Let $P = (\Omega, \mathcal{F}_{\Omega}, \mathbb{P})$ be a D1159: Probability space.
Let $M = (\Xi, \mathcal{F}_{\Xi})$ be a D1108: Measurable space such that
Let $M = (\Xi, \mathcal{F}_{\Xi})$ be a D1108: Measurable space such that
| (i) | $X : \Omega \to \Xi$ is a D202: Random variable from $P$ to $M$ |
| (ii) | $\mu_X : \mathcal{F}_{\Xi} \to [0, \infty]$ is a D204: Probability distribution measure for $X$ |
| (iii) | $f : \Xi \to [0, \infty]$ is a D313: Measurable function on $M_Y$ |
Then
\begin{equation}
\int_{\Omega} f(X) \, d \mathbb{P}
= \int_{\Xi} f \, d \mu_X
\end{equation}