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 \mu 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 f # \mu.

Properties

Change of variable formula

Theorem: A measurable function g on X2 is integrable with respect to the pushforward measure f∗(μ) if and only if the composition g \circ f is integrable with respect to the measure μ. In that case, the integrals coincide, i.e., Note that in the previous formula.

Functoriality

Pushforwards of measures allow to induce, from a function between measurable spaces f:X\to Y, a function between the spaces of measures. As with many induced mappings, this construction has the structure of a functor, on the category of measurable spaces. For the special case of probability measures, this property amounts to functoriality of the Giry monad.

Examples and applications

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.