nLab pushforward measure

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:

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

Properties

General

• The pushforward of a τ-additive measure along a continuous map is again τ-additive.

• If a continuous valuation $\nu$ on a topological space $X$ is extendable to a measure $\nu$, its pushforward (as a valuation) $f_*\mu$ along a continuous map $f:X\to Y$ is again extendable, and it extends to $f_*\nu$.

• The pushforward measure along a product projection is called a marginal measure.

Relation to entropy

Proposition

(entropy does not increase under pushforward) Let $f \colon X \longrightarrow Y$ be a measurable function between measure spaces, and let $\mu$ be a probability distribution on $X$. Then the entropy of $\mu$ is larger or equal to that of its pushforward distribution $f_\ast \mu$:

$$S(\mu) \;\geq\; S \big( f_\ast(\mu) \big) \,.$$

(e.g. Austin, Prop. 2.7)

Relation to the Giry monad

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 $f \colon X\to Y$, the pushforward gives a well-defined, measurable map $P X\to P Y$ (where $P$ denotes the Giry monad), making $P$ into a functor.

Related concepts

• Borel measure, Radon measure, τ-additive measure

• monads of probability, measures, and valuations

• Giry monad

• conditional expectation

• pullback of a function

• pullback in cohomology

References

See also

• Wikipedia, Pushforward measure

In relation to entropy:

• Tim D. Austin, Entropy and Sinai’s Theorem (pdf, pdf)