Pushforward measure

In measure theory, 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.

Definition

Given measurable spaces (X1, Σ1) and (X2, Σ2), a measurable mapping f : X1  X2 and a measure μ : Σ1  [0, +∞], the pushforward of μ is defined to be the measure f(μ) : Σ2  [0, +∞] given by

(f_{*} (\mu)) (B) = \mu \left( f^{-1} (B) \right) \mbox{ for } B \in \Sigma_{2}.

This definition applies mutatis mutandis for a signed or complex measure.

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 \circ f is integrable with respect to the measure μ. In that case, the integrals coincide, i.e.,

\int_{X_2} g \, d(f_* \mu) = \int_{X_1} g \circ f \, d\mu.

Examples and applications

f^{(n)} = \underbrace{f \circ f \circ \dots \circ f}_{n \mathrm{\, times}} : X \to 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.

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. This operator typically satisfies the requirements of the Frobenius–Perron theorem, and the maximal eigenvalue of this theorem 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.

Notes

  1. Sections 3.6-3.7 in Bogachev

References

See also

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