---- > [!theorem] Theorem. ([[integral with respect to a pushforward measure]]) > Let $(X, \Sigma, \mu)$ be a [[measure|measure space]], $(Y, \mathcal{T})$ a [[σ-algebra|measurable space]], and $P:X \to Y$ a [[measurable function]]. Denote by $P_{*}\mu$ the [[pushforward measure|pushforward]] on $(Y, \mathcal{T})$. > Then given a [[measurable function]] $f:Y \to \mathbb{R}$ that is [[integral|integrable]] wrt $P_{*}\mu$, we have $\int _{Y} f \, d(P_{*}\mu)= \int _{X}(f \circ P) \, d\mu . $ ^theorem > [!proof]- Proof. ([[integral with respect to a pushforward measure]]) > ~ ---- #### ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```