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For \(X\) a measurable space, \(\mu\) a probability measure on \(X\), \(V\) a finite dimensional vector space over \({\mathbb{R}}\) and \(f: X \rightarrow V\), \(\operatorname{E}_{x \sim \mu}[f(x)]\) will denote the expected value of \(f\) with respect to \(\mu\), i.e. \(\operatorname{E}_{x \sim \mu}[f(x)] := \int_X f(x) d\mu(x)\). We will the abbreviated notations \(\operatorname{E}_\mu[f(x)]\), \(\operatorname{E}[f(x)]\), \(\operatorname{E}_\mu[f]\), \(\operatorname{E}[f]\) when no confusion is likely to occur.
Given a topological space \(X\) and a Borel probability measure \(\mu\) on \(X\), \(\operatorname{supp}\mu\) will denote the support of \(\mu\). In particular when \(X\) is discrete, \(\operatorname{supp}\mu = \{x \in X \mid \mu(x) > 0\}\).