Pushforward measure
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
.
Definition
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 .
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 is integrable with respect to the measure μ. In that case, the integrals coincide, i.e.,
Note that in the previous formula .
Examples and applications
- A natural "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 lengthmeasure" 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.
- continuous dual spaceto 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, i.e one for which f∗(μ) = μ.
- One can also consider quasi-invariant measures for such a dynamical system: a measure on is called quasi-invariant under if the push-forward of by is merely equivalentto the original measure μ, not necessarily equal to it. A pair of measures on the same space are equivalent if and only if , so is quasi-invariant under if
- Many natural probability distributions, such as the chi distribution, can be obtained via this construction.
- Random variables induce pushforward measures. They map a probability space into a codomain space and endow that space with a probability measure defined by the pushforward. Furthermore, because random variables are functions (and hence total functions), the inverse image of the whole codomain is the whole domain, and the measure of the whole domain is 1, so the measure of the whole codomain is 1. This means that random variables can be composed ad infinitum and they will always remain as random variables and endow the codomain spaces with probability measures.
A generalization
In general, any
Frobenius–Perron theorem
, and the maximal eigenvalue of the operator corresponds to the invariant measure.
The adjoint to the push-forward is the
Koopman operator
.
See also
- Measure-preserving dynamical system
- Normalizing flow
- Optimal transport
Notes
References
- Bogachev, Vladimir I. (2007), Measure Theory, Berlin: ISBN 9783540345138
- Teschl, Gerald (2015), Topics in Real and Functional Analysis