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| In [[measure theory]], a '''pushforward measure''' (also '''push forward''', '''push-forward''' or '''image measure''' ) is obtained by transferring ("pushing forward") a [[measure (mathematics)|measure]] from one [[measurable space]] to another using a [[measurable function]].
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| ==Definition==
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| Given measurable spaces (''X''<sub>1</sub>, Σ<sub>1</sub>) and (''X''<sub>2</sub>, Σ<sub>2</sub>), a measurable mapping ''f'' : ''X''<sub>1</sub> → ''X''<sub>2</sub> and a measure ''μ'' : Σ<sub>1</sub> → [0, +∞], the '''pushforward''' of ''μ'' is defined to be the measure ''f''<sub>∗</sub>(''μ'') : Σ<sub>2</sub> → [0, +∞] given by
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| :<math>(f_{*} (\mu)) (B) = \mu \left( f^{-1} (B) \right) \mbox{ for } B \in \Sigma_{2}.</math>
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| This definition applies ''[[mutatis mutandis]]'' for a [[signed measure|signed]] or [[complex measure]].
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| ==Main property: Change of variables formula==
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| Theorem:<ref name="Boga" /> A measurable function ''g'' on ''X''<sub>2</sub> is integrable with respect to the pushforward measure ''f''<sub>∗</sub>(''μ'') if and only if the composition <math>g \circ f</math> is integrable with respect to the measure ''μ''. In that case, the integrals coincide, i.e.,
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| :<math>\int_{X_2} g \, d(f_* \mu) = \int_{X_1} g \circ f \, d\mu.</math>
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| ==Examples and applications==
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| * A natural "[[Lebesgue measure]]" on the [[unit circle]] '''S'''<sup>1</sup> (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, 2''π'') and let ''f'' : [0, 2''π'') → '''S'''<sup>1</sup> be the natural bijection defined by ''f''(''t'') = exp(''i'' ''t''). The natural "Lebesgue measure" on '''S'''<sup>1</sup> is then the push-forward measure ''f''<sub>∗</sub>(''λ''). The measure ''f''<sub>∗</sub>(''λ'') might also be called "[[arc length]] measure" or "angle measure", since the ''f''<sub>∗</sub>(''λ'')-measure of an arc in '''S'''<sup>1</sup> is precisely its arc length (or, equivalently, the angle that it subtends at the centre of the circle.)
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| * The previous example extends nicely to give a natural "Lebesgue measure" on the ''n''-dimensional [[torus]] '''T'''<sup>''n''</sup>. The previous example is a special case, since '''S'''<sup>1</sup> = '''T'''<sup>1</sup>. This Lebesgue measure on '''T'''<sup>''n''</sup> is, up to normalization, the [[Haar measure]] for the [[compact space|compact]], [[connected space|connected]] [[Lie group]] '''T'''<sup>''n''</sup>.
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| * [[Gaussian measure]]s 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 space|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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| * Consider a measurable function ''f'' : ''X'' → ''X'' and the [[Function composition|composition]] of ''f'' with itself ''n'' times:
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| ::<math>f^{(n)} = \underbrace{f \circ f \circ \dots \circ f}_{n \mathrm{\, times}} : X \to X.</math>
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| : 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''<sub>∗</sub>(''μ'') = ''μ''. | |
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| * 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 [[equivalence of measures|equivalent]] to the original measure ''μ'', not necessarily equal to it.
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| ==A generalization==
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| 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]]. This operator typically satisfies the requirements of the [[Frobenius–Perron theorem]], and the maximal eigenvalue of this theorem corresponds to the invariant measure. The adjoint to the push-forward is the [[pullback]]; as an operator on measurable spaces, it is the [[composition operator]] or [[Koopman operator]].
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| ==References== | |
| <references>
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| <ref name="Boga">V.I. Bogachev. ''Measure Theory''. Springer, 2007. Sections 3.6-3.7.</ref>
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| </references>
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| ==See also==
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| * [[Measure-preserving dynamical system]]
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| {{DEFAULTSORT:Pushforward Measure}}
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| [[Category:Measures (measure theory)]]
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