Time-dependent systems of generalized Young measures

In this paper some new tools for the study of evolution problems in the framework of Young measures are introduced. A suitable notion of time-dependent system of generalized Young measures is defined, which allows to extend the classical notions of total variation and absolute continuity with respect to time, as well as the notion of time derivative. The main results are a Helly type theorem for sequences of systems of generalized Young measures and a theorem about the existence of the time derivative for systems with bounded variation with respect to time.

In this paper we introduce some new tools in the theory of Young measures for the study of rate independent evolution problems. To describe the content of this paper, let us consider a problem defined on a time interval I , with space variable x in a compact metric space X , and state variable u in a finite dimensional Hilbert space Ξ. We assume that X is endowed with a given nonnegative Radon measure λ with supp λ = X . Given a sequence u k = u k (t, x) of functions from I×X to Ξ, satisfying suitable estimates, it is often possible to extract a subsequence converging, for every t ∈ I , to a Young measure µ t , which encodes information on the statistics of the space oscillations of u k (t, x) at time t.
To simplify the notation, the Young measure µ t will always be regarded as a measure on X×Ξ, whose projection on X coincides with λ. In this introduction we will never consider the standard disintegration (µ x t ) x∈X , which is usual in the classical presentation of the theory (see Remark 3.5).
If we want to extend some natural notions, like total variation, absolute continuity, or time derivative, from the original context of time dependent functions to the generalized context of time-dependent Young measures, we need to know the joint oscillations of u k (t 1 , x), . . . , u k (t m , x) for every finite sequence t 1 , . . . , t m of times. These are described by the Young measure µ t1...tm , with state space Ξ m , generated by the sequence of Ξ m -valued functions (u k (t 1 , x), . . . , u k (t m , x)). It is easy to see that µ t1...tm cannot be derived from the measures µ t1 , . . . , µ tm . Indeed, these measures give no information on the correlation between the oscillations at different times. The situation is similar to what happens in stochastic processes, where the knowledge of the distribution function of each single random variable is not enough to deduce their joint distribution.
This leads to the notion of system of Young measures, defined as a family (µ t1...tm ), where t 1 , . . . , t m run over all finite sequences of elements of I , with t 1 < · · · < t m , and each µ t1...tm is a Young measure on X with values in Ξ m . We assume that (µ t1...tm ) satisfies the following compatibility condition, which is always satisfied when µ t1...tm is generated by a sequence of time-dependent functions: if {s 1 , . . . , s n } ⊂ {t 1 , . . . , t m } and s 1 < · · · < s n , then µ s1...sn coincides with the corresponding projection of µ t1...tm .
The notions of total variation (Definition 8.1), absolute continuity (Definition 10.1), or time derivative (Definition 9.4) can be easily defined in the framework of systems of Young measures in such a way that they coincide with the standard notions in the case of timedependent functions. The main result of the paper is a version of Helly's Theorem for systems of Young measures (Theorem 8.10): if (µ k t1...tm ) has uniformly bounded variation, then there exist a system (µ t1...tm ) with bounded variation, a set Θ ⊂ I , with I \ Θ at most countable, and a subsequence, still denoted (µ k t1...tm ), such that µ k t1...tm ⇀ µ t1...tm weakly * for every finite sequence t 1 , . . . , t m ∈ Θ with t 1 < · · · < t m .
Another important result provides the existence of the time derivativeμ t for almost every t whenever the family (µ t1...tm ) has bounded variation (Theorem 9.7). The variation can be expressed by an integral involving the time derivatives when (µ t1...tm ) is absolutely continuous (Theorem 10.4).
Our motivation for studying systems of Young measures stems from the analysis of quasistatic evolution problems with nonconvex energies, which arise in the study of plasticity with softening [10]. Since in these applications the energy functionals have linear growth in some directions, we have to consider the case where the generating sequence (u k (t, x)) is bounded in L r λ (X; Ξ) only for r = 1 . It is well known that in this case Young measures should be replaced by more general objects, which take into account concentrations at infinity (see [12]). In [1] and [13] this is done by considering a pair (µ Y , µ ∞ ), where µ Y is a Young measure on X with values in Ξ and µ ∞ , called the varifold measure, is a measure supported on X×Σ Ξ , where Σ Ξ denotes the unit sphere in Ξ.
In the spirit of [12], we prefer to present these generalized Young measures in a different way, using homogeneous coordinates to describe the completion of Ξ obtained by adding a point at infinity for each direction. We replace the pair (µ Y , µ ∞ ) by a single nonnegative measure µ on X×Ξ×R (Definition 3.9), acting only on continuous functions f (x, ξ, η) which are positively homogeneous of degree one in (ξ, η). We assume that µ is supported on the set {η ≥ 0} and that the projection of ηµ onto X coincides with λ. We show that, if λ is nonatomic, then the space L 1 λ (X; Ξ) can be identified (Definition 3.1) with a dense subset of the space of generalized Young measures (Theorem 5.1).
Using this approach, we are able to prove the results on total variation and time derivatives for systems of Young measures in a context that is general enough for the applications considered in [10].

A space of homogeneous functions and its dual
If E is a locally compact space with a countable base and Ξ is a finite dimensional Hilbert space, M b (E; Ξ) denotes the space of bounded Radon measures on E with values in Ξ, endowed with the norm ν := |ν|(E), where |ν| denotes the variation of ν . When Ξ = R, the corresponding space will be denoted simply by M b (E). As usual, M + b (E) denotes the cone of nonnegative bounded Radon measures on E .
By the Riesz Representation Theorem M b (E; Ξ) can be identified with the dual of C 0 (E; Ξ), the space of continuous functions ϕ : E → Ξ such that {|ϕ| ≥ ε} is compact for every ε > 0 . The weak * topology of M b (E; Ξ) is defined using this duality.
Throughout the paper (X, d) is a given compact metric space and λ is a fixed nonnegative Radon measure on X with supp λ = X . The symbol Ξ will denote any finite dimensional Hilbert space. The spaces L r (X; Ξ), r ≥ 1 , will always refer to the measure λ. If µ ∈ M b (X; Ξ), µ a and µ s denote the absolutely continuous and the singular part of µ with respect to λ. Measures in M b (X; Ξ) which are absolutely continuous with respect to λ will always be identified with their densities, which belong to L 1 (X; Ξ). In this way L 1 (X; Ξ) is regarded as a subspace of M b (X; Ξ).
In order to define the notion of generalized Young measure on X with values in Ξ, it is convenient to introduce a space of homogeneous functions and to discuss some properties of its dual.
is positively homogeneous of degree one on Ξ for every x ∈ X ; i.e., f (x, tξ) = tf (x, ξ) for every x ∈ X , ξ ∈ Ξ, and t ≥ 0 . This space is endowed with the norm We introduce now two dense subspaces of C hom (X×Ξ) that will be useful in the proof of some properties of generalized Young measures.
Definition 2.2. Let C hom L (X×Ξ) be the space of all f ∈ C hom (X×Ξ) which satisfies the following Lipschitz condition: there exists a constant a ∈ R such that for every x ∈ X and every ξ 1 , ξ 2 ∈ Ξ.
and ω is the modulus of continuity of the restriction of f to X×Σ Ξ , then (2.1) and the homogeneity of f imply that Exchanging the roles of ξ 1 and ξ 2 we obtain for every x 1 , x 2 ∈ X and every ξ 1 , ξ 2 ∈ Ξ.
Proof. Let us fix f ∈ C hom (X×Ξ). For every k > f hom let us consider the Moreau-Yosida approximation f k : X×Ξ → R defined by Using the standard properties of Moreau-Yosida approximations it is easy to check that f k ∈ C hom L (X×Ξ) (with constant k ) and that the sequence f k is nondecreasing and converges pointwise to f (see, e.g., [8,Remark 9.6 and Theorem 9.13]). By Dini's Theorem we conclude that f k → f uniformly on X×Σ Ξ , hence f k → f in C hom (X×Ξ).
is Lipschitz continuous with respect to ξ and satisfies Proof. Thanks to the obvious density in C hom (X×Ξ) of the space of f ∈ C hom (X×Ξ) such that f (x, ·) belongs to C 2 (Ξ \ {0}) for every x, it is enough to prove that every such function can be written as To this aim it suffices to show that there exists a constant c := c(f ) such that f 2 (x, ξ) := c |ξ| − f (x, ξ) is convex in ξ for every x ∈ X . A simple calculation shows that the quadratic form corresponding to the Hessian matrix of f 2 with respect to ξ at a point (x, e), with e ∈ Σ Ξ , is given by By the Euler relation we have D ξ f (x, ξ) ξ = f (x, ξ). Taking the derivative with respect to ξ we obtain D 2 ξ f (x, ξ) ξ = 0 for every ξ , in particular D 2 ξ f (x, e) has an eigenvalue 0 with eigenvector e . This implies that there is a constant b(x, e) such that D 2 ξ f (x, e) ξ · ξ ≤ b(x, e) |ξ ⊥ e | 2 , where ξ ⊥ e := ξ − (ξ · e) e is the component of ξ orthogonal to e . As b(x, e) is bounded by the continuity of the second derivatives of f , and |ξ ⊥ e | 2 = |ξ| 2 − (ξ · e) 2 , by (2.2) there exists a constant c such that D 2 ξ f 2 (x, e) is positive definite for every x ∈ X and every e ∈ Σ Ξ , hence f 2 (x, ξ) is convex with respect to ξ for every x ∈ X .
Definition 2.8. The dual of the space C hom (X×Ξ) is denoted by M * (X×Ξ), and the corresponding dual norm by · * ; the weak * topology of M * (X×Ξ) is defined by using this duality. It is sometimes convenient to write the dummy variables explicitly and to use the notation f (x, ξ), µ(x, ξ) for the duality product f, µ . The positive cone M + * (X×Ξ) is defined as the set of all µ ∈ M * (X×Ξ) such that f, µ ≥ 0 for every f ∈ C hom (X×Ξ) with f ≥ 0 . Remark 2.9. It is easy to see that for every µ ∈ M + * (X×Ξ) we have µ * = |ξ|, µ(x, ξ) .
Strictly speaking, the elements µ of M * (X×Ξ) are not measures, because they act only on homogeneous functions. However, the notion of image of µ under a map ψ can be defined by duality, as in measure theory. Definition 2.10. Let Ξ and Ξ ′ be two finite dimensional Hilbert spaces and let ψ : X×Ξ → X×Ξ ′ be a continuous map of the form ψ(x, ξ) = (x, ϕ(x, ξ)), with ϕ : X×Ξ → Ξ ′ positively one-homogeneous in ξ . The image ψ(µ) of µ ∈ M * (X×Ξ) under ψ is defined as the element of M * (X×Ξ ′ ) such that Similarly we can define the notion of support of µ ∈ M * (X×Ξ). We say that a subset C of X×Ξ is a Ξ-cone if (x, ξ) ∈ C ⇒ (x, tξ) ∈ C for every t ≥ 0 .
Definition 2.11. The support supp µ of µ ∈ M * (X×Ξ) is defined as the smallest closed Ξ-cone C ⊂ X×Ξ such that f, µ = 0 for every f ∈ C hom (X×Ξ) vanishing on C .
Remark 2.12. For every µ ∈ M * (X×Ξ) there exists a measureμ ∈ M b (X×Ξ) with compact support such that for every f ∈ C hom (X×Ξ). A measure with this property can be constructed by considering the continuous linear map on C(X×Σ Ξ ) defined by g → |ξ|g(x, ξ/|ξ|), µ(x, ξ) . By the Riesz Representation Theorem there existsμ ∈ M b (X×Σ Ξ ) such that for every g ∈ C(X×Σ Ξ ). Regardingμ as a measure on X×Ξ supported by X×Σ Ξ , we obtain (2.3). This construction suggests that the measureμ satisfying (2.3) is not unique; indeed, we can repeat the same construction with Σ Ξ replaced by any other concentric sphere.
For the applications it is convenient to extend some of the previous results to a suitable space of possibly discontinuous functions. Definition 2.13. Given two finite dimensional Hilbert spaces Ξ and Ξ ′ , let B hom ∞ (X×Ξ; Ξ ′ ) be the space of Borel functions f : X×Ξ → Ξ ′ such that (a) for every x ∈ X the function ξ → f (x, ξ) is positively homogeneous of degree one on Ξ, (b) there exists a constant a ∈ R such that |f (x, ξ)| ≤ a|ξ| for every (x, ξ) ∈ X×Ξ. The smallest constant a satisfying the previous inequality is denoted by f hom . When Ξ ′ = R, the corresponding space will be denoted simply by B hom ∞ (X×Ξ). Definition 2.14. For every f ∈ B hom ∞ (X×Ξ) and every µ ∈ M * (X×Ξ) the duality product f, µ is defined by whereμ is any measure satisfying the conditions of Remark 2.12. By homogeneity the value of f, µ does not depend on the particular measureμ chosen in (2.3). The same definition (with values in R∪{+∞} this time) is adopted if µ ∈ M + * (X×Ξ) and f : X×Ξ → R∪{+∞} is a Borel function such that f (x, ξ) is positively homogeneous of degree one in ξ and f (x, ξ) ≥ −c|ξ| for some constant c ≥ 0 .
Let π X : X×Ξ → X be the projection onto X . We now define the image under π X of the product hµ of an element µ of M * (X×Ξ) by a homogeneous function h. Definition 2.15. Let Ξ and Ξ ′ be two finite dimensional Hilbert spaces, let µ ∈ M * (X×Ξ), and let h ∈ B hom ∞ (X×Ξ; Ξ ′ ). The measure π X (hµ) is the element of M b (X; Ξ ′ ) such that for every µ ∈ M * (X×Ξ) and every h ∈ B hom ∞ (X×Ξ; Ξ ′ ).

Generalized Young measures
As mentioned in the introduction, the notion of generalized Young measure is used to describe oscillation and concentration phenomena for sequences which are bounded in L r (X; Ξ) only for r = 1 . To study concentration phenomena, where the sequences tend to infinity along given directions in the space Ξ, it is useful to introduce homogeneous coordinates. This is done by replacing the space Ξ by Ξ×R, whose generic point is denoted by (ξ, η); the set of points with η = 1 is identified with Ξ, while points with η = 0 are interpreted as directions at infinity.
In our presentation the space of generalized Young measures will be a subset of the space M + * (X×Ξ×R), where Ξ×R plays the role of the Hilbert space Ξ of the previous section. Before describing this set, we first consider generalized Young measures associated with functions.
Definition 3.1. Given u ∈ L 1 (X; Ξ), the generalized Young measure associated with u is defined as the element δ u of M + * (X×Ξ×R) such that for every f ∈ C hom (X×Ξ×R).
In the spirit of [15] and [22] we extend this definition to measures p ∈ M b (X; Ξ).
Definition 3.2. Given p ∈ M b (X; Ξ), the generalized Young measure associated with p is defined as the element δ p of M + * (X×Ξ×R) such that for every f ∈ C hom (X×Ξ×R) where σ is an arbitrary nonnegative Radon measure on X with λ << σ and p << σ .
The homogeneity of f implies that the integral does not depend on σ and that the definitions coincide when p = u ∈ L 1 (X; Ξ). The norm of δ p is given by the following lemma.
Proof. Let us consider the Borel partition X = X a ∪ X s with λ(X s ) = 0 = |p s |(X a ) and let σ := λ + |p s |, so that σ = λ on X a and σ = |p s | on X s . By Remark 2.9 we have which concludes the proof.
We recall the definition of Young measure.
Definition 3.4. A Young measure on X with values in Ξ is a measure ν ∈ M + b (X×Ξ) such that π X (ν) = λ. The space of Young measures on X with values in Ξ is denoted by Y (X; Ξ). For every r ≥ 1 let Y r (X; Ξ) be the space of all ν ∈ Y (X; Ξ) whose r -moment X×Ξ |ξ| r dν(x, ξ) is finite.
Remark 3.5. By the Disintegration Theorem (see, e.g., [23, Appendix A2]) for every ν ∈ Y (X; Ξ) there exists a measurable family (ν x ) x∈X of probability measures on Ξ such that for every bounded Borel function g : X×Ξ → R. The probability measures ν x are uniquely determined for λ-a.e. x ∈ X .
Definition 3.6. Given ν ∈ Y 1 (X; Ξ), the generalized Young measure associated with ν is defined as the element ν of M + * (X×Ξ×R) such that for every f ∈ C hom (X×Ξ×R).
Remark 3.7. It follows from Remark 2.9 that Remark 3.8. If µ = δ p for some p ∈ M b (X; Ξ), the following properties hold: We will refer to (3.2) as the projection property. According to (2.4), it is equivalent to This motivates the following definition.  The sequential compactness of every bounded subset of GY (X; Ξ) is given by the following theorem.
Theorem 3.11. Every bounded sequence in GY (X; Ξ) has a subsequence which converges weakly * to an element of GY (X; Ξ).
Proof. Since GY (X; Ξ) is closed in the weak * topology of M * (X×Ξ×R), the result follows from the Banach-Alaoglu Theorem.
Indeed, any such f is the supremum of a family of functions in C hom (X×Ξ×R).
In the case of a generalized Young measure µ ∈ GY (X; Ξ) the duality product f, µ can be defined for every f in the space B hom ∞,1 (X×Ξ×R) introduced by the following definition, which is slightly larger than the space B hom ∞ (X×Ξ×R) considered in the previous section. Definition 3.14. Given two finite dimensional Hilbert spaces Ξ and Ξ ′ , we consider the for every (x, ξ, η) ∈ X×Ξ×R. When Ξ ′ = R, the corresponding space will be denoted simply by B hom ∞,1 (X×Ξ×R).
It follows from Fatou's Lemma that f isμ-integrable. Definition 3.16. Given f ∈ B hom ∞,1 (X×Ξ×R) and µ ∈ GY (X; Ξ), the duality product f, µ is defined by where a and b satisfy (3.4) and b 1 denotes the L 1 norm of b .
We now consider the image of a generalized Young measure.
Definition 3.18. Let Ξ and Ξ ′ be two finite dimensional Hilbert spaces and let ψ : Remark 3. 19. Under the assumptions of the definition the function f • ψ belongs to B hom ∞,1 (X×Ξ×R), so that the duality product f • ψ, µ is well defined. Moreover, by the particular form of the map ψ , the element of M + * (X×Ξ ′ ×R) defined by (3.6) satisfies (3.1) and (3.2), therefore it belongs to GY (X; Ξ ′ ).

Comparison with other presentations of the theory
In this section we show that every µ ∈ GY (X; Ξ) can be represented by a unique Young To introduce this representation, we recall that Y 1 (X; Ξ) can be identified with a suitable subset of GY (X; Ξ) (Definition 3.6). The following definition identifies the measures in M + b (X×Σ Ξ ) with particular elements of M * (X×Ξ×R).
The main result of this section is the following theorem.
Combining the compactness property (Theorem 3.11) and the representation formula (Theorem 4.3) we recover the following result, originally proved in [1] (see Remark 4.5).
Theorem 4.9. Let u k be a bounded sequence in L 1 (X; Ξ). Then there exist a subsequence, still denoted u k , a Young measure µ Y ∈ Y 1 (X; Ξ), and a measure µ for every continuous function g : X×Ξ → R such that for every (x 0 , ξ 0 ) ∈ X×Ξ the limit exists and is finite.
Proof. Let us consider the sequence δ u k in GY (X; Ξ) introduced in Definition 3.1. By Lemma 3.3 we have δ u k * ≤ sup j u j 1 + λ(X) < +∞. By Theorem 3.11 there exists a subsequence, still denoted u k , such that δ u k converge weakly * to an element µ of GY (X; Ξ). Let g be as in the statement of the theorem and let f : X×Ξ×R → R be defined by It is easy to check that f is continuous in (x, ξ, η) and homogeneous of degree one in (ξ, η). Therefore, the weak * convergence of δ u k to µ implies that By Theorem 4.3, taking into account the definition of f , we obtain that there exists a pair The conclusion follows from (4.8) and (4.9).

A density result
In this section we prove that, if λ is nonatomic, then the generalized Young measures of the form δ u associated with functions u ∈ L 1 (X; Ξ) are dense in GY (X; Ξ). The main result is the following approximation theorem.
Proof. We consider the decomposition µ = µ Y +μ ∞ of Theorem 4.3 and we fix a sequence σ n converging to 0 with 0 < σ n < min{1, λ(X)} . For every n we consider two countable partitions Ξ for every Borel set A ⊂ X . As λ = π X (µ Y ), the measure λ Y is absolutely continuous with respect to λ. Let us consider the partition X = X a ∪ X s , with λ(X s ) = 0 = λ ∞,s (X a ), where λ ∞,s is the singular part of λ ∞ with respect to λ. To prove the theorem, for every n we will consider a new partition X = X n,a ∪ X n,s , where X n,a and X n,s are suitable approximations of X a and X s such that λ(X n,a ) > 0 and λ(X n,s ) > 0 . We will construct the approximating sequence u n by defining it separately on X n,a and X n,s .
Step 1. Definition of u n on X n,s . We begin by constructing X n,s . For every n we can find a countable Borel partition X s = i X n,s i ∪ N n,s , where each X n,s i is closed, In the following, given a subset E ⊂ X and a radius r > 0 , the r -neighbourhood of E will be denoted by (E) r := {x ∈ X : d(x, E) < r} . Since λ(X n,s i ) = λ Y (X n,s i ) = 0 and λ((X n,s i ) r ) > 0 for every r > 0 , we can construct inductively a decreasing sequence r n i such that 0 < r n i ≤ σ n /2 , λ((X n,s i ) r n i ) ≤ σ n λ ∞ (X n,s i ) , λ((X n,s i+1 ) r n i+1 ) ≤ 1 3 λ((X n,s i ) r n i ) .
We define A n,s i := (X n,s i ) r n i \ j>i (X n,s j ) r n j .
while (5.2), together with the inequality 0 < r n i ≤ σ n /2 , yields diam A n,s i ≤ σ n . (5.7) Since λ is nonatomic we can find a countable Borel partition A n,s For every n we define X n,s := ij A n,s ij ∪ X s . . (5.12) By (5.8) we have that c n ij ≥ 1/σ n . (5.13) By (5.12) and by (5.13) we have X n,s 1 + |u n | 2 dλ ≤ λ ∞ (X s ) 1 + σ 2 n . (5.14) Step 2. Definition of u n on X n,a . We set X n,a := X \ X n,s .
In order to define u n on X n,a we consider a countable Borel partition X n,a = i A n,a i , with A n,a i satisfying 0 < diam A n,a i ≤ σ n . (5.15) As X n,a ⊂ X a by (5.9), λ ∞ is absolutely continuous with respect to λ on X n,a . Since λ is nonatomic, for every i we may choose 0 < ε n i ≤ σ n and two disjoint Borel sets A n,Y i and A n,∞ i in such a way that A n,a Since λ is nonatomic and we can also find a countable Borel partition A n,∞ Note also that by (5.16) we have ,a i ) and so there exists 0 < δ n i ≤ σ n such that λ(A n,Y i ) = (1 − δ n i )λ(A n,a i ). As λ = π X (µ Y ), arguing as before we may find a countable Borel partition A n,Y We are ready to define u n on X n,a by setting By (5.14) and (5.20) we have which implies that u n is bounded in L 1 (X; Ξ). It follows from Lemma 3.3 that δ un * is uniformly bounded.
Step 3. Proof of the convergence. Thanks to Lemma 2.4, to prove the weak * convergence of δ un it is enough to show that f, δ un → f, µ , By definition we have By (5.19) the first integral in the right-hand side can be written as where x n,a i are arbitrary points in A n,a i and the remainder r a,1 n tends to 0 as a consequence of (5.15), (5.17), (5.18), and (5.23), which lead to the estimate where the remainder r a,2 n tends to 0 as a consequence of (5.1), (5.10), (5.15), and (5.23), which lead to the estimate From (5.25) and (5.26) we obtain where r a n := r a,1 n + r a,2 n tends to 0 . By (5.11) and (5.12) the second integral in the right-hand side of (5.24) can be written as where the remainder r s,2 n tends to 0 as a consequence of (5.1), (5.2), (5.13), and (5.23), which lead to the estimate |r s,2 n | ≤ aσ n + ω(σ n ) 1 + σ 2 n λ ∞ (X s ) . From (5.28) and (5.29) we obtain where r s n := r s,1 n + r s,2 n tends to 0 . From (4.1), (5.24), (5.27), and (5.30) we obtain (5.22), which concludes the proof of the theorem.

The notion of barycentre
In this section we study some properties of the barycentre of a generalized Young measure.
We now prove the Jensen inequality for generalized Young measures. for every µ ∈ GY (X; Ξ).
Proof. Let us fix µ ∈ GY (X; Ξ), let p := bar(µ), and let σ ∈ M + b (X) be such that p << σ and λ << σ . We consider an increasing sequence of functions f k converging to f such that each f k has the form f k (x, ξ, η) = sup with a i : X → Ξ and b i : X → R bounded σ -measurable functions (see, e.g., [6, Theorem 2.
The opposite inequality requires special conditions on f and µ, as shown in the following lemma, that will be used in [10]. Lemma 6.7. Let µ ∈ GY (X; Ξ), let f : X×Ξ×R → [0, +∞] be a Borel function such that (ξ, η) → f (x, ξ, η) is positively one-homogeneous for every x ∈ X , and let co f be the lower semicontinuous convex envelope of f with respect to (ξ, η). Assume that f, µ ≤ co f, δ bar(µ) < +∞. Then supp µ is contained in the closure of {f = co f } .

Compatible systems of generalized Young measures
Let A ⊂ R and let t → p(t) be a function from A into M b (X; Ξ). For every finite sequence t 1 < t 2 < · · · < t m in A we consider the measure (p(t 1 ), . . . , p(t m )) ∈ M b (X; Ξ m ) and the corresponding generalized Young measure µ t1...tm := δ (p(t1),...,p(tm)) ∈ GY (X; Ξ m ) (7.1) introduced in Definition 3.2, with Ξ replaced by Ξ m . To describe an important property of this family of generalized Young measures it is convenient to introduce the following definition.
Definition 7.4. Given a function t → p(t) from A ⊂ R into M b (X; Ξ), the family (µ t1...tm ) defined in (7.1) is called the compatible system of generalized Young measures associated with t → p(t).
The notion of left continuity, introduced in the next definition, is very useful in the applications.
Definition 7.6. A system µ ∈ SGY (A, X; Ξ) is said to be left continuous if for every finite sequence t 1 , . . . , t m in A with t 1 < · · · < t m the following continuity property holds: µ s1...sm ⇀ µ t1...tm weakly * in GY (X; Ξ m ) (7.5) as s i → t i , with s i ∈ A and s i ≤ t i .
The following theorem proves the weak * compactness of the subsets of SGY (A, X; Ξ) defined by imposing bounds on the norms of µ t for every t ∈ A. To prove the theorem we need the following lemma which provides an estimate of the norm µ t1...tm * in terms of the norms µ ti * .
Proof of Theorem 7.7. By Lemma 7.8 for every function C : A → [0, +∞) the set defined in (7.6) is contained in the set of all µ ∈ SGY (A, X; Ξ) such that for every finite sequence t 1 , . . . , t m in A with t 1 < t 2 < · · · < t m . As the topology in SGY (A, X; Ξ) is induced by the product of the weak * topologies of the spaces GY (X; Ξ m ) corresponding to the projections µ t1...tm , the set (7.6) is compact in the weak * topology of SGY (A, X; Ξ) by Tychonoff's Theorem.
The notion of piecewise constant interpolation will be useful in the application to evolution problems.

The notion of variation
In this section we study the notion of variation on a time interval of a compatible system of generalized Young measures, and prove a compactness theorem which extends Helly's Theorem.
where the supremum is taken over all finite families t 0 , t 1 , . . . , t k in A such that a = t 0 < t 1 < · · · < t k = b (with the convention Var(µ; a, b) = 0 if a = b ).
where the supremum is taken over all finite families t 0 , t 1 , . . . , t k in A such that a = t 0 < t 1 < · · · < t k = b .
Remark 8.4. If t 1 , t 2 , t 3 ∈ A and t 1 < t 2 < t 3 , by the compatibility condition (7.2) and by the triangle inequality we have Using this inequality it is easy to deduce from Remark 8.3 that Var(µ; a, c) = Var(µ; a, b) + Var(µ; b, c) (8.1) for every a, b, c ∈ A with a ≤ b ≤ c. This implies in particular that the function t → Var(µ; a, t) is nondecreasing on A ∩ [a, +∞).
Remark 8.5. If A = {a 0 , . . . , a k } ⊂ R is a finite set, with a 0 < a 1 < · · · < a k , µ ∈ GY (X; Ξ k+1 ), and µ A ∈ SGY (A, X; Ξ) is the associated system defined in Remark 7.9, it follows from (8.1) that It is easy to see that, if µ [A] ∈ SGY ([a 0 , a k ], X; Ξ) is the piecewise constant interpolation of µ defined by (7.7), then where the supremum is taken over all finite families t 0 , t 1 , . . . , t k in A such that a = t 0 < t 1 < · · · < t k = b (with the convention Var h (µ; a, b) = 0 if a = b ). Using the compatibility condition it is easy to prove the following lemma. The proof is omitted, since it is similar to the proof of the following lemma, which will be used in Theorem 9.7. Proof. Let V (t) := Var(µ; 0, t) for every t ∈ [0, T ]. Let us fix f ∈ C hom L (X×Ξ 2 ×R) and let a be a constant satisfying (2.1). Let t 1 , t 2 with 0 ≤ t 1 < t 1 + c < t 2 < t 2 + c ≤ T . Using the compatibility condition (7.2) and (8.1), we obtain The same inequality can be proved if 0 ≤ t 1 < t 2 ≤ t 1 + c < t 2 + c ≤ T . As V is nondecreasing, we conclude that the total variation of Φ f c on [0, T − c] is less than or equal to V (T − c) + V (T ).
The following result can be considered as a version of Helly's Theorem for compatible systems of generalized Young measures. Note that this is a sequential compactness result, in contrast with Theorem 7.7.
Proof. The proof is divided in several steps.
Step 1. Boundedness of µ k t1...tm . We begin by proving that µ k t * is bounded uniformly with respect to t ∈ [0, T ] and k . Let us fix t < t 0 . By the compatibility condition (7.2) for every finite sequence t 1 , . . . , t m with t 1 < · · · < t m .
Step 2. Choice of the subsequence. Let D be a countable dense subset of [0, T ] containing 0 . By the compactness Theorem 3.11, using (8.8) and a diagonal argument, we can extract a subsequence, still denoted µ k , such that, for every s 1 , . . . , s m in D with 0 ≤ s 1 < · · · < s m ≤ T , the sequence µ k s1...sm converges weakly * in GY (X; Ξ m ).
Step 5. Extension to [0, T ]. It remains to show that we can define µ t1...tm when some t i does not belong to Θ , in such a way that the resulting system of generalized Young measures satisfies the compatibility conditions, inequalities (8.4) and (8.5), and the continuity property (7.5). To this purpose, it is enough to observe that, since V has a finite limit from the left at each point, we have for every sequence (s k For these sequences (s k 1 , . . . , s k m ) we can deduce from estimate (8.13) that f, µ s k 1 ...s k m satisfies a Cauchy condition for every f satisfying (8.10). By (8.12) we deduce from Lemma 2.4 the existence of the weak * -limit of µ s1...sm as s i → t i , s i ∈ Θ , and s i ≤ t i . We take such a weak * limit as the definition of µ t1...tm . Clearly µ t1...tm satisfies (8.12) and, by construction, from (8.13), we deduce that for every f satisfying (8.10) and every pair of finite sequences t 1 , . . . , t m and s 1 , . . . , where V − is the left-continuous representative of V defined by (8.14). The continuity property (7.5) follows easily from (8.15) and from Lemma 2.4. For every finite sequence t 1 , . . . , t m in Θ with t 1 < · · · < t m we have Passing to the limit as k → ∞, we obtain whenever t 1 , . . . , t m ∈ Θ . This restriction can be removed by an approximation argument, and this proves (8.4). The compatibility condition (7.2) for µ k implies that for every f ∈ C hom (X×Ξ h ×R) and every pair of finite sequences s 1 , . . . , s n and t 1 , . . . , t m in [0, T ] with s 1 < · · · < s n , t 1 < · · · < t m , and {s 1 , . . . , s n } ⊂ {t 1 , . . . , t m } . Passing to the limit as k → ∞, we obtain f, µ s1...sn = f • π t1...tm s1...s h , µ t1...tm , whenever s i and t j belong to Θ . This restriction can be removed by an approximation argument, therefore µ ∈ SGY ([0, T ], X; Ξ).
We conclude this section by proving the lower semicontinuity of the h-variation. for every finite sequence t 1 , . . . , t m in D with t 1 < · · · < t m . Then for every positively one-homogeneous function h : Ξ → [0, +∞) satisfying the triangle inequality.
Proof. Let us fix h. For every finite sequence t 1 , . . . , t m in D with t 1 < · · · < t m we have Since h is continuous (Remark 8.7), passing to the limit as k → ∞ we obtain whenever t 1 , . . . , t m ∈ D . The same inequality can be proved when t 1 , . . . , t m ∈ [0, T ] by an approximation argument, thanks to left continuity. The conclusion is obtained by taking the supremum with respect to t 1 , . . . , t m .

Weak * derivatives of systems with bounded variation
In this section we introduce the notion of weak * derivative of a compatible system of generalized Young measures on the time interval [0, T ], with T > 0 , and prove that, if Var(µ; 0, T ) < +∞, then the weak * derivative exists at almost every t ∈ [0, T ].
The following theorem is the main result of this section. Step 1. Boundedness of the difference quotients. By Remark 2.9 and by (3.1) for every t 1 , t 2 ∈ [0, T ], with t 1 < t 2 , we have Let t 0 ∈ [0, T ] be a point where the derivative of V exists. By the previous inequality we have that q tt0 (µ tt0 ) * and q t0t (µ t0t ) * are bounded uniformly with respect to t. By the separability of C hom (X×Ξ×R) there exists a countable dense subset F of the set C hom △ (X×Ξ×R) introduced in Definition 2.5. Therefore, since F is dense in C hom (X×Ξ×R) (see Lemma 2.7), to prove the existence of the weak * derivative of µ at t 0 it is enough to show that for every f ∈ F .
Step 2. Some auxiliary functions. In order to prove (9.7), let us fix f ∈ F and let τ i be a countable dense sequence in [0, T ]. For every i we define Let us prove that ϕ f i has bounded variation. Let us fix t 1 , t 2 ∈ [0, T ], with t 1 < t 2 . We consider first the case t 1 < τ i < t 2 . By the compatibility condition (7.2) we have using again (7.2) we obtain The same inequality can be proved when τ i ≤ t 1 or τ i ≥ t 2 . By (3.1), (3.2), (8.1), and (9.6) we conclude that We now prove that for every t 1 , t 2 ∈ [0, T ], with t 1 < t 2 , we have We consider first the case t 1 < τ i < t 2 . By (7.2) and (9.8) we have . From the triangle inequality and from (7.2) we get Step 5. Estimate from above. To prove the opposite inequality we show that for every t 1 , t 2 , with 0 < t 1 < t 2 < T , such that W is continuous at t 1 or t 2 . We prove (9.21) only when W is continuous at t 1 , the other case being analogous. For every ε > 0 there exists i such that τ i < t 1 and is measurable on [0, T − ε k ]. Since it converges to t → f,μ t for a.e. t ∈ [0, T ], we conclude that this function is measurable on [0, T ]. The same property can be proved for an arbitrary f ∈ C hom (X×Ξ×R) by approximation, thanks to Lemma 2.4. By (9.17) and (9.24) we have | f,μ t | ≤Ẇ (t) f hom for every f ∈ F . The same inequality holds for any f ∈ C hom (X×Ξ×R) by the density of F (see Lemma 2.7). SinceẆ is integrable, this concludes the proof of the integrability of t → f,μ t on [0, T ].

Absolute continuity
In this section we introduce the notion of absolutely continuous system of generalized Young measures on the time interval [0, T ], with T > 0 , and prove that for these systems the h-variation can be computed using the weak * derivative by the formula If t → u(t) is an absolutely continuous function from [0, T ] into L r (X; Ξ) for some r > 1 , then the derivativeu(t), defined as the strong L r limit of the difference quotients, exists at a.e. t ∈ [0, T ] (see, e.g., [5,Appendix]). By Remark 9.5 it follows that, if µ is the compatible system of generalized Young measures associated with t → u(t) according to (7.1), thenμ t = δu (t) for a.e. t ∈ [0, T ], and (10.1) follows from the classical theory (see, e.g., [5,Appendix]).
If t → p(t) is an absolutely continuous function with values in M b (X; Ξ), then the derivativeṗ(t), defined as the weak * limit of the difference quotients, exists at a.e. t ∈ [0, T ] (see [9,Appendix]). This is not enough to guarantee thatμ t = δṗ (t) for a.e. t ∈ [0, T ] when µ is the compatible system of generalized Young measure associated with t → p(t) (see Remark 9.5). Therefore, in this case (10.1) cannot be obtained directly from known results. Proof. Let W be defined by (9.12). By Remark 10.3 W is absolutely continuous on [0, T ]. By Remark 8.7 the function f (x, ξ, η) := h(ξ) belongs to C hom △ (X×Ξ×R). Therefore, we can add this function to the set F introduced in Step 1 of the proof of Theorem 9.7 and we can consider the corresponding function ω h : [0, W (T )] → R defined by (9.16). By (9.21) and (9.24) we have h(ξ 2 − ξ 1 ), µ t1t2 (x, ξ 1 , ξ 2 , η) ≤