The Egoroff Theorem for Operator-Valued Measures in Locally Convex Spaces

TL;DR

This paper extends the Egoroff theorem to measurable functions and operator-valued measures in locally convex spaces, proving convergence results under atomic measures and net convergence.

Contribution

It introduces a version of the Egoroff theorem applicable to operator-valued measures in locally convex spaces with atomic measures and net convergence.

Findings

Proves Egoroff theorem in locally convex spaces

Extends theorem to operator-valued measures

Addresses convergence with nets in this context

Abstract

The Egoroff theorem for measurable $\bold X$-valued functions and operator-valued measures $\bold m: \Sigma \to L(\bold X, \bold Y)$, where $\Sigma$ is a $\sigma$-algebra of subsets of $T \neq \emptyset$ and $\bold X$, $\bold Y$ are both locally convex spaces, is proved. The measure is supposed to be atomic and the convergence of functions is net.