---- > [!definition] Definition. ([[pushforward measure]]) > Given a [[measure|measure space]] $(X, \Sigma, \mu)$ and a [[σ-algebra|measurable space]] $(Y, \mathcal{T})$, any [[measurable function]] $f:X \to Y$ induces a [[measure]] $f_{*}\mu$ on $(Y, \mathcal{T})$ as $(f_{*}\mu)(E \in \mathcal{T}):= \mu\big( f ^{-1} (E) \in \Sigma\big).$ The [[measure]] $f_{*}\mu$ is called the **pushforward of $\mu$ by $f$**. > > [!basicproperties] > - [[integral with respect to a pushforward measure]] ^properties ---- #### ---- #### 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 > ```