By Mariano Giaquinta, Stefan Hildebrandt
This long-awaited booklet by way of of the major researchers and writers within the box is the 1st a part of a treatise that would hide the topic in breadth and intensity, paying distinctive awareness to the old origins, in part in purposes, e.g. from geometrical optics, of elements of the speculation. various aids to the reader are supplied: the precise desk of contents, an creation to every bankruptcy, part and subsection, an summary of the correct literature (in Vol. 2) plus the references within the Scholia to every bankruptcy, within the (historical) footnotes, and within the bibliography, and eventually an index of the examples used during the booklet. Later volumes will care for direct equipment and regularity thought. either separately and jointly those volumes will absolutely turn into commonplace references.
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If furthermore T is Polish, then (Y 1 , τYW1 ) = (Y 1 , τYS 1 ) is Polish. Proof. 14, the space (Y 1 , τYW1 ) is a closed subspace of M+,1 (Ω × T, τΩ ⊗ τT ) endowed with the narrow topology. But the space M+,1 (Ω × T, τΩ ⊗ τT ) is Suslin regular, thus (Y 1 , τYW1 ) is also Suslin regular. If furthermore T is Polish, then M+,1 (Ω × T, τΩ ⊗ τT ) is Polish, thus (Y 1 , τYW1 ) is Polish too. 4 CHAPTER 2. THE FOUR STABLE TOPOLOGIES Integrable Young measures and LpE spaces In this section, we are given a real number p ≥ 1 and we assume that T is completely regular and that its topology is deﬁned by a set D of semidistances.
Let Cbu (S, U) (resp. Cbu (T, U)) be the space of bounded U–uniformly continuous functions on S (resp. on T). g. 20]). Let ψ be the canonical injection from T to S. For each µ ∈ Y 1 and each µ–integrable ). Let (µα )α be a net integrand f on Ω × S, we have µ(f ) = µ(f ◦ ψ) = µ(f Ω×T in Y 1 (T) and let µ∞ ∈ Y 1 (T). 3 , we get µα −−−−−→ µ∞ ⇔ ∀A ∈ S ∀f ∈ Cbu (T, U) µα ( 1lA ⊗ f ) → µ∞ ( 1lA ⊗ f ) W–stably ⇔ ∀A ∈ S ∀f ∈ Cbu (T, U) µα ( 1lA ⊗ f ) → µ∞ ( 1lA ⊗ f ) ⇔ ∀A ∈ S ∀f ∈ Cbu (S, U) µα ( 1lA ⊗ f ) → µ∞ ( 1lA ⊗ f ) ⇔ µα −−−−−→ µ∞ .
So, we only need to prove the implication (11 ⇒ 11 ). We shall prove this implication by using a Functional Monotone Class Theorem. A set E of bounded measurable functions on Ω is called a monotone vector space [Sha88, Appendix A] if (i) it is a vector space over R, (ii) it contains the constant functions and (iii) it is stable under monotone limits of uniformly bounded sequences. The Monotone Class Theorem asserts that if E is a monotone vector space and if H is a subset of E which is stable under multiplication of two elements, then E contains all bounded functions which are measurable for the σ–algebra generated by H (this theorem is proved in [DM75, Th´eor`eme 21, page 20], with the extra hypothesis that E is stable under uniform convergence, but it appears that any monotone vector space satisﬁes this condition, see [DM83, page 231] or [Sha88]).