Criterion for convergence almost everywhere, with applications

TL;DR

This paper establishes the exact condition for almost sure convergence of measurable function sequences and explores applications in Fourier series and random fields.

Contribution

It provides the necessary and sufficient criterion for convergence almost everywhere, advancing theoretical understanding in measure theory.

Findings

Derived the criterion for almost sure convergence

Applied results to Fourier series

Applied results to random fields

Abstract

We derive the necessary and sufficient condition for almost sure convergence of the sequence of measurable functions, and consider some applications in the theory of Fourier series and in the theory of random fields.