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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.
Contents
1 Definition
2 Main property: change-of-variables formula
3 Examples and applications
4 A generalization
5 See also
6 Notes
7 References
Definition
Given measurable spaces (X1,Σ1){displaystyle (X_{1},Sigma _{1})}
![{displaystyle mu colon Sigma _{1}to [0,+infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8eb43a4e2444122692c2e2eb62d0797abdb2430f)
![{displaystyle f_{*}(mu )colon Sigma _{2}to [0,+infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f698a0b393a37c4d4e4f0e98938d414c6a6aaf)
(f∗(μ))(B)=μ(f−1(B)){displaystyle (f_{*}(mu ))(B)=mu left(f^{-1}(B)right)}for B∈Σ2.{displaystyle Bin Sigma _{2}.}
This definition applies mutatis mutandis for a signed or complex measure.
The pushforward measure is also denoted as f♯μ{displaystyle 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∘f{displaystyle gcirc f}
- ∫X2gd(f∗μ)=∫X1g∘fdμ.{displaystyle int _{X_{2}}g,d(f_{*}mu )=int _{X_{1}}gcirc f,dmu .}
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)=f∘f∘⋯∘f⏟ntimes:X→X.{displaystyle f^{(n)}=underbrace {fcirc fcirc dots circ f} _{nmathrm {,times} }:Xto X.}
- f(n)=f∘f∘⋯∘f⏟ntimes:X→X.{displaystyle f^{(n)}=underbrace {fcirc fcirc dots circ f} _{nmathrm {,times} }:Xto 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.
See also
- Measure-preserving dynamical system
Notes
^ Sections 3.6–3.7 in Bogachev
References
Bogachev, Vladimir I. (2007), Measure Theory, Berlin: Springer Verlag, ISBN 9783540345138.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
Teschl, Gerald (2015), Topics in Real and Functional Analysis
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