> [!definition] > > Let $(X, \cm, \mu)$ be a [[Measure Space|measure space]], $(Y, \cn)$ be a measurable space, and $f: X \to Y$ be a $(\cm, \cn)$-[[Measurable Function|measurable]] map. Define the **push-forward** $f_*\mu$ as > $ > f_\star \mu(E) = \mu(f^{-1}(E)) > $