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.

Definition

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

(f_{*}(\mu ))(B)=\mu \left(f^{-1}(B)\right)

for

B\in \Sigma _{2}.

This definition applies mutatis mutandis for a signed or complex measure. The pushforward measure is also denoted as $f_{\sharp }\mu$ .

Main property: change-of-variables formula

Theorem:^[1] A measurable function g on X₂ is integrable with respect to the pushforward measure f_∗(μ) if and only if the composition $g\circ f$ is integrable with respect to the measure μ. In that case, the integrals coincide, i.e.,

\int _{{X_{2}}}g\,d(f_{*}\mu )=\int _{{X_{1}}}g\circ f\,d\mu .

Examples and applications

A natural "Lebesgue measure" on the unit circle S¹ (here thought of as a subset of the complex plane C) may be defined using a push-forward construction and Lebesgue measure λ on the real line R. Let λ also denote the restriction of Lebesgue measure to the interval [0, 2π) and let f : [0, 2π) → S¹ be the natural bijection defined by f(t) = exp(i t). The natural "Lebesgue measure" on S¹ is then the push-forward measure f_∗(λ). The measure f_∗(λ) might also be called "arc length measure" or "angle measure", since the f_∗(λ)-measure of an arc in S¹ is precisely its arc length (or, equivalently, the angle that it subtends at the centre of the circle.)
The previous example extends nicely to give a natural "Lebesgue measure" on the n-dimensional torus Tⁿ. The previous example is a special case, since S¹ = T¹. This Lebesgue measure on Tⁿ is, up to normalization, the Haar measure for the compact, connected Lie group Tⁿ.
Gaussian measures on infinite-dimensional vector spaces are defined using the push-forward and the standard Gaussian measure on the real line: a Borel measure γ on a separable Banach space X is called Gaussian if the push-forward of γ by any non-zero linear functional in the continuous dual space to X is a Gaussian measure on R.
Consider a measurable function f : X → X and the composition of f with itself n times:

f^{{(n)}}=\underbrace {f\circ f\circ \dots \circ f}_{{n{\mathrm {\,times}}}}:X\to X.

This iterated function forms a dynamical system. It is often of interest in the study of such systems to find a measure μ on X that the map f leaves unchanged, a so-called invariant measure, one for which f_∗(μ) = μ.

One can also consider quasi-invariant measures for such a dynamical system: a measure μ on X is called quasi-invariant under f if the push-forward of μ by f is merely equivalent to the original measure μ, not necessarily equal to it.

A generalization

In general, any measurable function can be pushed forward, the push-forward then becomes a linear operator, known as the transfer operator or Frobenius–Perron operator. In finite-dimensional spaces this operator typically satisfies the requirements of the Frobenius–Perron theorem, and the maximal eigenvalue of the operator corresponds to the invariant measure.

The adjoint to the push-forward is the pullback; as an operator on spaces of functions on measurable spaces, it is the composition operator or Koopman operator.

Notes

↑ Sections 3.6–3.7 in Bogachev

References

Bogachev, Vladimir I. (2007), Measure Theory, Berlin: Springer Verlag, ISBN 9783540345138
Teschl, Gerald (2015), Topics in Real and Functional Analysis

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[1] Sections 3.6–3.7 in Bogachev