Let $M_X = (X, \mathcal{F}_X, \mu)$ be a D1158: Measure space.

Let $M_Y = (Y, \mathcal{F}_Y)$ be a D1108: Measurable space such that

Let $M_Y = (Y, \mathcal{F}_Y)$ be a D1108: Measurable space such that

(i) | $\phi : X \to Y$ is a D201: Measurable map from $M_X$ to $M_Y$ |

(ii) | \begin{equation} \phi_* \mu : \mathcal{F}_Y \to [0, \infty], \quad \phi_* \mu(E) = \mu(\phi^{-1} E) \end{equation} |

(iii) | $f : Y \to [0, \infty]$ is a D313: Measurable function on $M_Y$ |

Then
\begin{equation}
\int_X (f \circ \phi) \, d \mu
= \int_Y f \, d \phi_* \mu
\end{equation}