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.
Given measurable spaces and , a measurable mapping and a measure , the pushforward of is defined to be the measure given by
Main property: change-of-variables formula
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.,
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:
- 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.
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.
- Sections 3.6–3.7 in Bogachev