# Pushforward measure

In

mathematics , a**pushforward measure**(also**push forward**or**push-forward**) is obtained by transferring ("pushing forward") a measure from onemeasurable space to another using ameasurable function .**Definition**Given measurable spaces ("X"

_{1}, Σ_{1}) and ("X"_{2}, Σ_{2}), a measurable function "f" : "X"_{1}→ "X"_{2}and a measure "μ" : Σ_{1}→ [0, +∞] , the**pushforward**of "μ" is defined to be the measure "f"_{∗}("μ") : Σ_{2}→ [0, +∞] given by:$(f\_\{*\}\; (mu))\; (B)\; =\; mu\; left(\; f^\{-1\}\; (B)\; ight)\; mbox\{\; for\; \}\; B\; in\; Sigma\_\{2\}.$

This definition applies "

mutatis mutandis " for a signed orcomplex measure .**Examples and applications*** A natural "

Lebesgue measure " on theunit circle **S**^{1}(here thought of as a subset of thecomplex plane **C**) may be defined using a push-forward construction and Lebesgue measure "λ" on thereal line **R**. Let "λ" also denote the restriction of Lebesgue measure to the interval [0, 2"π") and let "f" : [0, 2"π") →**S**^{1}be the natural bijection defined by "f"("t") = exp("i" "t"). The natural "Lebesgue measure" on**S**^{1}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**S**^{1}is precisely is 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 **T**^{"n"}. The previous example is a special case, since**S**^{1}=**T**^{1}. This Lebesgue measure on**T**^{"n"}is, up to normalization, theHaar measure for the compact, connectedLie group **T**^{"n"}.*

Gaussian measure s on infinite-dimensional vector spaces are defined using the push-forward and the standard Gaussian measure on the real line: aBorel measure "γ" on a separableBanach space "X" is called**Gaussian**if the push-forward of "γ" by any non-zerolinear functional in thecontinuous 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:

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

: This forms a measurable

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-calledinvariant measure , one for which "f"_{∗}("μ") = "μ".* One can also consider

quasi-invariant measure s 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.

*Wikimedia Foundation.
2010.*

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