Pushforward measure
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 .