On Baire property of spaces of compact-valued measurable functions

TL;DR

This paper characterizes when the space of measurable compact-valued functions is Baire, proving it is Baire for any metrizable compact space, thus advancing understanding of functional space topologies.

Contribution

It provides a complete characterization of the Baire property for spaces of measurable compact-valued functions, specifically proving they are Baire when the codomain is metrizable and compact.

Findings

The space of measurable compact-valued functions is Baire for any metrizable compact space.

The Baire property is characterized through the topological support of functions.

The result applies to spaces with the topology of pointwise convergence.

Abstract

A topological space $X$ is Baire if the Baire Category Theorem holds for $X$, i.e., the intersection of any sequence of open dense subsets of $X$ is dense in $X$. One of the interesting problems in the theory of functional spaces is the characterization of the Baire property of a functional space through the topological property of the support of functions. In the paper this problem is solved for the space $M(X, K)$ of all measurable compact-valued ($K$-valued) functions defined on a measurable space $(X,\Sigma)$ with the topology of pointwise convergence. It is proved that $M(X, K)$ is Baire for any metrizable compact space $K$.