An Analytic Construction of Random Variables in Lebesgue Spaces

TL;DR

The paper addresses the problem of defining random variables that take values in Lebesgue spaces $L^{p}$ and extending the notion of expectation to this vector-valued setting. It develops a functional-analytic framework based on Pettis measurability and the Bochner integral, connecting measurability, duality with $L^{q}$, and vector-valued integration. A key contribution is the concrete construction of $L^{p}$-valued random variables and their Bochner expectation, including a detailed analysis of the indicator-function example $\chi(\omega)=\mathbf{1}_{(0,\omega)}$ which yields the distribution $(0,\varepsilon^{p})$ and the expectation $\mathbb{E}(\chi)(t)=1-t$. This framework provides a rigorous approach to randomness in function spaces, with potential applications in stochastic partial differential equations and Malliavin calculus across separable Banach targets.

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.

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Theorems & Definitions (18)

• Theorem 1.3
• Theorem 2.1: Riesz--Fischer
• Theorem 2.2: Hölder’s Inequality
• Theorem 2.3: Riesz Representation Theorem
• Definition 2.4
• Theorem 2.5
• Theorem 2.6: Hahn--Banach
• Definition 3.1
• Theorem 3.2
• proof