Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
Let $M = (\mathbb{R}, \mathcal{B}(\mathbb{R}))$ be the D5072: Standard real borel measurable space such that
| (i) | $X : \Omega \to \mathbb{R}$ is a D3161: Random real number on $P$ |
| (ii) | $\mu_X$ is a D204: Probability distribution measure for $X$ |
| (i) | $f : \mathbb{R} \to [0, \infty]$ is an D5610: Unsigned basic Borel function on $M$ |
Then
\begin{equation}
\mathbb{E} f(X)
= \int_{\mathbb{R}} f \, d \mu_X
\end{equation}