 2.14

- (a)
\(L^{\Phi }(\mu )\) is lower locally uniformly monotone,

- (b)
\(L^{\Phi }(\mu )\) is strictly monotone,

- (c)
\(\Phi (t,.) \) is strictly increasing on \(\mathbb {R}_+\) for \(\mu -\)a.e. \(t\in \Omega \) and \(\Phi \in \Delta _2\).

### Proof

By virtue of Theorem 2.8, statements (b) and (c) are equivalent. Moreover, it is obvious that statement (a) implies statement (b). Assume that \(L^{\Phi }(\mu )\) is strictly monotone. Then, by Theorem 2.8, \(\Phi \) satisfies condition (c). By Theorem 1.9, we obtain that \(L^\Phi (\mu )=E^\Phi (\mu )\) and it follows from condition (c) that \(\mu (\Omega \backslash \Omega _{oc})=0\). By virtue of Theorem 2.13 the proof is finished. \(\square \)

### Theorem 2.15

- (a)
\(E^{\Phi }(\mu )\) is orthogonally lower locally uniformly monotone,

- (b)
\(E^{\Phi }(\mu )\) is orthogonally strictly monotone,

- (c)
\(\mu (\Omega _{oc})=0\) or if \(\mu (\Omega _{oc})>0\), then \(\mu (\Omega _{oc}\cap {\text {supp}}a_\Phi )=0\).

### Proof

### Corollary 2.16

- (a)
\(L^\Phi (\mu )\) is orthogonally lower locally uniformly monotone,

- (b)
\(L^\Phi (\mu )\) is orthogonally strictly monotone,

- (c)
\(a_\Phi (t)=0\) for \(\mu -\)a.e. \(t\in \Omega \) and \(\Phi \in \Delta _2\).

### Proof

The implication \((a)\Rightarrow (b)\) is clear. Assume that (b) holds. Then, the condition \(\Phi \in \Delta _2\) follows from the proof of necessity of Theorem 2.8, where it was proved that if \(\Phi \notin \Delta _2\), then \(L^\Phi (\mu )\) is not orthogonally strictly monotone. If \(\Phi \in \Delta _2\), then \(L^\Phi (\mu )=E^\Phi (\mu )\) and \({\text {supp}}E^\Phi (\mu )=\Omega \). By virtue of Theorem 2.15, we get that \(a_\Phi (t)=0\) for \(\mu -\)a.e. \(t\in \Omega \).

Finally, assume that statement (c) holds true. Note that if \(\Phi \in \Delta _2\), then \(b_\Phi (t)=\infty \) for \(\mu -\)a.e. \(t\in \Omega \) and \(L^\Phi (\mu )=E^\Phi (\mu )\), so we can apply Theorem 2.15 and the proof is finished. \(\square \)

### Theorem 2.17

- (a)
\(L^\Phi (\mu )\) is upper locally uniformly monotone,

- (b)
\(\Phi (t,.) \) is strictly increasing on \(\mathbb {R}_+\) for \(\mu -\)a.e. \(t\in \Omega \) and \(\Phi \in \Delta _2\).

### Proof

*k*large enough, we get

### Theorem 2.18

- (a)
\(E^\Phi (\mu )\) is upper locally uniformly monotone,

- (b)
\(\mu (\Omega _{oc})=0\) or if \(\mu (\Omega _{oc})>0\), then \(\Phi \in \Delta _2(\Omega _{oc})\) and \(\mu (\Omega _{nsi})=0\).

### Proof

Assume that statement (b) holds. If \(\mu (\Omega _{oc})=0\), then \(E^\Phi (\mu )=\{0\}\) is obviously upper locally uniformly monotone. If \(\Phi \in \Delta _2(\Omega _{oc})\) and \(\mu (\Omega _{nsi})=0\) under the assumption that \(\mu (\Omega _{oc})>0\), then Theorems 2.17 and 1.9 yield that \(E^\Phi (\mu )\) is upper locally uniformly monotone.

Assume that statement (a) holds. Then \(\mu (\Omega _{oc})=0\) or if \(\mu (\Omega _{oc})>0\), then \(\mu (\Omega _{oc}\cap \Omega _{nsi})=0\), by Theorem 2.13 (see also Corollary 2.10). In what follows, we will restrict the measure space \((\Omega ,\Sigma ,\mu )\) to the measure space \((\Omega _{oc}, \Sigma |_{\Omega _{oc}}, \mu |_{\Omega _{oc}})\), that is, to the support of \(E^\Phi (\mu )\). Then \(E^\Phi (\mu )\ne \{0\}\). Although the beginning of the proof starts similarly to the proof of Lemma 1 on page 65 of the paper [6] by Kamińska (the application of the same technique is enough), we will present it for the convenience of the reader. Suppose that \(\Phi \notin \Delta _2(\Omega _{oc})\).

*A*of positive measure such that \(A\subset \Omega _{oc}\backslash \underset{k=1}{\overset{\infty }{\bigcup }}A_k\) and \(\mu (A\cap \Omega _{\widetilde{m}})>0\), where \(\Omega _{\widetilde{m}}\) is some set from the sequence of sets \(\{\Omega _m\}_{m=1}^\infty \) described earlier in the proof (see condition (2.12)). Passing to a subset

*B*of positive measure of the set \(A\cap \Omega _{\widetilde{m}}\), if necessary, we can assume that \(\mu (B)<\infty \). Take a nonnegative real-valued function

*a*(.) such that \(\int _{D}\Phi (t,a(t))d\mu =1\), where \(D\in \Sigma \), \(D\subset B\), \(\mu (D)>0\) and define \(x(t)=a(t)\chi _{D}(t)\). Then \(x\in E^\Phi (\mu )\), \(I_\Phi (x)=1\), whence \(\Vert x\Vert _\Phi =1\). Define

Therefore, the \(\Delta _2(\Omega _{oc})\)-condition is necessary for the upper local uniform monotonicity of \(E^\Phi (\mu )\). Consequently, as it was shown before, \(\Omega =\Omega _{oc}\) up to a measure zero set, so statement (b) holds true. \(\square \)

### Theorem 2.19

- (a)
\(E^\Phi (\mu )\) is orthogonally upper locally uniformly monotone,

- (b)
\(\mu (\Omega _{oc})=0\) or if \(\mu (\Omega _{oc})>0\), then \(\Phi \in \Delta _2(\Omega _{oc})\) and \(\mu ({\text {supp}}a_\Phi )=0\).

### Proof

Assume that \(E^\Phi (\mu )\) is orthogonally upper locally uniformly monotone. Since orthogonal upper local uniform monotonicity implies orthogonal strict monotonicity, by virtue of Theorem 2.15, we obtain that \(\mu (\Omega _{oc})=0\) or if \(\mu (\Omega _{oc})>0\), then \(\mu (\Omega _{oc}\cap {\text {supp}}a_\Phi )=0\). In the case when \(\mu (\Omega _{oc})>0\), the necessity of the \(\Delta _2(\Omega _{oc})\)-condition follows from the proof of Theorem 2.18, implication (a)\(\Rightarrow \)(b), where it was shown that if \(\Phi \notin \Delta _2(\Omega _{oc})\), then \(E^\Phi (\mu )\) is not orthogonally upper locally uniformly monotone. Since \(\Phi \in \Delta _2(\Omega _{oc})\), we have that \(\mu (\Omega \backslash \Omega _{oc})=0\), so statement (b) is true.

*x*have orthogonal supports, we obtain

*k*large enough and all \(t\in \Omega \). Since \(\varepsilon _k\rightarrow 0\) as \(k\rightarrow \infty \), we obtain that \(I_\Phi \left( \frac{x_{n_k}}{2\Vert x\Vert _\Phi }\right) \rightarrow 0\) as \(k\rightarrow \infty \). Applying the \(\Delta _2(\Omega _{oc})\)-condition, we get that \(I_\Phi \left( \lambda x_{n_k}\right) \rightarrow 0\) as \(k\rightarrow \infty \) for any \(\lambda >0\), whence \(\Vert x_{n_k}\Vert _\Phi \rightarrow 0\) as \(k\rightarrow \infty \). The double extract subsequence theorem finishes the proof. \(\square \)

### Corollary 2.20

- (a)
\(L^\Phi (\mu )\) is orthogonally upper locally uniformly monotone,

- (b)
\(L^\Phi (\mu )\) is orthogonally strictly monotone,

- (c)
\(\mu ({\text {supp}}a_\Phi )=0\) and \(\Phi \in \Delta _2\).

### Proof

It is obvious that (a) implies (b). Implication (b)\(\Rightarrow \)(c) follows from Corollary 2.16. Assume that statement (c) holds true. Since \(\Phi \in \Delta _2\) and \(a_\Phi (t)=0\) for \(\mu -\)a.e. \(t\in \Omega \) then \(E^\Phi (\mu )=L^\Phi (\mu )\), so it is enough to apply Theorem 2.19. \(\square \)

## Notes

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