Uniform Integrability and Young Measures

Young Measures on Topological Spaces

“…, when T ⊂ R n , has been considered (with different definitions) by Schonbek [Sch82], Ball [Bal89c], Kinderlehrer and Pedregal [KP94] (see also [Rou97] about these three papers), by Piccinini and Valadier [PV95], by Artstein [Art01a, Art01b], Artstein and Popa [AP03], and by Dedecker and Merlevède [DM02].…”

“…We say that K is relatively compact if it is contained in a compact subset of T. Thus every relatively compact subset of T is net-compact. The converse implication is true if T is regular (see the proof in [PV95] or [OW98]). We say that K is sequentially relatively compact if every sequence of elements of K admits a convergent subsequence.…”

“…2) The norm u n − u L 1 equals Ω×E ψ dδ u n , where ψ is the integrand ψ(ω, ξ) := ξ − u(ω) . This is a Carathéodory integrand with linear growth (for some applications of these integrands see [PV95]). Theorem 6.1.11 Let (u n ) n be a sequence in L 1 E (Ω, S, P) which is bounded and tight relatively to the norm of E (again note that flexibly and strictly are equivalent).…”

Compact convergence of σ-fields and relaxed conditional expectation

“…Kružik and Roubiček [23]. According to Piccinini and Valadier [29], the L p -Young measures would be Young measures of order p as the measures have a p-moment. It is clear that a degenerate Young measure, namely a function f (·), is in Y p if and only if it is in L p , and the L p -norm and the Y p -norm coincide.…”

“…See Berliocchi and Lasry [12] and Piccinini and Valadier [29]. It is clear that the Young measures in a uniformly p-integrable family share a common bound on their norms.…”