A bang-bang principle for the conditional expectation vector measure

TL;DR

This paper establishes a bang-bang principle for conditional expectation vector measures, showing that the expectation of a set-valued map equals that of its extreme points, leading to a purification principle.

Contribution

It introduces a novel bang-bang principle for conditional expectation vector measures and derives a straightforward purification principle.

Findings

Conditional expectation of set-valued maps equals that of their extreme points.

Established a purification principle as a consequence.

Provided theoretical insights into the properties of conditional expectation vector measures.

Abstract

We prove a conditional expectation bang-bang principle. Based on properties of the conditional expectation vector measure, we establish that the conditional expectation of a set-valued mapping coincides with the conditional expectation of the set of selections of its extreme points part. As a by-product, we obtain straightforwardly a purification principle.