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

(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}