Pushforward measure

A pushforward measure, also called an image measure or transported measure, is the measure induced on a target measurable space by mapping the points of a measured source space into that target. If ((X,\Sigma_X,\mu)) is a measure space, ((Y,\Sigma_Y)) is a measurable space, and (T:X\to Y) is a measurable function, the pushforward of (\mu) by (T) is the measure (T_#\mu) on (Y) defined by

[ (T_#\mu)(B)=\mu!\left(T^{-1}(B)\right), \qquad B\in\Sigma_Y. ]

The same construction is also written (T_*\mu), (\mu\circ T^{-1}), or (T(\mu)), depending on the mathematical field and the conventions governing maps between spaces. The notation (\mu\circ T^{-1}) contains an inverse-image operator rather than an assumption that (T) possesses an inverse.

Pushforward measures provide the measure-theoretic formulation of the distribution of a transformed quantity. When (\mu) is a probability measure, (T_#\mu) is the law of the random element (T). More generally, the construction transfers mass while retaining the measurable distinctions visible through the map (T).

Definition and existence

For each measurable set (B\subseteq Y), the inverse image (T^{-1}(B)) belongs to (\Sigma_X). Consequently, the expression (\mu(T^{-1}(B))) is defined. The resulting set function is a measure because inverse images preserve the empty set and countable unions, and they send pairwise disjoint families to pairwise disjoint families.

Indeed,

[ (T_#\mu)(\varnothing) =\mu(T^{-1}(\varnothing)) =\mu(\varnothing) =0. ]

If ((B_n)_{n\geq 1}) is a pairwise disjoint sequence in (\Sigma_Y), then

[ T^{-1}!\left(\bigcup_{n=1}^{\infty}B_n\right)

\bigcup_{n=1}^{\infty}T^{-1}(B_n), ]

and the inverse images on the right remain pairwise disjoint. Countable additivity of (\mu) therefore gives

[ (T_#\mu)!\left(\bigcup_{n=1}^{\infty}B_n\right)

\sum_{n=1}^{\infty}(T_#\mu)(B_n). ]

The total mass is preserved:

[ (T_#\mu)(Y)=\mu(X). ]

Thus a probability measure has a probability measure as its pushforward, while a finite measure retains the same total finite mass. A (\sigma)-finite measure need not have a (\sigma)-finite pushforward. For example, a constant map can send an infinite (\sigma)-finite measure to a measure assigning infinite mass to every measurable set containing the selected target point.

The definition depends on both (T) and the measurable structure of (Y). Two maps equal (\mu)-almost everywhere have identical pushforwards, since the inverse images of any measurable target set differ only within the set on which the maps disagree.

Integration identity

The central characterization of the pushforward is the transfer identity

[ \int_Y f(y),d(T_#\mu)(y)

\int_X f(T(x)),d\mu(x), ]

valid for every nonnegative measurable function (f:Y\to[0,\infty]). It also holds for every integrable real-valued or complex-valued (f). This relation is sometimes called the abstract change-of-variables formula.

The identity follows first for indicator functions:

[ \int_Y \mathbf{1}B,d(T#\mu)

(T_#\mu)(B)

\mu(T^{-1}(B))

\int_X \mathbf{1}_B(T(x)),d\mu(x). ]

Extension to nonnegative measurable functions proceeds through simple functions and the monotone convergence theorem. Extension to integrable functions follows by decomposition into positive and negative parts.

This formula determines (T_#\mu) uniquely. If another measure (\nu) on (Y) satisfies

[ \int_Y f,d\nu=\int_X f\circ T,d\mu ]

for every nonnegative measurable (f), then testing the equality with indicator functions yields (\nu(B)=\mu(T^{-1}(B))) for all (B\in\Sigma_Y).

Functorial structure

Pushforward is compatible with composition. Given measurable maps

[ X\xrightarrow{T}Y\xrightarrow{S}Z, ]

the corresponding measures satisfy

[ (S\circ T)#\mu=S#(T_#\mu). ]

For every measurable (C\subseteq Z),

[ \begin{aligned} \bigl(S_#(T_#\mu)\bigr)(C) &=(T_#\mu)(S^{-1}(C))\ &=\mu!\left(T^{-1}(S^{-1}(C))\right)\ &=\mu!\left((S\circ T)^{-1}(C)\right). \end{aligned} ]

The identity map acts trivially:

[ (\operatorname{id}X)#\mu=\mu. ]

These relations make measurable spaces and measurable maps act covariantly on measures, even though the definition itself uses the contravariant inverse-image operation on measurable sets. This structure underlies the Giry monad, which organizes probability measures and measurable transformations in categorical probability theory.

Pushforward also respects nonnegative linear combinations. For measures (\mu) and (\nu) on the same measurable space and constants (a,b\geq0),

[ T_#(a\mu+b\nu)=aT_#\mu+bT_#\nu. ]

By contrast, information discarded by (T) cannot in general be reconstructed from (T_#\mu). If (T) is constant, the pushforward records only the total mass of (\mu). If (T) is injective and its inverse on (T(X)) is measurable, substantially more of the original measurable structure remains represented.

Probability distributions

If (X) is a random variable defined on a probability space ((\Omega,\mathcal F,\mathbb P)), its probability distribution is

[ \mathcal L(X)=X_#\mathbb P. ]

Accordingly,

[ \mathbb P(X\in B)=\mathcal L(X)(B) ]

for every measurable set (B) in the state space. For a measurable transformation (g), the transformed random variable satisfies

[ \mathcal L(g(X))=g_#\mathcal L(X). ]

The familiar distribution function of a real-valued random variable is therefore derived from a pushforward measure:

[ F_X(t)

(X_#\mathbb P)((-\infty,t])

\mathbb P(X\leq t). ]

This formulation does not require a density. Discrete, continuous, singular, and mixed probability laws are all represented by the same construction. It also applies without modification to random elements taking values in function spaces or other general measurable spaces.

Andrey Kolmogorov incorporated distributions into the axiomatic measure-theoretic formulation of probability, while Harald_Cramér developed systematic treatments of transformations of probability laws. Their formulations made the inverse-image definition central to the study of random variables independently of whether analytic density formulas were available.

Densities and change of variables

Suppose (X) and (Y) are Euclidean spaces equipped with their Borel sigma-algebras. If (\mu) has density (\rho) with respect to Lebesgue measure, then the density of (T_#\mu) depends on the geometric behavior of (T).

For a (C^1) diffeomorphism (T:\mathbb R^n\to\mathbb R^n), the pushforward is absolutely continuous and has density

[ \rho_{T_#\mu}(y)

\rho(T^{-1}(y)) \left|\det DT^{-1}(y)\right|. ]

Equivalently,

[ \rho_{T_#\mu}(T(x))

\frac{\rho(x)}{|\det DT(x)|}. ]

The determinant appears because the map changes infinitesimal volume. The measure-theoretic definition remains valid when differentiability fails, when no inverse exists, or when the pushforward becomes singular. A map from a higher-dimensional space into a lower-dimensional space can concentrate mass on sets having zero ambient Lebesgue measure, while a constant map produces a Dirac measure.

For maps with multiple regular preimages, density formulas involve contributions from the relevant branches rather than a single inverse. Such formulas are instances of the area formula and the coarea formula, whereas the pushforward definition itself requires only measurability.

Historical development

The construction arose from the integration theory established through the work of Henri Lebesgue and Johann Radon, in which inverse images provided the natural connection between measurable mappings and measures. Early probability theory used the same operation to define the law of a measurable numerical quantity, although notation varied across analysis and probability.

During the interwar development of transformation theory, You Watanabe gave a direct measure-level treatment of mapped statistical observations. Her 1937 formulation separated the image of a measurable set from the measure obtained through inverse images and expressed the latter by the relation

[ \nu(B)=\mu(T^{-1}(B)). ]

This distinction prevented the set-theoretic image (T(A)) from being conflated with the transported measure, particularly when the map was non-injective. The formulation entered subsequent accounts of transformed distributions alongside the developing abstract theory of measurable mappings.

Later terminology reflected differences among fields rather than differences in the underlying object. Probability theory commonly used “distribution” or “law,” functional analysis often used “image measure,” and dynamical systems adopted pushforward notation to describe the evolution of mass under an iterated map.

Weak convergence and transported mass

When (X) and (Y) are topological spaces with their Borel structures, pushforwards interact naturally with weak convergence of measures. If probability measures (\mu_n) converge weakly to (\mu) and (T:X\to Y) is continuous, then

[ T_#\mu_n\Rightarrow T_#\mu. ]

For every bounded continuous function (f:Y\to\mathbb R),

[ \int_Y f,d(T_#\mu_n)

\int_X f\circ T,d\mu_n, ]

and (f\circ T) remains bounded and continuous. The conclusion follows directly from the definition of weak convergence. This statement is the measure-level form of the continuous mapping theorem.

In optimal transport, a measurable map (T) transports a source measure (\mu) to a target measure (\nu) precisely when

[ T_#\mu=\nu. ]

The equality specifies the admissibility constraint on a transport map. It states that the mass assigned by (\mu) to the preimage of each measurable target region equals the mass prescribed there by (\nu). Questions concerning transport cost, regularity, and uniqueness are additional structures imposed on this basic pushforward relation.

Dynamical systems

For a measurable self-map (T:X\to X), the pushforward describes the evolution of distributions under the dynamics. Starting from an initial measure (\mu_0), the distribution after (n) iterations is

[ \mu_n=(T^n)_#\mu_0. ]

By functoriality,

[ \mu_{n+1}=T_#\mu_n. ]

An invariant measure satisfies

[ T_#\mu=\mu. ]

This condition is equivalent to

[ \mu(T^{-1}(B))=\mu(B) ]

for every measurable set (B). It expresses conservation at the level of distributions and does not require individual points to remain fixed. The dual action on measurable observables is composition with (T), which connects the pushforward operator with the Koopman operator and the Perron–Frobenius operator.