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# Pushforward measure

In mathematics, a pushforward measure (also push forward or push-forward) 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 function "f" : "X"1 &rarr; "X"2 and a measure "&mu;" : &Sigma;1 &rarr; [0, +&infin;] , the pushforward of "&mu;" is defined to be the measure "f"&lowast;("&mu;") : &Sigma;2 &rarr; [0, +&infin;] given by

:$\left(f_\left\{*\right\} \left(mu\right)\right) \left(B\right) = mu left\left( f^\left\{-1\right\} \left(B\right) ight\right) mbox\left\{ for \right\} B in Sigma_\left\{2\right\}.$

This definition applies "mutatis mutandis" for a signed or complex measure.

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 "&lambda;" on the real line R. Let "&lambda;" also denote the restriction of Lebesgue measure to the interval [0, 2"&pi;") and let "f" : [0, 2"&pi;") &rarr; 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"&lowast;("&lambda;"). The measure "f"&lowast;("&lambda;") might also be called "arc length measure" or "angle measure", since the "f"&lowast;("&lambda;")-measure of an arc in S1 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 S1 = T1. This Lebesgue measure on T"n" is, up to normalization, the Haar measure for the compact, connected Lie group T"n".

* 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 "&gamma;" on a separable Banach space "X" is called Gaussian if the push-forward of "&gamma;" 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" &rarr; "X" and the composition of "f" with itself "n" times:

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

: This forms a measurable dynamical system. It is often of interest in the study of such systems to find a measure "&mu;" on "X" that the map "f" leaves unchanged, a so-called invariant measure, one for which "f"&lowast;("&mu;") = "&mu;".

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

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