On lower and upper bounds for probabilities of unions and the Borel--Cantelli lemma

TL;DR

This paper introduces improved, sharp bounds for probabilities of unions of events, generalizes classical Borel--Cantelli lemmas, and enhances existing methods applicable in arbitrary measurable spaces.

Contribution

It provides new lower and upper bounds for union probabilities and generalizes key Borel--Cantelli results, advancing the theoretical framework.

Findings

Bounds are sharper than previous results

Method improvements lead to broader applicability

Generalizations of Borel--Cantelli lemmas derived

Abstract

We obtain new lower and upper bounds for probabilities of unions of events.These bounds are sharp. They are stronger than earlier ones. General bounds maybe applied in arbitrary measurable spaces.We have improved the method that has been introduced in previous papers. We derive new generalizations of the first and second part of the Borel--Cantelli lemma.