Describing limits of integrable functions as grid functions of nonstandard analysis

In functional analysis, there are different notions of limit for a bounded sequence of $L^1$ functions. Besides the pointwise limit, that does not always exist, the behaviour of a bounded sequence of $L^1$ functions can be described in terms of its weak-$\star$ limit or by introducing a measure-valued notion of limit in the sense of Young measures. Working in Robinson's framework of analysis with infinitesimals, we show that for every bounded sequence $\{z_n\}_{n \in \mathbb{N}}$ of $L^1$ functions there exists a function of a hyperfinite domain (i.e.\ a grid function) that represents both the weak-$\star$ and the Young measure limits of the sequence. This result has relevant applications to the study of nonlinear PDEs.


Introduction
The lack of a nonlinear theory of distributions, first established by Schwartz [23], poses some limitations in the study of nonlinear PDEs: while some nonlinear problems can be solved by studying the limit of suitable regularized problems, other are ill-posed in the sense that they do not allow for solutions in the space of distributions. For some of these problems, the notion of admissible solution can be meaningfully extended to include measure-valued solutions (we refer to [16] for a theoretical discussion on the issue, and to [5,8,15,20,21,24,25] for some examples of measure valued solutions to ill-posed PDEs). These measure-valued solutions are obtained as suitable limits of approximate solution in the presence of some estimates. For instance, a uniformly bounded sequence of integrable functions has a weak-⋆ limit that corresponds to a Radon measure and has also limit in the sense of Young measures. For some PDEs, both notions of limit must be considered in order to recover a measure-valued solution. An example is studied in detail in [25].
In order to overcome the absence of a nonlinear theory for distributions, some authors have embedded the space of distributions in a differential algebra with a good nonlinear theory. This approach has been used for instance by Colombeau, Todorov & Vernaeve, and Benci & Luperi Baglini (see [4,12,27] and references therein).
The algebra of grid functions G(Ω) defined over an open domain Ω ⊆ R k , denoted by G(Ω), has been proposed as another algebra of generalized functions for the study of partial differential equations [7]. This algebra seem particularly suitable for this purpose, mainly due to the following results (proved in [7]).
(1) There exists an embedding from the space of distributions over Ω to the algebra of grid functions that satisfies the following conditions: • the pointwise product of grid functions extends the product over C 0 functions; • D, the discrete derivative of grid functions, extends the distributional derivative; • the following product rule holds: where ε is an infinitesimal of a hyperreal field of Robinson's nonstandard analysis. (2) It is possible to determine a real vector subspace D ′ X (Ω) ⊂ G(Ω) and a surjective homomorphism of vector spaces π : D ′ X (Ω) → D ′ (Ω) which is coherent with the above embedding. This correspondence is also coherent with the homomorphism π.
Thus the algebra of grid functions provides a generalization both of the space of distributions and the space of Young measures, two spaces of generalized functions customarily used for the study of linear and nonlinear PDEs. In this paper we will prove that a single grid function simultaneously represents two different notions of limit (namely, the weak-⋆ and the Young measure limit) of a sequence of integrable functions. Thus a number of classical concepts (such as different notions of limits and of generalized solutions) can be successfully unified in a relatively elementary but nontrivial hyperfinite setting.

Terminology and preliminary notions
In this section, we will define the notation and recall some results on grid functions and on Young measures that will be useful in the proof of the main results of this paper.
2.1. Terminology. In the sequel, Ω ⊆ R k will be an open set.
We will often reference the following real vector spaces: • The duality between a vector space X and its dual X ′ is denoted by ·, · X . • A real distribution over Ω is an element of D(Ω) ′ , i.e. a continuous linear functional T : D(Ω) → R. If T is a distribution and ϕ is a test function, according to the notation introduced above we denote the action of T over ϕ by T, ϕ D(Ω) . • M(R) = {ν : ν is a Radon measure over R satisfying |ν|(R) < +∞}.
• Following [2,3,28] and others, measurable functions ν : Ω → M P (R) will be called Young measures. Measurable functions ν : Ω → M(R) will be called parametrized measures, even though in the literature the term parametrized measure is used as a synonym for Young measure. If ν is a parametrized measure and if x ∈ Ω, we will write ν x instead of ν(x).

Grid functions.
Throughout the paper, we will work with a |D(Ω) ′ |saturated hyperreal field * R, and we will assume familiarity with the basics of Robinson's theory of analysis with infinitesimals. For an introduction on the subject, we refer for instance to Goldblatt [17], but see also [1,14,18,19,22].
For any x, y ∈ * R we will write x ≈ y to denote that x − y is infinitesimal, we will say that x is finite if there exists a standard M ∈ R satisfying |x| < M , and we will say that x is infinite whenever x is not finite. We will denote by * R f in the set of finite numbers in * R, i.e. * R f in = {x ∈ * R : x is finite}.
The notion of finiteness can be extended componentwise to elements of * R k whenever k ∈ N. For any A ⊆ * R k , • X will denote the set of the standard parts of the finite elements of X.
We will now recall the definition and some properties of grid functions studied in [7]. Grid functions over Ω are functions defined over a hyperfinite domain that represents the open domain Ω.
Define Ω X = * Ω ∩ X k . Notice that Ω X is an internal subset of X k , and in particular it is hyperfinite.
We will say that x ∈ Ω X is nearstandard in Ω iff there exists y ∈ Ω such that x ≈ y. Proposition 1.5 of [7] and the hypothesis that Ω is open ensure that • Ω X = Ω. Grid functions over Ω are internal functions over Ω X . Definition 2.2 (Grid functions over Ω). We will say that a grid function over Ω is an internal function f : Ω X → * R. The space of grid functions over Ω is defined as Since grid functions are defined on a discrete domain, the derivative is represented by a finite difference operator of an infinitesimal step.
For a grid function f ∈ G(Ω), we define the i-th forward finite difference of step ε as ) and, if α is a multi-index, then D α is defined as expected: For a discussion of the relevance of the operator D in representing the distributional derivative, we refer to [7].
In the same spirit, integrals are replaced by hyperfinite sums.
Definition 2.4 (Grid integral and inner product). Let f, g : * Ω → * R and let A ⊆ Ω X ⊆ X k be an internal set. We define For further details about the properties of the grid derivative and the grid integral, we refer to [7,10,11,18,19].
By using the operator D, it is possible to introduce a grid function counterpart of the space of test functions.
We define the algebra of grid test functions as follows: In Lemma 2.2 of [7] it is proved that the algebra of test function is the grid function counterpart of the space of standard test functions D(Ω) in the following sense: , then the restriction of * ϕ to Ω X belongs to D X (Ω). The duality with grid test functions allows for the definition of a meaningful equivalence relation on the algebra of grid functions. Definition 2.6. Let f, g ∈ G(Ω). We say that f ≡ g iff f, ϕ ≈ g, ϕ for all ϕ ∈ D X (Ω). We will call π the projection from G(Ω) to the quotient G(Ω)/ ≡, and we will denote by [f ] the equivalence class of f with respect to ≡.
In [7] it is proved that the space of grid functions generalizes the space of distributions. In particular, there exists a real subspace of G(Ω)/ ≡ that is isomorphic to the space of distributions.
is an isomorphism of real vector spaces.
In addition to the above result, in Theorem 3.16 of [7] it is also proved that the finite difference operator induces the distributional derivative on the quotient D ′ X (Ω)/ ≡.

2.3.
Grid functions of a finite L 1 norm. In the sequel, we will use the following L p norms over the space of grid functions. . For all f ∈ G(Ω), define In Lemma 4.1 of [7] it is proved that if f p ∈ * R f in for some p, then f ∈ D ′ X (Ω), i.e. that [f ] is a well-defined distribution. In this paper we will use also the following property: if it can be identified with a continuous linear functional over C 0 c (Ω), that we will still denote by [f ], defined by By the discrete Hölder's inequality, This estimate and linearity of the hyperfinite sum over Ω X allow to conclude that [f ] is a linear functional over C 0 c (Ω). In order to prove continuity it is sufficient to notice that if ϕ, ψ ∈ S 0 (Ω) satisfy ϕ − ψ ∞ ≈ 0, then As a consequence, [f ] is a continuous linear functional over C 0 c (Ω), as desired.
2.4. Young measures. We find it useful to recall some definitions and results on Young measures.
It is well-known that Young measures are able to express the weak-⋆ limit in L ∞ of the composition between a bounded sequence of L 1 functions with a function in C 0 b (R). This result is a consequence of the fundamental theorem of Young measures.
Theorem 2.11. For every bounded sequence of L 1 (Ω) functions {z n } n∈N , there exists a subsequence {z n k } k∈N of {z n } n∈N and a Young measure ν such that for all g ∈ C 0 b (R) and for all ϕ ∈ C 0 c (Ω), In other words, g(z n ) ⋆ ⇀ g(ν) in L ∞ (Ω) for all g ∈ C 0 b (R). Proof. See e.g. [2,3,6] and references therein.
In the last statement of Theorem 2.11, we have used density of C 0 c (Ω) in L 1 (Ω). Definition 2.12. If {z n } n∈N is a bounded sequence of L 1 (Ω) functions and if ν is a Young measure that satisfies Theorem 2.11, we will say that {z n } n∈N converges to ν in the sense of Young measure and we will write z n Y ⇀ ν.
The relations between grid functions and parametrized measures (including Young measures), are studied in depth in [7]. We recall the main results that will be useful for this paper. (1)

Moreover,
(1) for every Young measure ν over Ω there exists a grid function f such that ν f = ν; (2) for all x ∈ Ω and for all Borel Proof. For the proof of point (1), see Theorem 2.9 of [13]. The other statements are proved in Theorem 4.12, Theorem 4.14 and Proposition 4.17 of [7].
The difference between ν f x (R) and 1 is due to f being not finite at some non-negligible fraction of the monad of x. Point (3) of Theorem 2.11 can be rephrased in the following way: if f p ∈ * R f in for some 1 ≤ p ≤ ∞, then f assumes infinite values only on a (possibly empty) set Ω inf ⊆ Ω X of Loeb measure 0.
Lemma 2.14. For every f ∈ G(Ω), let ν f : Ω → M(R) the parametrized measure satisfying Theorem 2.13, and let f r : Ω → R be its barycentre, defined by Then f r is a measurable function. Moreover, if f 1 ∈ * R f in , then f r ∈ L 1 (Ω) and f r 1 ≤ f 1 .

The main results
We are now ready to prove that grid functions are expressive enough to describe simultaneously both the weak-⋆ limit and the Young measure limit of bounded sequences of integrable functions. Theorem 3.1. For every bounded sequence {z n } n∈N in L 1 (Ω) such that Proof. Since D(Ω) ⊆ C 0 c (Ω), recall that z ∞ ∈ C 0 c (Ω) ′ ⊆ D ′ (Ω) can be identified with a distribution (that we will still denote by z ∞ ) by posing for every ϕ ∈ D(Ω).
Let also b : Ω → R be the barycentre of ν: b(x) = R τ dν x . The hypotheses over ν are sufficient to entail b ∈ L 1 (Ω) (see e.g. Corollary 3.13 of [28]). Thus the function b can be identified with a distribution (that we will still denote by b) by posing Thanks to Theorem 2.7, there exists a grid function z D ∈ G(Ω) that corresponds to the distribution z ∞ . The grid function z D might not correspond to the Young measure ν; however ν z D and ν have the same barycentre b.
To see that this is the case, consider the grid functions defined for every n ∈ N, and let b n be the barycentre of ν z D n : b n (x) = R τ dν z D n . By this definition it is easy to see that b n (x) = b(x) for every x ∈ Ω such that |z D (y)| ≤ n for every y ∈ Ω X , y ≈ x. We have already observed that the hypothesis z D 1 ∈ * R f in ensures that the set Ω inf = {x : z D (x) is infinite} has Loeb measure 0. As a consequence, lim n→∞ b n (x) = b(x) for a.e. x ∈ Ω. Now let g n ∈ C 0 b (R) with g(τ ) = τ for every τ ∈ [−n, n]. By Theorem 2.13, we have that for all ϕ ∈ C 0 From the previous equalities we obtain Since g ∈ C 0 b (R) entails that g is bounded, sup |x|>n |g(x) is well-defined. Thus From the last estimate and from equation (2) we obtain Since lim |x|→∞ g(x) = 0, lim n→∞ sup |x|>n |g(x)| = 0. As a consequence, taking the limit as n → ∞ in equation (3) and taking into account the arbitrariness of ϕ, we obtain that the barycentre of ν z D is lim n→∞ b n = b, as desired.
By Theorem 2.13, there exists a grid function z 0 ∈ G(Ω) that corresponds to the Young measure ν −ν z D . By the previous part of the proof, this Young measure has null barycentre, i.e.
for every x ∈ R.
We claim that the grid function z = z D +z 0 satisfies the desired properties. In fact, for all ϕ ∈ D(Ω) Then, since D(Ω) is dense in C 0 c (Ω), we conclude that also z, * ϕ ≈ z ∞ , ϕ C 0 as desired.
The following result is an immediate consequence of Theorem 2.7 and of Theorem 3.1.
Proof. The function Ψ is well-posed: let f, g ∈ L 1 (Ω) satisfy f ≡ g. Then • f, * ϕ = • g, * ϕ for every ϕ ∈ D(Ω). Since D(Ω) is dense in C 0 c (Ω), we deduce that Ψ([f ]) = Ψ([g]) also in C 0 c (Ω) ′ . Similarly, injectivity of Ψ is a consequence of the injectivity of Φ (see Theorem 2.7) and of density of D(Ω) in C 0 c (Ω). Finally, surjectivity of Ψ is a consequence of Theorem 3.1 and of the fact that L 1 (Ω) is dense in C 0 c (Ω) ′ with respect to the weak-⋆ topology, so that every µ ∈ C 0 c (Ω) ′ can be obtained as the weak-⋆ limit of a sequence of functions in L 1 (Ω).
We will now prove the converse of Theorem 3.1, namely that every grid function with a finite L 1 norm corresponds simultaneously to the weak-⋆ limit and the Young measure limit of a sequence of integrable functions.
Proof. The proof is based upon the following results on distributions and Young measures, respectively. See e.g. Section 6.6 of [26]. Notice that, since C ∞ c (Ω) ⊆ L 1 (Ω), {d n } n∈N is also weakly-⋆ convergent in C 0 c (Ω) ′ to a continuous linear functional, still denoted by T , defined by (Y) For every Young measure ν there is a sequence {y n } n∈N in L 1 (Ω) such that for every g ∈ C 0 b and for all ϕ ∈ D(Ω) Denote by ν d the Young measure limit of {d n } n∈N , i.e. d n Y ⇀ ν d . Notice that it is not necessary that ν d = ν z . However, an argument similar to that of the proof of Theorem 3.1 allows to conclude that the Young measures ν z and ν d have the same barycentre. For the purposes of this proof, it is more convenient to rephrase this result by saying that the barycentre of ν z − ν d is null.
Let {y n } n∈N be a sequence in D(Ω) satisfying condition (Y) with ν = ν z − ν d . Finally, define z n = d n + y n for every n ∈ N. We claim that {z n } n∈N satisfies the desired conditions.
(1) Let ϕ ∈ D(Ω): by defintion of {z n } n∈N , by linearity of the limit and by recalling that the Young measure limit of {y n } n∈N has a null barycentre, This equality and density of D(Ω) in C 0 c (Ω) allow to conclude that {z n } n∈N is also weakly-⋆ convergent in C 0 c (Ω) ′ to [z]. (2) By definition, z n Y ⇀ ν d + ν z − ν d = ν z , as desired.
The proof of Theorem 3.3 provides an interpretation of the infinite and finite part of a grid function f ∈ G(Ω) with f 1 ∈ * R f in : • the finite part of f corresponds to a Young measure ν f over Ω; the barycentre f r of ν f belongs to L 1 (Ω); • the infinite part of f , that corresponds to the distribution [f ] − f r , is a grid function representative of a Radon measure whose support is a null subset of Ω.
4. An application of Theorems 3.1 and 3.3 An application of Theorems 3.1 and 3.3, i.e. of the correspondence between grid functions and the two measure-valued limits of integrable functions, to the study of nonlinear PDEs is discussed in [8]. In the paper, we studied a grid function formulation of the Neumann initial value problem    ∂ t u = ∆φ(u) in Ω ∂φ(u) ∂n = 0 in [0, T ] × ∂Ω u(0, x) = u 0 (x) with a non-monotone φ : R → R. Recall that, under this hypothesis, the above problem is ill-posed forward in time in the intervals where φ is decreasing and only has measure-valued solutions [8,21,25]. Under the additional assumptions that • φ ∈ C 1 (R); • φ(x) ≥ 0 for all x ≥ 0 and φ(0) = 0; • there exists u − ∈ R with 0 < u − such that φ ′ (u) > 0 if u ∈ (0, u − ) and φ ′ (u) < 0 for u ∈ (u − , +∞); • lim x→+∞ φ(x) = 0. we have shown that the solution to the grid function formulation corresponds to the sum of the weak-⋆ limit and the Young measure limit of a sequence of L 1 solutions of a regularized problem. For a more precise statement, we refer to Theorem 5.7 of [8]. Further applications to PDEs will be discussed in [9].