nLab
pushforward measure

Contents

Contents

Definition

In measure theory, the pushforward f *μf_\ast \mu of a measure μ\mu on a measurable space XX along a measurable function f:XYf \colon X \to Y to another measure space YY assigns to a subset the original measure of the preimage under ff of that subset:

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

Properties

General

Relation to entropy

Proposition

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

S(μ)S(f *(μ)). 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:XYf \colon X\to Y, the pushforward gives a well-defined, measurable map PXPYP X\to P Y (where PP denotes the Giry monad), making PP into a functor.

References

See also

In relation to entropy:

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

Last revised on May 20, 2021 at 05:02:34. See the history of this page for a list of all contributions to it.