Measurable Selection Theorem has wide applications on stochastic control theory, and other fields. We will describe this theorem briefly, see details in [Wag77] (Download).
Suppose is a measurable space, is a topological space, and for . Denote . The problems is that of existence of -measurable such that for all .
The principle conditions that yield such are one of the following two conditions:
- is Polish, each is closed, and whenever .
- is a Hausdorff space, is a continuous image of a Polish space, and is the -algebra of sets measurable with respect to an outer measure, among which are the open sets of .