In measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a measure from one measurable space to another using a measurable function.
Definition Given measurable spaces ( X 1 , Σ 1 ) {\displaystyle (X_{1},\Sigma _{1})} and ( X 2 , Σ 2 ) {\displaystyle (X_{2},\Sigma _{2})} , a measurable function f : X 1 → X 2 {\displaystyle f\colon X_{1}\to X_{2}} and a measure μ : Σ 1 → [ 0 , + ∞ ] {\displaystyle \mu \colon \Sigma _{1}\to [0,+\infty ]} , the pushforward of μ {\displaystyle \mu } by f {\displaystyle f} is defined to be the measure f ∗ ( μ ) : Σ 2 → [ 0 , + ∞ ] {\displaystyle f_{*}(\mu )\colon \Sigma _{2}\to [0,+\infty ]} given by
f ∗ ( μ ) ( B ) = μ ( f − 1 ( B ) ) {\displaystyle f_{*}(\mu )(B)=\mu \left(f^{-1}(B)\right)} for B ∈ Σ 2 . {\displaystyle B\in \Sigma _{2}.}
This definition applies mutatis mutandis for a signed or complex measure. The pushforward measure is also denoted as μ ∘ f − 1 {\displaystyle \mu \circ f^{-1}} , f ♯ μ {\displaystyle f_{\sharp }\mu } , f ♯ μ {\displaystyle f\sharp \mu } , or f # μ {\displaystyle f\#\mu } .
Properties
Change of variable formula Theorem: A measurable function g on X2 is integrable with respect to the pushforward measure f∗(μ) if and only if the composition g ∘ f {\displaystyle g\circ f} is integrable with respect to the measure μ. In that case, the integrals coincide, i.e.,
∫ X 2 g d ( f ∗ μ ) = ∫ X 1 g ∘ f d μ . {\displaystyle \int _{X_{2}}g\,d(f_{*}\mu )=\int _{X_{1}}g\circ f\,d\mu .}
Note that in the previous formula X 1 = f − 1 ( X 2 ) {\displaystyle X_{1}=f^{-1}(X_{2})} .
Functoriality Pushforwards of measures allow to induce, from a function between measurable spaces f : X → Y {\displaystyle f:X\to Y} , a function between the spaces of measures M ( X ) → M ( Y ) {\displaystyle M(X)\to M(Y)} . As with many induced mappings, this construction has the structure of a functor, on the category of measurable spaces. For the special case of probability measures, this property amounts to functoriality of the Giry monad.
… excerpt ends here. Continue reading the full article.
