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A variational problem to calculate probabilities
Hugo Guadalupe Reyna-Castañeda, María de los Ángeles Sandoval Ro · 2025-03-21 · via math.PR updates on arXiv.org

In this paper, we prove the existence and uniqueness of the conditional expectation of an event $A$ given a $σ$-algebra $\mathcal{G}$ as a linear problem in the Lebesgue spaces $L^{p}$ associated with a probability space through the Riesz Representation Theorems. For the $L^{2}$ 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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