Survey of Measurable Selection Theorems

Suppose $(T,\mathcal{M})$ is a measurable space, X is a topological space, and $\emptyset \ne F(t) \subset X$ for $t \in T$. Denote ${\operatorname {Gr}}F = \{ (t,x):x \in F(t)\} $. The problem surveyed (reviewing work of others) is that of existence off: $f:T \to X$ such that $f(t) \in F(t)$ for $t \in T$ and $f^{ - 1} (U) \in \mathcal{M}$ for open $U \subset X$. The principal conditions that yield such f are (i) X is Polish, each $F(t)$ is closed, and $\{ t:F(t) \cap U \ne \emptyset \} \in \mathcal{M}$ a .tit whenever $U \subset X$ is open (Kuratowski and Ryll-Nardzewski and, under stronger assumption, Castaing), or (ii) T is a Hausdorff space, ${\operatorname {Gr}}F$ is a continuous image of a Polish space, and M is the $\sigma $-algebra of sets measurable with respect to an outer measure, among which are the open sets of T (primarily von Neumann). The latter result follows from the former by lifting F in a natural way to a map into the closed sets of a Polish space. This procedure leads to the theory ...