An Analytic Construction of Random Variables in Lebesgue Spaces

TL;DR

This paper presents a functional analytic approach to constructing random variables in Lebesgue spaces, extending classical probabilistic concepts like measurability and expectation using advanced theorems.

Contribution

It introduces a novel framework for defining random variables in L^p spaces via Pettis's and Riesz's theorems, generalizing classical probabilistic notions.

Findings

Extended classical measurability to L^p-valued functions

Defined Bochner integral as a generalization of expectation

Provided a rigorous functional analytic foundation for random variables in Lebesgue spaces

Abstract

This work develops, from a functional analytic perspective, the construction of random variables in Lebesgue spaces L^p. It extends classical notions of measurability, integrability, and expectation to L^p valued functions, using Pettis's theorem and the Riesz representation theorem to define the Bochner integral as a natural generalization of classical expectation.