Closed Graph Theorem For Linear Random Operators

TL;DR

This paper introduces a probabilistic characterization of continuous linear random operators, establishing a link between continuity and closed graph properties, and proposes the foundation for a new field called Random Analysis.

Contribution

It provides the first characterization of probably continuous operators in the context of linear random operators and initiates the development of Random Analysis.

Findings

Probability of continuity equals probability of having a closed graph.

Established foundational concepts for Random Analysis.

Inspired by previous work on operator properties.

Abstract

In this work, which was inspired by the article [2] by M. V. Velasco and A. R. Villena, we obtain a characterization for probably continuous operators and show that the probability of a linear random operator being continuous coincides with the probability of having a closed graph. Furthermore, we laid the foundations to start a new branch of mathematics, namely, Random Analysis.