Journal of Convex Analysis Volume 2 (1995), No.1/2, 345–357

Extraction of a “Good” Subsequence From a Bounded Sequence of Integrable Functions

Mohammed Saadoune

Département de Mathématiques, Université Ibnou Zohr, Lot. Addakhala B.P. 28/S, Agadir, Marocco.

Michel Valadier

Département de Mathématiques — Case 051, Université Montpellier II, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France. e-mail: lacan@math.univ-montp2.fr

Received 9 June 1994 Revised manuscript received 21 September 1994

Dedicated to R. T. Rockafellar on his 60th Birthday

For a uniformly integrable sequence, the Young measures allow to precise the Dunford-Pettis theorem: there exists a subsequence and two complementary subsets above which one has strong convergence and “pure” weak L1-convergence. For a bounded sequence in L1, the “biting lemma” permits the extraction of a subsequence presenting, besides the foregoing behaviors, concentration of mass. This structural result allows us to prove very quickly some known results.

Keywords : Bounded sequences in L1, biting lemma, Young measures, uniform integrability

1991 Mathematics Subject Classification: 46E30, 28A20

1. Introduction

We consider functions belonging to L^{1}(Ω, μ; IR^{d}), where (Ω, F, μ) is a measured space with μ a positive bounded measure. Bounded sequences in L^{1} may have weak limit points in the bidual (L^{1})′′ — identified to (L^{∞})′ — out of L^{1}. This appears when uniform integrability (briefly UI) does not hold: see J. Diestel [10, p.45]. The decomposition of such elements in the sum of a function of L^{1} and of a non null singular linear form ℓ_{s} (see on this subject K. Yosida & E. Hewitt [33], [9, Ch.VIII]) has been very efficient in the study of convex integral functionals: at least four papers by R.T. Rockafellar [21–24], several Russian works, among them V.L. Levin [18], Ioffe-Tihomirov [15] (this last one has been resumed using the compact stonian spectrum of L^{∞} in [29]! For a strange singular linear form see [30]). Then, the “biting lemma” proved several times: by V.F. Gaposhkin [13], Brooks-Chacon [7], Acerbi-Fusco [1], M. Słaby [26], L. Egghe [12], Ball-Murat [6], C. Castaing [8], has showed its efficiency and P.-L. Lions [19] defined the concentration (of mass).

ISSN 0944-6532 / $ 2.50 ◯^{c} Heldermann Verlag

346 M. Saadoune, M. Valadier / Extraction of a “good” subsequence

Other interesting and useful weak limit points come from Young measures theory. This notion, introduced by L.C. Young [34] in 1937 to define generalized curves (and then generalized surfaces), presents a renewal of interest these last years, specially with the works of E.J. Balder [3–4] and J.M. Ball [5]. When the bounded sequence (u_{n})_{n∈IN} is UI, there exists a subsequence (u′_{n})_{n} whose associated Young measures converge and the limit Young measure, τ, has a disintegration (τ_{x})_{x∈Ω} such that the function ^{1} x ↦ bar(τ_{x}) is the weak limit of u′_{n} ([31, Th.19 p.169], [32, Th.9]). Moreover τ contains some precise information about the possible asymptotic oscillatory behavior of the u′_{n}. When the sequence (u_{n})_{n} is not UI it may happen that all subsequences are not UI (think of (n 1_{[0,1/n]})_{n≥1} on [0, 1] equipped with the Lebesgue measure). If ℓ_{s} ≠ 0 is the singular part of a weak limit point ℓ ∉ L^{1}, there exists a sequence (B_{p})_{p} in F decreasing to ∅ such that ∀p, B_{p} carries ℓ_{s} (see [9, VIII.9 p.239]). But, while Young measures allow to work with subsequences, here only a filter on IN finer than the Fréchet filter^{2} gives the σ((L^{∞})′, L^{∞}) convergence u_{n} → ℓ. The biting lemma does not give a precise limit but allows the extraction of a subsequence presenting concentration of mass on sets B_{p} decreasing to ∅. Combining the biting lemma and Young measures techniques, we will prove a structural result (Theorem 4.5) which roughly speaking says: there exists a subsequence (u′_{n})_{n} and a partition of Ω in two measurable sets, one above which one has strong convergence, the other above which the functions oscillates in such a manner that^{3} not any subsequence of (u′_{n})_{n} strongly converges on any non negligible subset and thirdly the possible non uniform integrability gives rise to concentration of mass on some B_{p} sets. The following example seems typical of the general case. Let

u_{n}(x, y) = n 1_{[0, \frac{1}{n}]}(x) + 1_{[0, \frac{1}{2} ]}(y) sin(n x)

on Ω = [0, 1]^{2} equipped with the Lebesgue measure. This example is studied in Section 4. In Section 5, Theorem 4.5 will be used to prove quickly some theorems by Hewitt-Stromberg, H.-A. Klei and Klei-Miyara. One word about the infinite dimensional case. Surely our result does not extend to a bounded sequence in L^{1}(Ω, μ; E). But one can notice some formulation analogy with the parametric versions of Rosenthal’s theorem by M. Talagrand [27, Theorem 1] and Diestel-Ruess-Schachermayer [11, Theorem 2.4] : there are different behaviors on complementary subsets of Ω.

2. Notations and backgrounds

For u ∈ L^{1}(Ω, μ; IR^{d}) (to be correct one should write u ∈ L^{1}(Ω, μ; IR^{d}) ), the Young measure associated to u is the measure on Ω×IR^{d} which is the image of μ by x ↦ (x, u(x)); it is denoted by ν. If one considers functions u_{n}, their Young measures will be denoted by ν^{n}. A general Young measure is a positive measure on Ω × IR^{d}, usually denoted by τ,

1 bar(τ_{x}) denotes the barycenter of τ_{x}. 2 The extraction of a convergent subsequence is usually impossible. 3 This will be called “pure” weak L^{1}-convergence. See section 4. such that ∀A ∈ F, τ(A × IR^{d}) = μ(A) (i.e. the projection of τ on Ω is μ). The set of all Young measures on Ω × IR^{d} is denoted by Y(Ω, μ; IR^{d}). A Carathéodory integrand is a real function on Ω × IR^{d} separately measurable on Ω and continuous on IR^{d}. The narrow topology on Y(Ω, μ; IR^{d}) is the weakest topology making continuous the maps τ ↦ ∫_{Ω×IR^{d}} ψ(x, ξ) dτ(x, ξ) where ψ runs through the set of all bounded Carathéodory integrands. If the Young measures (ν_{n})_{n} are associated to the functions u_{n}, they converge narrowly to τ iff

∫_{Ω} ψ(x, u_{n}(x)) μ(dx) =: ∫_{Ω×IR^{d}} ψ dν^{n} ⟶ ∫_{Ω×IR^{d}} ψ dτ  (2.1)

for any bounded Carathéodory integrand ψ. The convergence in (2.1) still holds when ψ is a Carathéodory integrand with linear growth (i.e. |ψ(x, ξ)| ≤ α(x) + K∥ξ∥, where α ∈ L^{1}) as soon as (u_{n})_{n} is UI. More generally there is a lower semi-continuity theorem ([3], [31, Th.16 p.166], [32, Th.6]): if the Young measures ν^{n} associated to u_{n} converge to τ, if ψ Ω × IR^{d} → \overline{IR} is measurable in (x, ξ) and l.s.c. in ξ and if the sequence of negative parts ((ψ(., u_{n}(.))^{−})_{n} is UI, then

∫_{Ω×IR^{d}} ψ dτ ≤ lim inf_{n→+∞} ∫_{Ω} ψ(x, u_{n}(x)) μ(dx) .  (2.2)

For ψ ≥ 0, this is an easy preliminary result ([31, Th.7 p.161], [32, Lemma 5]). If u_{n} (n ∈ IN∪{+∞}) are measurable functions, one has ([31, Prop.6 p.160], [32, Prop.1]):

ν^{n} → ν^{∞} narrowly ⟺ u_{n} → u_{∞} in measure .  (2.3)

If a sequence (τ^{n})_{n} of Young measures satisfies

sup_{n} ∫_{Ω×IR^{d}} ∥ξ∥ dτ^{n}(x, ξ) < +∞ ,

— this happens when the Young measures are associated to the functions of a bounded sequence in L^{1} — there exists a narrowly convergent subsequence (more generally there is an extension of the Prokhorov theorem to Young measures: [3], [31, Th.11 p.162], [32, Th.7]). The disintegration of a Young measure τ is a measurable family of probabilities on IR^{d}, (τ_{x})_{x∈Ω}, characterized by: for any ψ Ω × IR^{d} → \overline{IR} measurable and ≥ 0 or τ-integrable,

∫_{Ω×IR^{d}} ψ dτ = ∫_{Ω}[∫_{IR_{d}} ψ(x, ξ) τ_{x}(dξ)] μ(dx) .

If ν is associated to u, ν_{x} = δ_{u(x)}, where δ_{u(x)} denotes the Dirac mass at u(x). If ∫_{Ω×IR^{d}} ∥ξ∥ dτ(x, ξ) < +∞, then for μ-almost every x, τ_{x} is of first order, that is satisfies ∫_{IR}_{d} ∥ξ∥ τ_{x}(dξ) < +∞, hence has a barycenter: bar(τ_{x}) := ∫_{IR}_{d} ξ τ_{x}(dξ).

3. Preliminary results

Definition 3.1. The modulus of uniform integrability of H.P. Rosenthal, η((u_{n})), of the sequence (u_{n})_{n}, is:

η((u_{n})) := lim_{ε→0, ε>0} [ sup{∫_{A} ∥u_{n}(x)∥ μ(dx) : n ∈ IN, μ(A) ≤ ε} ] .

The sequence (u_{n})_{n} is UI iff η((u_{n})) = 0.

Lemma 3.2. Let (u_{n})_{n} be a bounded sequence in L^{1}(Ω, μ; IR^{d}). Then

η((u_{n})) = lim_{t→+∞} [ sup_{n∈IN} ∫_{{∥u_{n}(.)∥>t}} ∥u_{n}(x)∥ μ(dx) ] .

Proof. The two limits are decreasing limits, hence infima. Let us denote η_{1} the modulus of the statement and M := sup_{n} ∥u_{n}∥_{L^{1}}. By definition of η_{1}, ∀δ > 0 one has ∀t, ∃n such that

∫_{{∥u_{n}(.)∥>t}} ∥u_{n}∥ dμ > η_{1} − δ .

Let ε > 0. There exists t large enough such that \frac{M}{t} is ≤ ε. Then if n corresponds

to such a t, A := {∥u_{n}(.)∥ > t} satisfies μ(A) ≤ ε and ∫_{A} ∥u_{n}∥ dμ ≥ η_{1} − δ. Hence η((u_{n})) ≥ η_{1} − δ, and this holds for all δ > 0, so η((u_{n})) ≥ η_{1}. Let us prove the converse inequality. Let t > 0 and δ > 0 be given. Take ε > 0 small enough such that t ε < δ. By definition of η((u_{n})) there exist A and n such that μ(A) ≤ ε and ∫_{A} ∥u_{n}∥ dμ > η((u_{n})) − δ. Then

∫_{{∥u_{n}∥>t}} ∥u_{n}∥ dμ = ∫_{A} ∥u_{n}∥ dμ − ∫_{A∩{∥u_{n}∥≤t}} ∥u_{n}∥ dμ + ∫_{{∥u_{n}∥>t}\A} ∥u_{n}∥ dμ

≥ ∫_{A} ∥u_{n}∥ dμ − ∫_{A∩{∥u_{n}∥≤t}} ∥u_{n}∥ dμ
≥ (η((u_{n})) − δ) − t ε
≥ η((u_{n})) − 2 δ .

So one has η_{1} ≥ η((u_{n})) − 2 δ for all δ > 0, hence η_{1} ≥ η((u_{n})).

The expression of η((u_{n})) given in Lemma 3.2 is used in Słaby’s proof of the biting lemma (reproduced in L. Egghe [12, VIII.1.17 pp.303–305] and [31, Th.23 pp.173–174]). We reproduce this proof here in order to be able to refer to its notations.

Theorem 3.3. (Biting lemma) Let (u_{n})_{n∈IN} be a bounded sequence in L^{1}(Ω, μ; IR^{d}). There exist a subsequence (u_{n_{k}})_{k∈IN} and a sequence (B_{p})_{p∈IN} in F decreasing to ∅ such that the sequence (1_{B_{k}^{c}} u_{n_{k}})_{k∈IN} is UI.

Proof. Denote, for i ∈ IN,

g_{i}(t) := sup_{n≥i} ∫_{{∥u_{n}(.)∥>t}} ∥u_{n}(x)∥ μ(dx) .

The functions g_{i} are decreasing with values in [0, +∞[. Hence, lim_{t→+∞} g_{i}(t) exists. By Lemma 3.2, η((u_{n})) = lim_{t→+∞} g_{0}(t). There exists a sequence increasing to +∞, (t_{q})_{q≥1} such that ∀q, g_{0}(t_{q}) ≤ η((u_{n})) + \frac{1}{q}. Since ∀i, \{u_{0}, …, u_{i−1}\} is UI, it is easy to check that lim_{t→+∞} g_{i}(t) = η((u_{n})). Hence ∀i, ∀t, g_{i}(t) ≥ η((u_{n})), so there exists a strictly increasing sequence (m_{q})_{q} such that

∫_{\{∥u_{mq} (.)∥>t_{q}\}} ∥u_{m_{q}}(x)∥ μ(dx) ≥ η((u_{n})) − \frac{1}{q} .  (3.1)

Let A_{p} = {∥u_{m_{p}}(.)∥ > t_{p}}. Then t_{p} μ(A_{p}) ≤ sup_{n∈IN} ∥u_{n}∥_{L^{1}} implies μ(A_{p}) → 0. Now we show that (1_{A_{q}^{c}} u_{m_{q}})_{q∈IN} is UI. Let

g(t) := sup_{q∈IN} ∫_{A_{q}^{c}∩{∥u_{mq} (.)∥>t}} ∥u_{m_{q}}(x)∥ μ(dx) .

We have to prove that g(t) → 0 when t → +∞. One has

g(t_{j}) = sup_{q>j} ∫_{{t_{j}<∥u_{mq} (.)∥≤t_{q}}} ∥u_{m_{q}}(x)∥ μ(dx)

= sup_{q>j}[∫_{{∥u_{mq} (.)∥>t_{j}}} ∥u_{m_{q}}(x)∥ μ(dx) − ∫_{{∥u_{mq} (.)∥>t_{q}}} ∥u_{m_{q}}(x)∥ μ(dx)]

≤ sup_{q>j}[g_{0}(t_{j}) − ∫_{{∥u_{mq} (.)∥>t_{q}}} ∥u_{m_{q}}(x)∥ μ(dx)]

≤ sup_{q>j}[(η((u_{n})) + \frac{1}{j}) − (η((u_{n})) − \frac{1}{q})]

≤ \frac{2}{j} .

This proves: t → +∞ ⇒ g(t) → 0. It remains to replace (A_{p})_{p} by a sequence decreasing to ∅ (or to a negligible set). There exists a strictly increasing sequence (l_{k})_{k} such that μ(A_{l_{k}}) ≤ 2^{−k}. Then B_{p} := ∪_{k≥p}A_{l_{k}} decreases to a negligible set. The subsequence u_{n_{k}} := u_{m_{l_{k}}} (or u_{m_{(l_{k})}}) has the required properties. The uniform integrability of (1_{B_{k}^{c}} u_{n_{k}})_{k∈IN} follows from the inclusion B_{k}^{c} ⊂ A_{l_{k}}^{c}.

4. Main results

Definition 4.1. Let u_{n} (n ∈ IN ∪ {+∞}) be functions in L^{1}(Ω, μ; IR^{d}).

1) One says that u_{n} converges strongly to u_{∞} if ∥u_{n} − u_{∞}∥_{L^{1}} → 0.

2) One says that u_{n} converges weakly to u_{∞} if

∀p ∈ L^{∞}, ∫_{Ω}⟨p, u_{n}⟩ dμ ⟶ ∫_{Ω}⟨p, u_{∞}⟩ dμ .

3) One says that u_{n} converges purely weakly to u_{∞} on W ∈ F if the restrictions u_{n}|_{W} converge weakly to u_{∞}|_{W} and, for any non negligible measurable subset A of W, not any subsequence of (u_{n}|_{A})_{n} converges strongly to u_{∞}|_{A}.

Remark 4.2. In 3) it is equivalent to say: not any subsequence of (u_{n}|_{A})_{n} converges in measure (to any function). For clarity we give first the simplified version of the general result in the case when the sequence is UI. All assertions are consequence of the Prokhorov theorem, of (2.3) and of (2.1) extended to Carathéodory integrands with linear growth.

Theorem 4.3. Let (u_{n})_{n} be a uniformly integrable sequence in L^{1}(Ω, μ; IR^{d}). There exist a subsequence (u′_{n})_{n}, a function u_{∞} ∈ L^{1}(Ω, μ; IR^{d}) and M ∈ F such that^{4}: 1b) u′_{n} converges weakly to u_{∞}, 1c) the restrictions u′_{n}|_{M} converge strongly to u_{∞}|_{M}, 1d) the restrictions u′_{n}|_{M^{c}} converge purely weakly to u_{∞}|_{M^{c}}.

2) There exists a Young measure τ, such that

M^{c} = {x ∈ Ω : ∫_{IR_{d}} ∥ξ − u_{∞}(x)∥ τ_{x}(dξ) > 0}

and such that, for any Carathéodory integrand ψ with linear growth (i.e. |ψ(x, ξ)| ≤ α(x) + K∥ξ∥, where α ∈ L^{1})

∫_{Ω} ψ(x, u′_{n}(x)) μ(dx) ⟶ ∫_{Ω×IR^{d}} ψ dτ .  (4.1)

Remark 4.4. We will not give the proof since the statement, except (4.1), is a particular case of Theorem 4.5. First versions of this statement appear in [31, Th.19 p.169] and [32, Th.9]. The novelty is the introduction of the partition (M, M^{c}).

Theorem 4.5. Let (u_{n})_{n} be a bounded sequence in L^{1}(Ω, μ; IR^{d}). There exist a subsequence (u′_{n})_{n}, a function u_{∞} ∈ L^{1}(Ω, μ; IR^{d}), a sequence (B_{p})_{p} decreasing to ∅ (if (u_{n})_{n} is UI, one may take ∀p, B_{p} = ∅) and M ∈ F such that: 1a) (1_{B_{n}^{c}} u′_{n})_{n} is UI 1b) ∀p, the restrictions u′_{n}|_{B_{p}^{c}} converge weakly to u_{∞}|_{B_{p}^{c}}, 1c) the restrictions u′_{n}|_{M} converge in measure to u_{∞}|_{M} and ∀p, the restrictions u′_{n}|_{(M\B_{p})}

converge strongly to u_{∞}|_{(M\B_{p})}, 1d) for any non negligible A ∈ F contained in W := Ω \ M, not any subsequence of

(u′_{n}|_{A})_{n} converges in measure to u_{∞}|_{A} (nor to another function).

4 Here 1a) of Theorem 4.5 is obviously satisfied.

1e) for any subsequence (u′′_{n})_{n} of (u′_{n})_{n}, η((u′′_{n})) = η((u_{n})).

2) There exists a Young measure τ whose disintegration is a measurable family of first order probabilities on IR^{d}, (τ_{x})_{x∈Ω}, such that τ_{x} is carried by the set Ls(u_{n}(x)) of limit points of the sequence (u_{n}(x))_{n}, such that for x ∈ W, τ_{x} is not a Dirac mass and bar(τ_{x}) = u_{∞}(x), such that for x ∈ M, τ_{x} = δ_{u_{∞}(x)} and such that, for any

u ∈ L^{1}(Ω, μ; IR^{d}), ∥u′_{n} − u∥_{L^{1}} converges to

\begin{aligned}η((u_{n}))+ & ∫_{Ω×IR^{d}} ∥ξ − u(x)∥ dτ(x, ξ) \\ & = η((u_{n})) + ∫_{W ×IR^{d}} ∥ξ − u(x)∥ dτ(x, ξ) + ∥1_{M} (u_{∞} − u)∥_{L^{1}} . \end{aligned}

Moreover for any bounded Carathéodory integrand ψ

∫_{Ω} ψ(x, u_{n}(x)) μ(dx) ⟶ ∫_{Ω×IR^{d}} ψ dτ .

Remark 4.6.

1) Assertion 1e) means that, for the subsequence (u′_{n})_{n}, the concentration of mass (non null iff η((u_{n})) is > 0) equals the maximum of mass concentrations among all the subsequences of (u_{n})_{n}. Maybe the following idea is a good motivation: often to prove an implication where there is a regularity assumption, it is useful to be able to extract an irregular subsequence. In this line the second author has already given the following result ([31, Th.20 p.169], [32, Th.9]): if (u_{n})_{n} converges weakly (hence is UI) and does not converge strongly, there exists a subsequence whose associated Young measures converge to a Young measure τ not associated to a function.

2) The norm ∥u′_{n} − u∥_{L^{1}} equals ∫_{Ω×IR^{d}} ψ dν^{n} where ψ is the integrand ψ(x, ξ) := ∥ξ − u(x)∥. This is a Carathéodory integrand with linear growth (for some application of these integrands see [20]).

Example 4.7. Here is an example where B_{p} necessarily bites both M and W. Moreover this example seems to be a rather general one. Let Ω = [0, 1]^{2} with the Lebesgue measure, d = 1 and

u_{n}(x, y) = n 1_{[0, \frac{1}{n}]}(x) + 1_{[0, \frac{1}{2} ]}(y) sin(n x) .

Then u_{∞} = 0, W = [0, 1] × [0, \frac{1}{2}] and ^{5} M = [0, 1] × ]\frac{1}{2}, 1]. Roughly speaking, mass concentration appears on {0} × [0, 1]. In any way the B_{p} are chosen, they meet M and W: since (1_{B_{k}^{c}} u_{n_{k}})_{k} is UI, ∥1_{[0,1/n_{k}]×[0,1]} 1_{B_{k}^{c}} u_{n_{k}}∥_{L^{1}} tends to 0 when k → +∞, hence

∫∫_{[0,1/n_{k}]×[0,1]} 1_{B_{k}^{c}}(x, y) [n_{k} + 1_{[0, \frac{1}{2} ]}(y) sin(n_{k} x)] dx dy ⟶ 0 .

5 One can check (see [32, Th.4]), that for (x, y) ∈ W, τ_{(x,y)} is the probability on IR with

the density ξ ↦ (π \sqrt{1 − ξ^{2}} )^{−1} on ]−1, 1[.

Thus n_{k} μ(B_{k}^{c} ∩ ([0, 1/n_{k}] × [0, 1])) → 0 and

\frac{μ(B_{k} ∩ ([0, 1/n_{k}] × [0, 1]))}{μ([0, 1/n_{k}] × [0, 1])} ⟶ 1 .

Consequently

\frac{μ(B_{k} ∩ W)}{μ([0, 1/n_{k}] × [0, 1/2])} ⟶ 1 and \frac{μ(B_{k} ∩ M)}{μ([0, 1/n_{k}] ×]1/2, 1])} ⟶ 1 .

Hence B_{k} covers a big part of W ∩([0, 1/n_{k}]×[0, 1]) and a big part of M ∩([0, 1/n_{k}]×[0, 1]). Note that sin(n y) in place of sin(n x) would give the same W and τ despite the change of directions of the “waves.” When Ω is an open subset of IR^{N}, there exist more powerful tools than Young measures: see G. Allaire [2] and L. Tartar [28].

Proof of Theorem 4.5

1) Let (B_{p})_{p} denote the sequence of the proof of the biting lemma and (u′_{n})_{n} the subsequence denoted by (u_{n_{k}})_{k} in that proof. Let (u′′_{n})_{n} be a further subsequence. A priori η((u′′_{n})) ≤ η((u_{n})). Using the notations of the proof of Theorem 3.3, u′′_{n} is some u_{m_{q}}. For all t, there exists q large enough such that t_{q} ≥ t and such that the function u_{m_{q}} is some u′′_{n}. By (3.1) η((u_{n})) is approximated with a gap less than 1/q by

∫_{{∥u′′_{n}(.)∥>t_{q}}} ∥u′′_{n}(x)∥ μ(dx)

hence a fortiori by

∫_{{∥u′′_{n}(.)∥>t}} ∥u′′_{n}(x)∥ μ(dx) .

This proves η((u′′_{n})) = η((u_{n})).

2) Since (u_{n})_{n} is bounded, thanks to the Prokhorov theorem extended to Young measures, one may assume that the Young measures associated to the u′_{n} converge to a Young measure τ. Thanks to the lower semi-continuity result (2.2),

∫_{Ω×IR^{d}} ∥ξ∥ dτ(x, ξ) ≤ sup_{n} ∥u_{n}∥_{L^{1}} .  (4.2)

Then μ-a.e. the disintegration τ_{x} has a barycenter bar(τ_{x}) and, with u_{∞}(x) := bar(τ_{x}), u_{∞} is integrable. Moreover τ_{x} is carried by Ls(u_{n}(x)) ([31, Prop.5 p.159]). For a fixed p, since u′_{n}|_{B_{p}^{c}} is UI, u′_{n}|_{B_{p}^{c}} → u_{∞}|_{B_{p}^{c}} weakly (this is again a consequence of (2.1) extended to Carathéodory integrands with linear growth. For a reference see [31, Th.19 p.169]; see also the proof of Th.9 in [32]). The set W of all x where τ_{x} is not a Dirac mass, is {x ∈ Ω : ∫_{IR}_{d} ∥ξ − u_{∞}(x)∥ τ_{x}(dξ) > 0}. By the Fubini theorem it is measurable. Let M := Ω \ W. Note that μ-almost everywhere on M, τ_{x} = δ_{u_{∞}(x)}. From (2.3), u′_{n}|_{M} → u_{∞}|_{M} in measure and, for any non negligible A contained in W, not any subsequence of (u′_{n})_{n} converges in measure on A. Thanks to UI and to the Lebesgue-Vitali theorem, strong convergence holds on M \ B_{p}.

M. Saadoune, M. Valadier / Extraction of a “good” subsequence 353

3) Let u ∈ L^{1}(Ω, μ; IR^{d}) and ε > 0. Thanks to (4.2), for p large enough one has:

∫_{B_{p}×IR^{d}} ∥ξ − u(x)∥ dτ(x, ξ) ≤ ∫_{B_{p}×IR^{d}} ∥ξ∥ dτ(x, ξ) + ∫_{B_{p}×IR^{d}} ∥u(x)∥ dτ(x, ξ)

= ∫_{B_{p}×IR^{d}} ∥ξ∥ dτ(x, ξ) + ∫_{B_{p}} ∥u(x)∥ μ(dx)

≤ \frac{ε}{4} .

Let ν^{n} denotes the Young measure associated to u′_{n}. On B_{p}^{c} × IR^{d} the integrand ψ defined by ψ(x, ξ) := ∥ξ − u(x)∥ is Carathéodory with linear growth and the u′_{n} are UI on B_{p}^{c}, so by (2.1)

∥1_{B_{p}^{c}} (u′_{n} − u)∥_{L^{1}} = ∫_{B_{p}^{c}×IR^{d}} ∥ξ − u(x)∥ dν^{n}(x, ξ)

converges to

∫_{B_{p}^{c}×IR^{d}} ∥ξ − u(x)∥ dτ(x, ξ) .

Hence, p being fixed, for n large enough,

|∥1_{B_{p}^{c}} (u′_{n} − u)∥_{L^{1}} − ∫_{Ω×IR^{d}} ∥ξ − u(x)∥ dτ(x, ξ)| ≤ \frac{ε}{2} .

As for ∥1_{B_{p}} (u′_{n} − u)∥_{L^{1}}, firstly, if p is large enough,

|∥1_{B_{p}} (u′_{n} − u)∥_{L^{1}} − ∥1_{B_{p}} u′_{n}∥_{L^{1}} | ≤ ∥1_{B_{p}} u∥_{L^{1}} ≤ \frac{ε}{4} .

It remains to show that, for n large enough, one has

|∥1_{B_{p}} u′_{n}∥_{L^{1}} − η((u_{n}))| ≤ \frac{ε}{4} .

Recall (notations of the proof of Theorem 3.3) that B_{p} = A_{l_{p}} ∪ A_{l_{p+1}} ∪ … and that, from (3.1),

∫_{A_{l_{k}}} ∥u_{m_{l_{k}}} (x)∥ μ(dx) ≥ η((u_{n})) − \frac{1}{l_{k}} ,

hence

∀k ≥ p, ∫_{B_{p}} ∥u_{m_{l_{k}}} (x)∥ μ(dx) ≥ η((u_{n})) − \frac{1}{l_{p}} .

If one has chosen p such that l_{p} ≥ 4/ε, then, for n large enough, ∥1_{B_{p}} u′_{n}∥_{L^{1}} ≥ η((u_{n}))− \frac{ε}{4}. Finally, setting

η((u_{n}); δ) := sup{∫_{A} ∥u_{n}(x)∥ μ(dx) : n ∈ IN, μ(A) ≤ δ} ,

one has, as soon as μ(B_{p}) ≤ δ,

sup_{n} ∫_{B_{p}} ∥u′_{n}∥ dμ ≤ η((u_{n}); δ) .

Consequently, if one has chosen δ small enough such that η((u_{n}); δ) ≤ η((u_{n})) + \frac{ε}{4} and

then p large enough such that μ(B_{p}) ≤ δ, one has

∀n, ∫_{B_{p}} ∥u′_{n}∥ dμ ≤ η((u_{n})) + \frac{ε}{4} .

Finally, for a given ε, a good choice of p is possible and one gets, for n large enough,

|∥u′_{n} − u∥_{L^{1}} − [η((u_{n})) + ∫_{Ω×IR^{d}} ∥ξ − u(x)∥ dτ(x, ξ)]| ≤ ε .

5. Applications

Theorem 5.1. (Hewitt & Stromberg [14, Th.13.47 p.208]) Let u_{n} (n ∈ IN ∪ {+∞}) be functions of L^{1}(Ω, μ; IR^{d}). Suppose that u_{n} converge in measure to u_{∞} and that

lim sup_{n→+∞} ∥u_{n}∥_{L^{1}} ≤ ∥u_{∞}∥_{L^{1}} .  (5.1)

Then u_{n} converge strongly to u_{∞}.

Proof. If u_{n} does not converge strongly to u_{∞}, one may, extracting a further subsequence, assume ∀n, ∥u_{n} − u_{∞}∥_{L^{1}} ≥ ε > 0. Then there exists a subsequence satisfying part 1) of Theorem 4.5. Necessarily W is negligible and the function u_{∞} is the same. From part 2) of Theorem 4.5 applied with u = 0,

∥u′_{n}∥_{L^{1}} ⟶ η((u_{n})) + ∥u_{∞}∥_{L^{1}} .

By (5.1), η((u_{n})) = 0, hence UI holds and, thanks to Lebesgue-Vitali, u′_{n} converges strongly to u_{∞}, which gives a contradiction.

Theorem 5.2. (Klei [16, Th.6]) Let (u_{n})_{n} be a bounded sequence in L^{1}(Ω, μ; IR^{d}) and u ∈ L^{1}(Ω, μ; IR^{d}). Suppose lim sup ∥u_{n} − u∥_{L^{1}} ≤ η((u_{n})). Then there exists a subsequence (u′_{n})_{n} with the following properties: u′_{n} converges in measure to u, η((u_{n})) = lim ∥u′_{n}−u∥_{L^{1}} and, for any subsequence (u′′_{n})_{n} of (u′_{n})_{n}, η((u′′_{n})) = η((u_{n})).

Proof. Part 1) of Theorem 4.5 gives a subsequence (u′_{n})_{n}, u_{∞} et W. And for any subsequence (u′′_{n})_{n} of (u′_{n})_{n}, η((u′′_{n})) = η((u_{n})). From part 2) of Theorem 4.5,

∥u′_{n} − u∥_{L^{1}} ⟶ η((u_{n})) + ∫_{W ×IR^{d}} ∥ξ − u(x)∥ dτ(x, ξ) + ∥1_{M} (u_{∞} − u)∥_{L^{1}} ,

hence

η((u_{n})) + ∫_{W ×IR^{d}} ∥ξ − u(x)∥ dτ(x, ξ) + ∥1_{M} (u_{∞} − u)∥_{L^{1}} ≤ η((u_{n})) .

So ∥u′_{n} − u∥_{L^{1}} → η((u_{n})) and

∫_{W ×IR^{d}} ∥ξ − u(x)∥ dτ(x, ξ) + ∥1_{M} (u_{∞} − u)∥_{L^{1}} = 0 .  (5.2)

For x ∈ W, τ_{x} is not a Dirac mass, so _{∫_{IR_{d}}}∥ξ − u(x)∥ τ_{x}(dξ) is > 0. By (5.2) and the formula

∫_{W ×IR^{d}} ∥ξ − u(x)∥ dτ(x, ξ) = ∫_{W} [∫_{IR_{d}} ∥ξ − u(x)∥ τ_{x}(dξ)] μ(dx) ,

the set W is μ-negligible, which implies 1_{M} = 1_{Ω} μ-a.e. By (5.2) we have also ∥1_{M} (u_{∞} − u)∥_{L^{1}} = 0. Hence u = u_{∞} μ-a.e. and u′_{n} → u in measure.

Theorem 5.3. (Klei-Miyara [17]) Let (u_{n})_{n} be a bounded sequence in L_{+}^{1}(Ω, μ; IR) such that ∫_{Ω} u_{n} dμ → ℓ. Then

∫_{Ω}(lim inf u_{n}) dμ ≤ ℓ − η((u_{n})) .

Remark 5.4. This means that in the Fatou lemma, after extraction of a subsequence whose integrals converge^{6}, the gap is at least η((u_{n})), the modulus of the subsequence. One can observe that the negative parts of the u_{n} are UI, so the possible concentration of mass concerns only positive mass.

Proof. Let (u′_{n})_{n}, u_{∞}, M, W and τ given by Theorem 4.5. Here τ_{x} is carried by [0, +∞[ and τ is carried by Ω × [0, +∞[, so in the following formulas ξ replaces |ξ|. Part 2) of Theorem 4.5 applied with u = 0 gives

ℓ = lim ∫_{Ω} u′_{n} dμ = η((u_{n})) + ∫_{W ×IR} ξ dτ(x, ξ) + ∫_{M} u_{∞} dμ
= η((u_{n})) + ∫_{W} [∫_{IR} ξ τ_{x}(dξ)] μ(dx) + ∫_{M} u_{∞} dμ .

On M, u_{n} → u_{∞} in measure, hence lim inf u_{n}(x) ≤ u_{∞}(x). For x ∈ W, μ-a.e. τ_{x} is carried by Ls(u_{n}(x)), hence by [lim inf u_{n}(x), +∞[. Consequently^{7} lim inf u_{n}(x) ≤ ∫_{IR} ξ τ_{x}(dξ) and

∫_{Ω}(lim inf u_{n}) dμ ≤ ∫_{W} [∫_{IR} ξ τ_{x}(dξ)] μ(dx) + ∫_{M} u_{∞} dμ = ℓ − η((u_{n})) .

This proves the expected inequality.

6 Without extraction the result does not hold: let Ω = [0, 1], u_{n} = 0 if n is even, u_{n} =

n 1_{[0,1/n]} if n is odd. Then lim inf(∫ u_{n}) − ∫ (lim inf u_{n}) = 0 but η((u_{n})) = 1. 7 This holds for all x ∈ Ω, but we give a direct argument in the case x ∈ M, because it is

so easy and W = ∅ is possible !

356 M. Saadoune, M. Valadier / Extraction of a “good” subsequence

References

[1] E. Acerbi, N. Fusco : Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal., Vol. 86, 1984, 125–145.

[2] G. Allaire : Homogenization and two-scale convergence, SIAM J. Math. Anal., Vol. 23, 1992, 1482–1518.

[3] E.J. Balder : A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optim., Vol. 22, 1984, 570–598.

[4] E.J. Balder : On weak convergence implying strong convergence in L_{1}-spaces, Bull. Austral. Math. Soc., Vol. 33, 1986, 363–368.

[5] J.M. Ball : A version of the fundamental theorem for Young measures in: Partial differential equations and continuum models of phase transitions, M. Rascle, D. Serre, M. Slemrod, Lecture Notes in Physics, Vol. 344, Springer Verlag, Berlin 1989, 207–215.

[6] J.M. Ball, F. Murat : Remarks on Chacon’s biting lemma, Proc. Amer. Math. Soc., Vol. 107, 1989, 655–663.

[7] J.K. Brooks, R.V. Chacon : Continuity and compactness of measures, Adv. in Math., Vol. 37, 1980, 16–26.

[8] C. Castaing : Méthode de compacité et de décomposition applications: minimisation, convergence des martingales, lemme de Fatou multivoque, Ann. Mat. Pura Appl., Vol. 164, 1993, 51–75.

[9] C. Castaing, M. Valadier : Convex Analysis and Measurable Multifunctions, Lecture Notes in Math., Vol. 580, Springer-Verlag, Berlin, 1977.

[10] J. Diestel : Uniform integrability: an introduction, School on Measure Theory and Real Analysis, Grado (Italy), October 14-25, 1991, Rend. Istit. Mat. Univ. Trieste, Vol. XXIII, 1991, 41–80.

[11] J. Diestel, W.M. Ruess, W. Schachermayer : Weak compactness in L^{1}(μ, X), Proc. Amer. Math. Soc., Vol. 118, 1993, 447–453.

[12] L. Egghe : Stopping time techniques for analysts and probabilists, Cambridge University Press, Cambridge, 1984.

[13] V.F. Gaposhkin : Convergence and limit theorems for sequences of random variables, Theory Probab. Appl., Vol. 17-3, 1972, 379–400 (translated from Russian).

[14] E. Hewitt, K. Stromberg : Real and abstract analysis, Springer-Verlag, Berlin, 1965.

[15] A.D. Ioffe, V.M. Tihomirov : The minimization of integral functionals, Funktsional Anal. i Prilozhen., Vol. 3, 1969, 61–70 (in Russian. In English: Functional Anal. Appl. 3, 1969, 218–229).

[16] H.-A. Klei : Convergences faible, en mesure et au sens de Cesaro dans L^{1}(IR), C. R. Acad. Sci. Paris Sér. I, Vol. 315, 1992, 9–12.

[17] H.-A. Klei, M. Miyara : Une extension du lemme de Fatou, Bull. Sci. Math. 2^{ème} série, Vol. 115, 1991, 211–221.

[18] V.L. Levin : Convex analysis in spaces of measurable functions and its application to mathematics and economics, Nauka, Moscow, 1985.

[19] P.-L. Lions : The concentration-compactness principle in the Calculus of Variations. The locally compact case, Part 1, Ann. Inst. H. Poincaré Anal. Non Linéaire, Vol. 1, 1984, 109–145.

[20] L. Piccinini, M. Valadier : Uniform integrability and Young measures, Prépublication No. 03 (10 pages), Départ. de Math., Université de Montpellier, France, 1994, to appear: J. Math. Anal. Appl.

[21] R.T. Rockafellar : Weak compactness of level sets of integral functionals, Troisième Colloque d’Analyse Fonctionnelle (CBRM, Liège, Belgium, 1970), H.G. Garnir, 85–98, 1971.

[22] R.T. Rockafellar : Integrals which are convex functionals, II, Pacific J. Math., Vol. 39, 1971, 439–469.

[23] R.T. Rockafellar : Convex integral functionals and duality, Contributions to Nonlinear Functional Analysis, E. Zarantonello, Academic Press, New York, 215–236, 1971.

[24] R.T. Rockafellar : Integral functionals, normal integrands and measurable selections, Nonlinear Operators and the Calculus of Variations, J.P. Gossez, E.J. Lami Dozo, J. Mawhin,, L. Waelbroeck, Lecture Notes in Math., Vol. 543, Springer-Verlag, Berlin, 157–207, 1976.

[25] M. Saadoune : Lemme de Fatou multivoque. Ensembles tendus, Thèse de 3ème cycle, Montpellier, 1986.

[26] M. Słaby : Strong convergence of vector-valued pramarts and subpramarts, Probab. Math. Statist., Vol. 5, 1985, 187–196.

[27] M. Talagrand : Weak Cauchy sequences in L^{1}(E), Amer. J. Math., Vol. 106, 1984, 703–724.

[28] L. Tartar : H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, ProcṘoyṠocĖdinburgh Sect. A, Vol. 115, 1990, 193–230.

[29] M. Valadier : Intégration de convexes fermés, notamment d’épigraphes, inf-convolution continue, Rev. Française Informat. Recherche Opérationnelle, Vol. R-2, 1970, 57–73.

[30] M. Valadier : Une singulière forme linéaire singulière sur L^{∞}, Sém. Anal. Convexe, Vol. 17, 1987, 4.1–4.3.

[31] M. Valadier : Young measures, in: Methods of Nonconvex Analysis, A. Cellina, Lecture Notes in Math., Vol. 1446, Springer-Verlag, Berlin, 152–188, 1990.

[32] M. Valadier : A course on Young measures, Prépublication No. 08 (33 pages), Départ. de Math., Université de Montpellier, France, 1994, to appear: Rend. Istit. Mat. Univ. Trieste, Workshop on Measure Theory and Real Analysis, Grado (Italy), September 19 - October 2, 1993.

[33] K. Yosida, E. Hewitt : Finitely additive measures, 1952, 46–66, Vol. 72, Trans. Amer. Math. Soc.

[34] L.C. Young : Generalized curves and the existence of an attained absolute minimum in the Calculus of Variations, Comptes Rendus de la Société des Sc. et des lettres de Varsovie cl. III, Vol. 30, 1937, 212–234.

358 M. Saadoune, M. Valadier / Extraction of a “good” subsequence HIER :

Leere Seite