Definition 18.1.8 (Pushforward Measure).label Let $(X, \cm, \mu)$ be a measure space, $(Y, \cn)$ be a measurable space, and $f: X \to Y$ be a $(\cm, \cn)$-measurable map, then:
- (1)
The mapping
\[f_{*}\mu: \cn \to [0, \infty] \quad A \mapsto \mu\bracs{f \in A}\]is a measure on $(Y, \cn)$.
- (2)
For any Banach space $E$ and $g \in L^{1}(f_{*}\mu; E)$,
\[\int g df_{*}\mu = \int g \circ f d\mu\]
Proof. (1): Preimage commutes with unions, intersections, and complements.
(2): By definition and linearity, (2) holds for $L^{1}(f_{*}\mu; E) \cap \Sigma(f^{*}\mu; E)$, which is dense in $L^{1}(f_{*}\mu; E)$ by Proposition 14.1.9. By continuity of the integral, (2) holds on $L^{1}(\mu_{*}\mu; E)$.$\square$