Pushforward measure

In measure theory, a discipline within mathematics, a pushforward measure (also 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.

Given measurable spaces ${\displaystyle (X_{1},\Sigma _{1})}$ and ${\displaystyle (X_{2},\Sigma _{2})}$, a measurable mapping ${\displaystyle f\colon X_{1}\to X_{2}}$ and a measure ${\displaystyle \mu \colon \Sigma _{1}\to [0,+\infty ]}$, the pushforward of ${\displaystyle \mu }$ is defined to be the measure ${\displaystyle f_{*}(\mu )\colon \Sigma _{2}\to [0,+\infty ]}$ given by

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