Survey of Measurable Selection Theorems

Suppose (T,M) is a measurable space, X is a topological space, and ∅ ≠ F(t) ⊂ X for t ∈ T. Denote GrF = \ (t,x):x ∈ F(t)\. The problem surveyed (reviewing work of others) is that of existence off: f:T → X such that f(t) ∈ F(t) for t ∈ T and f- 1 (U) ∈ M for open U ⊂ X. The principal conditions that yield such f are (i) X is Polish, each $F(t)$ is closed, and \ t:F(t) ∩ U ≠ ∅ \ ∈ M a .tit whenever U ⊂ X is open (Kuratowski and Ryll-Nardzewski and, under stronger assumption, Castaing), or (ii) T is a Hausdorff space, GrF is a continuous image of a Polish space, and M is the σ-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 of set-valued functions of Suslin type (Leese), which extends the , result (i) to comprehend a considerable portion of the results on the problem surveyed. Among the topics addressed, measurable implicit functions and the case where X is a linear space and each $F(t)$ is convex and compact are particularly important to control theory, for example. With $T = X = [0,1]$ and GrF Borel, an elegant partition of GrF into Lebesgue measurable maps from T to X, parameterized by Borel functions, has been found (Wesley) via Cohen forcing methods. Other topics discussed include pointwise optimal selections, selections of partitions, uniformization, non-σ-algebras in place of M, Lusin measurability, and set-valued measures. Substantial historical comments and an extensive bibliography are included. (See addenda (i)–(iii).)