# nLab pushforward measure

Contents

### 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:

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

## As a functor

The construction of pushforward measures is how one can make the Giry monad, or other measure monads, functorial. Given a measurable (or continuous, etc.) map $f:X\to Y$, the pushforward gives a well-defined, measurable map $P X\to P Y$ (where $P$ denotes the Giry monad), making $P$ into a functor.