Measurable selectors for the metric projection

✍ Scribed by B. Cascales; M. Raja

• Publisher: John Wiley and Sons
• Year: 2003
• Tongue: English
• Weight: 136 KB
• Volume: 254-255
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310050

✦ Synopsis

Abstract

Let Y and Z be two topological spaces and F : Y × Z → ℝ a function that is upper semi–continuous in the first variable and lower semi–continuous in the second variable. If Z is Polish and for every y ∈ Y there is a point z ∈ Z with F(y, z) = inf~w∈Z~ F(y, w) we prove that there is a nice measurable function h : Y → Z satisfying F(y, h(y)) = inf~z∈Z~ F(y, z) for every y ∈ Y . As an application we obtain the existence of universally measurable selectors for the metric projection onto weakly K–analytic convex proximinal subsets of a Banach space, which then allows us to prove that L^p^(μ, Y ) is proximinal in L^p^(μ, X) for every proximinal weakly K–analytic subspace Y of a Banach space X.