On computability and disintegration

TL;DR

This paper investigates the computational complexity of measure disintegration in metric spaces, establishing equivalences with the limit operator and analyzing cases with non-unique disintegrations.

Contribution

It characterizes the Weihrauch degree of the disintegration operator, linking it to the limit operator and analyzing computable bases of continuity sets.

Findings

Disintegration operator is Weihrauch equivalent to the limit operator Lim.

Disintegration with a computable basis is reducible to Lim.

A specific distribution exemplifies the upper bound of computational complexity.

Abstract

We show that the disintegration operator on a complete separable metric space along a projection map, restricted to measures for which there is a unique continuous disintegration, is strongly Weihrauch equivalent to the limit operator Lim. When a measure does not have a unique continuous disintegration, we may still obtain a disintegration when some basis of continuity sets has the Vitali covering property with respect to the measure; the disintegration, however, may depend on the choice of sets. We show that, when the basis is computable, the resulting disintegration is strongly Weihrauch reducible to Lim, and further exhibit a single distribution realizing this upper bound.