Given a Random Variable $X : \Omega \rightarrow R$, where $(\Omega, \mathcal{F}, P)$ is a probability space and $(R, \mathcal{R})$ is a measurable space, where $\mathcal{R}$ is a borel space.
we define a density $f_X$ of $X$ as a function s.t. for every $x \in R$,
$P(X \leq x) = P(X^{-1}(-\infty, x])= \int_{\infty}^{x}f_X(x)dx$.
Given this density of $X$, we can compute a expected value of $g(X)$ by
$E[g(X)] = \int g(x)f_X(x)dx$.
I tried to find the Theorem for this result, but I couldn't. Do we have a name for this Theorem?