nLab pushforward measure

Context

Measure and probability theory

measure theory

probability theory

Contents

Definition

In measure theory, the pushforward $f_\ast \mu$ of a measure $\mu$ on a measurable space $X$ along a measurable function $f \colon X \to Y$ to another measure space $Y$ assigns to a subset the original measure of the preimage under $f$ of that subset:

$(f_\ast\mu)( - ) \coloneqq \mu(f^{-1}(-)) \,.$