Verification of Probability Measure or Inner Product Properties
Verify that a given construction defines a valid probability measure, inner product, or reproducing kernel on a function space associated with random variables.
Let $\left( X _ { i } \right) _ { i \in \mathbf { N } }$ be a sequence of independent random variables all following a Rademacher distribution. Let the map $\psi : u \in \mathbf { R } ^ { ( \mathbf { N } ) } \mapsto \sum _ { i = 0 } ^ { + \infty } u _ { i } X _ { i }$. Show that $\psi$ takes its values in $L ^ { 0 } ( \Omega )$, then that $\psi$ preserves the inner product.