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Concept
Pushforward measure
Summary
In measure theory, a pushforward measure (also known as 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.
Given measurable spaces and , a measurable mapping and a measure , the pushforward of is defined to be the measure given by
for
This definition applies mutatis mutandis for a signed or complex measure.
The pushforward measure is also denoted as , , , or .
Theorem: A measurable function g on X2 is integrable with respect to the pushforward measure f∗(μ) if and only if the composition is integrable with respect to the measure μ. In that case, the integrals coincide, i.e.,
Note that in the previous formula .
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.
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