A variational problem to calculate probabilities

2025/03/20 by Hugo Guadalupe Reyna-Castañeda, Reyna-Castañeda, Hugo Guadalupe, Marisol López-Romero +1
Decision Sciences · Mathematics · Physics and Astronomy · #46E15 #46E20 #49J50 #49K35 #60A05 #60A10 #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Control (math.OC) #Probability (math.PR) #Risk and Portfolio Optimization #Statistical Mechanics and Entropy

paper · pdf · doi:10.48550/arxiv.2503.16727

openalex publication_date 2025/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove the existence and uniqueness of the conditional expectation of an event A given a σ-algebra G as a linear problem in the Lebesgue spaces L^p associated with a probability space through the Riesz Representation Theorems. For the L² case, we state the Dirichlet's principle. Then, we extend this principle for specific values of p, framing the existence of the conditional expectation as a variational problem. We conclude with a proof of the law of total probability using these tools.

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