The Young Measure Representation for Weak Cluster Points of Sequences in M-spaces of Measurable Functions

Hôǹg Thái Nguyêñ; Dariusz Pączka

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 56 (2008), p. 109-120 / Harvested from The Polish Digital Mathematics Library

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Résumé

Let ⟨X,Y⟩ be a duality pair of M-spaces X,Y of measurable functions from Ω ⊂ ℝ ⁿ into $ℝ^{d}$ . The paper deals with Y-weak cluster points ϕ̅ of the sequence $ϕ (\cdot, z_{j} (\cdot))$ in X, where $z_{j} : Ω \to ℝ^{m}$ is measurable for j ∈ ℕ and $ϕ : Ω \times ℝ^{m} \to ℝ^{d}$ is a Carathéodory function. We obtain general sufficient conditions, under which, for some negligible set $A_{ϕ}$ , the integral $I (ϕ, ν_{x}) : = \int_{ℝ^{m}} ϕ (x, λ) d ν_{x} (λ)$ exists for $x \in Ω ∖ A_{ϕ}$ and $ϕ ̅ (x) = I (ϕ, ν_{x})$ on $Ω ∖ A_{ϕ}$ , where $ν = {ν_{x}}_{x \in Ω}$ is a measurable-dependent family of Radon probability measures on $ℝ^{m}$ .

Publié le : 2008-01-01

Zbl 1146.28003

EUDML-ID : urn:eudml:doc:281301

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     title = {The Young Measure Representation for Weak Cluster Points of Sequences in M-spaces of Measurable Functions},
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     volume = {56},
     year = {2008},
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Hôǹg Thái Nguyêñ; Dariusz Pączka. The Young Measure Representation for Weak Cluster Points of Sequences in M-spaces of Measurable Functions. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 56 (2008) pp. 109-120. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba56-2-3/