# A Note on Uniform Integrability of Random Variables in a Probability Space and Sublinear Expectation Space

## TL;DR

This paper explores uniform integrability of random variables in probability and sublinear expectation spaces, introducing new notions and criteria, and establishing their equivalence to classical concepts.

## Contribution

It introduces two new notions of uniform integrability in probability spaces and provides a de La Vallée Poussin criterion in sublinear expectation spaces, expanding theoretical understanding.

## Key findings

- New notions of uniform integrability are equivalent to classical ones in probability spaces.
- De La Vallée Poussin criterion is established for sublinear expectation spaces.
- Discusses properties and implications of uniform integrability in different frameworks.

## Abstract

In this note we discuss uniform integrability of random variables. In a probability space, we introduce two new notions on uniform integrability of random variables, and prove that they are equivalent to the classic one. In a sublinear expectation space, we give de La Vall\'ee Poussin criterion for the uniform integrability of random variables and do some other discussions.