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


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

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.

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 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


  1. ^ Sections 3.6–3.7 in Bogachev