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Vadim Kosoy added subsection_Measures_and_Probabilities_For__.tex
about 8 years ago
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\subsection{Measures and Probabilities}
For $X$ a measurable space, $\mu$ a probability measure on $X$, $V$ a finite dimensional vector space over $\Reals$ and $f: X \rightarrow V$, $\E_{x \sim \mu}[f(x)]$ will denote the expected value of $f$ with respect to $\mu$, i.e. $\E_{x \sim \mu}[f(x)] := \int_X f(x) d\mu(x)$. We will the abbreviated notations $\E_\mu[f(x)]$, $\E[f(x)]$, $\E_\mu[f]$, $\E[f]$ when no confusion is likely to occur.
Given a topological space $X$ and a Borel probability measure $\mu$ on $X$, $\Supp \mu$ will denote the support of $\mu$. In particular when $X$ is discrete, $\Supp \mu = \{x \in X \mid \mu(x) > 0\}$.
