In mathematics, a pushforward measure (also push forward or push-forward) 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 function 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.

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 is 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:

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

This forms a measurable 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.

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