Disintegration theorem

The disintegration theorem is a cornerstone result in measure theory and probability theory that asserts the existence of a decomposition of a measure \mu on a measurable space (X, \mathcal{A}) with respect to a measurable map f: X \to Y into a family of probability measures \{\mu_y\}_{y \in Y} on X, such that for every measurable set E \subseteq X, \mu(E) = \int_Y \mu_y(f^{-1}(y) \cap E) \, \nu(dy), where \nu = \mu \circ f^{-1} is the pushforward measure on Y, and each \mu_y is supported on the fiber f^{-1}(y).^[1] This decomposition, known as a disintegration of \mu over \nu, requires appropriate regularity conditions, such as the spaces being Radon or countably compact, and ensures the family \{\mu_y\} is measurable in y.^[2] Originating from efforts to rigorize conditional probabilities, the theorem traces its roots to Andrey Kolmogorov's 1933 axiomatic foundations of probability, where he introduced conditional expectations and extended them to arbitrary index sets, laying the groundwork for handling conditional distributions in general measure spaces.^[3] Subsequent developments by David Blackwell in 1956 formalized aspects of conditional distributions under Polish space assumptions, while later generalizations by Jan Pachl (1978) and Michel Talagrand (1981) extended existence results to broader classes of measures, including non-locally compact spaces.^[1] David Fremlin's comprehensive treatment in his 2004 Measure Theory (Volume 4) synthesizes these advances, proving the theorem in settings like Radon spaces and probability measures with almost strong liftings, emphasizing its analogy to Fubini's theorem for iterated integrals.^[1] The theorem's importance lies in its role as a bridge between abstract measure theory and applied probability, enabling the precise definition of conditional distributions and expectations in continuous spaces, where naive conditioning on events of measure zero fails.^[2] It underpins key applications, including the analysis of stochastic processes like Markov chains and Brownian motion via stopping times and \sigma-algebras, the study of Gaussian processes through covariance-preserving decompositions, and the handling of exchangeable random variables on infinite product spaces.^[1] In statistics, it supports algorithms like the expectation-maximization (EM) method by providing rigorous conditional measures for sufficiency and inference.^[2] Existence is not guaranteed in all settings—counterexamples exist for certain Borel partitions without integrable disintegrations—but under standard assumptions like separability or completeness, the decomposition is essentially unique almost everywhere with respect to \nu.^[4]

Background and motivation

Historical development

The foundations of the disintegration theorem lie in the early development of measure theory by Émile Borel and Henri Lebesgue. Borel laid the groundwork with his 1898 introduction of measure for Borel sets, providing a systematic way to assign sizes to subsets of the real line. Lebesgue built upon this in 1902 by developing the Lebesgue integral, which extended integration to a broader class of functions and enabled rigorous treatment of limits and products in analysis. Early ideas precursor to disintegration emerged through theorems on product measures. Guido Fubini's 1907 theorem established conditions under which double integrals over product spaces could be evaluated as iterated integrals, facilitating the decomposition of measures on Cartesian products. Leonida Tonelli extended this in 1909 with a version for non-negative measurable functions, removing some restrictions and emphasizing the role of σ-finiteness in such decompositions. These results provided essential tools for breaking down joint measures into marginal and conditional components, analogous to later disintegration frameworks.^[5] John von Neumann's 1932 work on conditional expectations in Hilbert spaces further influenced the theorem's conceptual development. In his foundational text on quantum mechanics, von Neumann defined conditional expectations as projections onto subspaces, offering a rigorous probabilistic interpretation that paralleled the idea of disintegrating measures with respect to a sub-σ-algebra. This approach connected operator theory to probability and laid groundwork for general measure-theoretic disintegrations. Andrey Kolmogorov's 1933 axiomatization of probability theory introduced conditional expectations more broadly, extending them to general measure spaces and providing a foundation for conditional distributions.^[3] Vladimir Rokhlin's 1949 theorem established the existence of disintegrations with respect to measurable partitions in standard probability spaces, marking a key formalization in ergodic theory.^[6] Laurent Schwartz provided a treatment of the disintegration theorem for general measures in his 1974 lectures, extending it beyond special cases like L² spaces to arbitrary σ-finite measures on standard spaces. His work emphasized the existence and uniqueness of disintegrations under suitable separability conditions, integrating it into the broader theory of distributions and Radon measures.^[7] A comprehensive modern reference is David Fremlin's "Measure Theory" (Volume 4, Chapter 45, 2003), which provides a detailed exposition of disintegrations, including proofs of existence and uniqueness, while highlighting connections to perfect measures and Fubini's theorem as a motivating analogy.^[1]

Intuitive explanation

The disintegration theorem provides a conceptual framework for decomposing a measure on a product space into simpler, fiber-wise components, akin to slicing a cake to analyze each layer individually while preserving the overall structure. Imagine a measure μ on X × Y; the theorem posits that μ can be expressed as an integral over a family of measures {μ_y} on the fibers X × {y}, where each μ_y describes the "conditional" distribution along that fixed y-slice, valid for almost every y with respect to the marginal measure on Y. This fiber decomposition allows one to study the global measure by examining local behaviors on these one-dimensional slices, much like averaging densities over parallel lines to understand a two-dimensional density.^[1] A straightforward illustration arises with the uniform Lebesgue measure on the unit square [0,1] × [0,1]. Disintegrating with respect to the horizontal coordinate yields, for each fixed y in [0,1], a uniform measure μ_y on the vertical line segment {0 ≤ x ≤ 1} × {y}, such that the original area measure is recovered by integrating these line measures over y. This process transforms the two-dimensional uniform distribution into a collection of one-dimensional uniforms, highlighting how disintegration partitions complexity into manageable parts.^[2] The theorem generalizes Fubini's theorem, which enables iterated integrals for product measures, by providing analogous fiber decompositions for measures with respect to arbitrary measurable maps, even in non-product settings. By yielding these fiber-wise disintegrations, it facilitates averaging over slices to compute expectations or probabilities, proving invaluable for conceptualizing conditional phenomena and simplifying analysis in higher dimensions.^[1]

Formal statement

Prerequisites and assumptions

The disintegration theorem is set in the framework of measure theory, beginning with two measurable spaces (X, \Sigma_X) and (Y, \Sigma_Y), where X and Y are nonempty sets equipped with \sigma-algebras \Sigma_X and \Sigma_Y of subsets, respectively. These \sigma-algebras determine the measurable functions and sets for integration and probability. The product space X \times Y is then considered with the product \sigma-algebra \Sigma_X \otimes \Sigma_Y, which is the smallest \sigma-algebra containing all rectangles of the form A \times B where A \in \Sigma_X and B \in \Sigma_Y; this structure allows for the extension of measures from components to the joint space via Fubini-type results.^[1] A key measure \mu is defined on the measurable space (X \times Y, \Sigma_X \otimes \Sigma_Y) and is required to be a probability measure. This condition, which implies \sigma-finiteness and total mass 1, prevents pathological behaviors and enables the theorem's constructions by ensuring finite approximations and normalized conditionals.^[1] The projection map \pi: X \times Y \to X given by \pi(x, y) = x plays a central role and is measurable from (X \times Y, \Sigma_X \otimes \Sigma_Y) to (X, \Sigma_X), since for any A \in \Sigma_X, the preimage \pi^{-1}(A) = A \times Y lies in \Sigma_X \otimes \Sigma_Y. This measurability ensures that the pushforward measure \lambda = \mu \circ \pi^{-1} on (X, \Sigma_X) inherits properties from \mu and serves as the marginal measure with respect to which conditional properties are defined.^[1]^[6] For regularity in the disintegration process, particularly to guarantee the existence and separability of conditional measures along fibers \pi^{-1}(x), the space X is assumed to be a standard Borel space. Standard Borel spaces include Polish spaces—complete separable metric spaces endowed with their Borel \sigma-algebras generated by the open sets—as well as countable discrete spaces; these ensure that \Sigma_X is countably generated, separates points, and admits a rich class of measurable selections.^[1]^[8] All relevant properties of the disintegration, such as equalities or consistencies between measures, are required to hold almost everywhere with respect to the pushforward measure \lambda on X. This means that for any two objects satisfying the conditions (e.g., versions of conditional measures), their difference vanishes on a \lambda-negligible set, i.e., a set N \in \Sigma_X with \lambda(N) = 0, allowing for non-uniqueness up to such null sets while preserving integrals and expectations.^[1]^[6]

Theorem formulation

The disintegration theorem provides a rigorous decomposition of a measure on a product space into conditional measures along the fibers of the projection map. Specifically, let (X, \Sigma_X) and (Y, \Sigma_Y) be measurable spaces with X a standard Borel space, and let \mu be a probability measure on the product \sigma-algebra \Sigma_X \otimes \Sigma_Y on X \times Y. Let \pi: X \times Y \to X denote the canonical projection onto the first factor, and let \lambda = \mu \circ \pi^{-1} be the pushforward (marginal) measure on (X, \Sigma_X). There exists a family \{\nu_x\}_{x \in X} of probability measures on (Y, \Sigma_Y) such that, for every B \in \Sigma_Y, the map x \mapsto \nu_x(B) is \Sigma_X-measurable, and for all A \in \Sigma_X, B \in \Sigma_Y,

\mu(A \times B) = \int_A \nu_x(B) \, d\lambda(x).

^[1] This decomposition is unique up to \lambda-almost everywhere equality: if \{\tilde{\nu}_x\}_{x \in X} is another such family, then \lambda\{x : \nu_x \neq \tilde{\nu}_x\} = 0.^[1] An equivalent notational formulation expresses the disintegration of \mu as

\mu = \int_X \nu_x \otimes \delta_x \, d\lambda(x),

where \delta_x is the Dirac measure at x \in X.^[1] The theorem extends to more general settings. For \sigma-finite measures, disintegrations exist locally on sets of finite marginal measure, yielding probability measures on those restrictions; globally, the fiber measures \nu_x are \sigma-finite with \nu_x(Y) finite \lambda-a.e., and probabilities obtained by normalization where possible. The general disintegration theorem applies to arbitrary measurable maps f: (Z, \Sigma_Z) \to (W, \Sigma_W) with W standard Borel and probability measure \mu on Z, yielding \{\mu_w\}_{w \in W} probability measures on Z supported on f^{-1}(w) such that \mu(E) = \int_W \mu_w(E) \, d\nu(w) for \nu = \mu \circ f^{-1}, unique \nu-a.e.^[1]

Proof overview

Existence of disintegration

The existence of a disintegration for a measure \mu on a measurable space (X, \Sigma) with respect to a measurable projection \pi: X \to Y and the induced measure \nu = \pi_* \mu on (Y, \mathcal{T}) relies on constructing a family of probability measures \{\nu_x\}_{x \in Y} (or \sigma-finite versions thereof) such that each \nu_x is supported on the fiber \pi^{-1}(x) and satisfies the integral condition \int_Y \left( \int_{X} f \, d\nu_x \right) d\nu(x) = \int_X f \, d\mu for all measurable f: X \to [0, \infty).^[1] This construction proceeds via conditional expectations in L^1(\mu), assuming \nu is \sigma-finite and the spaces are standard Borel to ensure measurability.^[9] The core step defines \nu_x(B) for measurable B \subset X as the Radon-Nikodym derivative of the restricted measure \mu|_{B \cap \pi^{-1}(Y)} with respect to the pushforward \nu, evaluated at x; more precisely, for integrable indicators $1_B, the map x \mapsto \int_{B \cap \pi^{-1}(x)} d\mu is \mathcal{T}-measurable, and \nu_x(B) is obtained as the conditional expectation E^\mu[1_B \mid \pi](x), normalized if necessary to account for the fiber measure.^[7] This leverages the Radon-Nikodym theorem applied to the \sigma-finite measure \nu, ensuring the derivative exists \nu-almost everywhere since the conditional expectation operator T: L^1(\mu) \to L^1(\nu) is well-defined under \sigma-finiteness, which decomposes \nu into countable finite parts for iterative application.^[1] The \sigma-finiteness of \nu is crucial here, as it guarantees the existence of densities for subprobability measures on fibers without infinite mass accumulation.^[9] To ensure the family \{\nu_x\} is measurable in x, the Kuratowski–Ryll-Nardzewski measurable selection theorem is invoked, particularly in standard Borel spaces where the fibers admit measurable parametrizations; this selects a measurable version of the multifunction x \mapsto \{\eta : \eta \ll \mu|_{\pi^{-1}(x)}, \int d\eta = 1\} from the weak* topology on measures.^[1] The proof outline begins with simple functions: for a partition of X into finitely many measurable sets A_i with \pi(A_i) = Y_i, disintegrate indicators $1_{A_i} directly via pointwise Radon-Nikodym derivatives on each Y_i, yielding \nu_x(A_i) = E^\mu[1_{A_i} \mid \pi](x) \nu-a.e.^[7] Extension to general nonnegative measurable functions follows by monotone approximation: the set of functions f for which x \mapsto \int f \, d\nu_x equals E^\mu[f \mid \pi](x) forms a monotone class containing indicators of a \pi-system generating \Sigma, hence closed under limits by the monotone class theorem, assuming \sigma-finiteness to control integrability.^[1] Alternatively, a Carathéodory extension approach constructs \nu_x on the algebra generated by compact sets (in Polish spaces), starting from finite premeasures on cylinders and extending uniquely due to inner regularity of Radon measures, with \sigma-finiteness ensuring countable additivity holds almost everywhere.^[9] This family is well-defined \nu-almost everywhere, as \sigma-finiteness partitions Y into sets of finite \nu-measure where the derivatives are finite and the selection is uniform.^[7]

Uniqueness properties

The uniqueness of a disintegration of a measure \mu on a product space (X \times Y, \mathcal{A} \otimes \mathcal{B}) with respect to the projection \pi: X \times Y \to X is a fundamental property under standard assumptions. Suppose \{\nu_x\}_{x \in X} and \{\nu'_x\}_{x \in X} are two families of measures on (Y, \mathcal{B}) such that both disintegrate \mu, meaning \mu(E) = \int_X \nu_x(F_x) \, d(\mu \circ \pi^{-1})(x) for all measurable E \subset X \times Y with F_x = \{y \in Y : (x,y) \in E\}, and similarly for \{\nu'_x\}. Then, \nu_x = \nu'_x for \mu \circ \pi^{-1}-almost every x \in X.^[1]^[9] This uniqueness follows from the integral equation defining the disintegration and the Radon-Nikodym theorem applied to the measures induced on measurable sections. Specifically, for any bounded measurable function f: Y \to \mathbb{R}, the equality \int_{X \times Y} f(y) \, d\mu(x,y) = \int_X \left( \int_Y f(y) \, d\nu_x(y) \right) d(\mu \circ \pi^{-1})(x) holds, and the same for \nu'. Subtracting these yields \int_X \left( \int_Y f(y) \, d(\nu_x - \nu'_x)(y) \right) d(\mu \circ \pi^{-1})(x) = 0 for all such f, implying \nu_x = \nu'_x \mu \circ \pi^{-1}-a.e. by the uniqueness of Radon-Nikodym derivatives in the \sigma-finite case.^[1] When \mu is a probability measure, each \nu_x in the disintegration can be chosen as a probability measure on Y, normalized so that \nu_x(Y) = 1 for \mu \circ \pi^{-1}-almost every x. This follows from the \sigma-finiteness of \mu \circ \pi^{-1} allowing normalization via Radon-Nikodym derivatives.^[1] Disintegrations exhibit stability under measure-preserving transformations. If \phi: X \to X is a measurable map preserving \mu \circ \pi^{-1}, then the pushed-forward family \{\nu_{\phi(x)}\}_{x \in X} disintegrates the pushforward measure \mu \circ (\mathrm{id}_X \times \mathrm{id}_Y) \circ (\phi \times \mathrm{id}_Y)^{-1}, preserving uniqueness up to the null sets of the transformed base measure. In the context of group actions, such as a compact group G acting measurably on X with invariant probability measure \mu, the disintegration into measures on orbits is unique and G-invariant.^[1] Uniqueness fails in non-\sigma-finite settings without additional normalization. Another counterexample involves a non-\sigma-finite measure on a product space where mixing measures are not unique, leading to distinct families satisfying the disintegration equation.

Applications

Product measures and integration

The disintegration theorem provides a foundational tool for evaluating integrals over product spaces by decomposing a measure \mu on X \times Y with respect to the projection \pi: X \times Y \to X. Specifically, when \mu is the product measure \lambda \times \rho, where \lambda is a measure on X and \rho on Y, the disintegration \{\nu_x\}_{x \in X} yields \nu_x = \rho for \lambda-almost every x \in X.^[1] This decomposition ensures that for any measurable function f: X \times Y \to [0, \infty),

\int_{X \times Y} f \, d\mu = \int_X \left( \int_Y f(x,y) \, d\nu_x(y) \right) d\lambda(x) = \int_X \left( \int_Y f(x,y) \, d\rho(y) \right) d\lambda(x),

recovering the Fubini-Tonelli theorem under the standard assumptions of \sigma-finiteness and non-negativity.^[1] This iterated integral representation holds even when f is integrable, allowing the interchange of integration order provided the integrals converge absolutely. For measures \mu that are not product measures, the disintegration theorem extends the ability to compute integrals via iteration, as long as \mu is \lambda-absolutely continuous with respect to the projection. In this case, the family \{\nu_x\} consists of probability measures (or finite measures, normalized appropriately) supported on the fibers \pi^{-1}(x), enabling

\int_{X \times Y} f \, d\mu = \int_X \left( \int_Y f(x,y) \, d\nu_x(y) \right) d\lambda(x)

for suitable f, without requiring \mu = \lambda \times \rho.^[1] This framework is particularly useful in abstract measure spaces where direct product structure is absent, yet fiberwise integration remains well-defined almost everywhere. A concrete illustration arises with the Lebesgue measure \lambda_2 on the unit square [0,1]^2, projected onto the base [0,1] via \pi(x,y) = x. Here, \lambda_2 disintegrates with respect to the one-dimensional Lebesgue measure \lambda_1 on [0,1], yielding \nu_x as the uniform (Lebesgue) measure on the vertical fiber \{x\} \times [0,1] for \lambda_1-almost every x.^[10] This setup confirms the integral formula, as \int_{[0,1]^2} f \, d\lambda_2 = \int_0^1 \left( \int_0^1 f(x,y) \, dy \right) dx for integrable f, aligning with the classical double integral over rectangles.^[10]

Conditional distributions in probability

In probability theory, the disintegration theorem provides a rigorous foundation for conditional distributions on a probability space (\Omega, \mathcal{F}, P), where P is a probability measure. Consider random variables X and Y taking values in measurable spaces (E, \mathcal{E}) and (F, \mathcal{F}), respectively, with joint distribution \mu = P_{X,Y} on the product space E \times F. The theorem asserts the existence of a family of probability measures \{\nu_x\}_{x \in E} on F, known as regular conditional distributions, such that \nu_x(\cdot) = P(Y \in \cdot \mid X = x) for \mu-almost every x \in E, and \mu disintegrates as \mu(A \times B) = \int_A \nu_x(B) \, dP_X(x) for all A \in \mathcal{E}, B \in \mathcal{F}.^[9]^[11] This disintegration directly connects to conditional expectations. For a measurable function f: F \to \mathbb{R} that is integrable with respect to \nu_x for almost every x, the conditional expectation satisfies E[f(Y) \mid X = x] = \int_F f(y) \, d\nu_x(y), and more generally, E[g(X) f(Y)] = \int_E g(x) \left( \int_F f(y) \, d\nu_x(y) \right) dP_X(x) for bounded measurable g: E \to \mathbb{R}. This formulation ensures that conditional expectations can be computed via integration against the disintegrated measures, providing a measure-theoretic justification for manipulations of conditional probabilities.^[9]^[11] A concrete illustration arises in the case of a bivariate normal distribution. Suppose (X, Y) follows a standard bivariate normal distribution with correlation \rho \in (-1, 1), so the joint density is

f_{X,Y}(x,y) = \frac{1}{2\pi \sqrt{1 - \rho^2}} \exp\left( -\frac{x^2 - 2\rho x y + y^2}{2(1 - \rho^2)} \right).

The marginal P_X is standard normal N(0,1), and the disintegration yields conditional distributions \nu_x = P_{Y \mid X=x} that are normal N(\rho x, 1 - \rho^2), with density

f_{Y \mid X}(y \mid x) = \frac{1}{\sqrt{2\pi (1 - \rho^2)}} \exp\left( -\frac{(y - \rho x)^2}{2(1 - \rho^2)} \right).

Thus, the conditional mean E[Y \mid X = x] = \rho x follows directly from integrating against \nu_x.^[11] Regarding uniqueness, while the family \{\nu_x\} may not be unique pointwise for every x, it is unique in the L^1 sense with respect to expectations: for any integrable f, the map x \mapsto \int f \, d\nu_x is unique P_X-almost everywhere, ensuring that all disintegrations yield the same conditional expectations almost surely. This property holds under standard assumptions like Polish spaces and \sigma-finite measures, as guaranteed by the theorem's existence results via Radon-Nikodym derivatives.^[9]^[7]

Change of variables in vector calculus

The disintegration theorem provides a framework for understanding change of variables in multiple integrals over Euclidean spaces, particularly when transforming Lebesgue measure under smooth mappings. Consider a diffeomorphism \phi: \mathbb{R}^k \to \mathbb{R}^k that is C^1 with \det D\phi(u) \neq 0 for all u. The pushforward measure \phi_* \lambda, where \lambda is the Lebesgue measure on \mathbb{R}^k, admits a disintegration with respect to the Lebesgue measure on the domain, yielding conditional measures \nu_u that scale the volume elements. Specifically, the Jacobian determinant |\det D\phi(u)| emerges as the Radon-Nikodym derivative governing this scaling, ensuring that the integral of a nonnegative measurable function f transforms as \int_{\mathbb{R}^k} f(\phi(u)) \, d\lambda(u) = \int_{\mathbb{R}^k} f(x) \, |\det D\phi(\phi^{-1}(x))| \, d\lambda(x). This follows from the absolute continuity of the pushforward with respect to Lebesgue measure and the uniqueness of disintegration, where the conditional measures \nu_x on the fibers are adjusted by the local volume distortion induced by D\phi.^[12] In the context of product spaces, the disintegration theorem extends this to transformations \phi: \mathbb{R}^{n+m} \to \mathbb{R}^n \times \mathbb{R}^m, where the projection onto the first factor induces a disintegration of the Lebesgue measure \lambda_{n+m}. The resulting family of measures \{\nu_x\}_{x \in \mathbb{R}^n} on the fibers \mathbb{R}^m incorporates the Jacobian adjustment for the density. For an integrable function f: \mathbb{R}^n \times \mathbb{R}^m \to [0, \infty), the change of variables formula becomes

\int_{\mathbb{R}^{n+m}} f(\phi(u)) \, |\det D\phi(u)| \, d\lambda_{n+m}(u) = \int_{\mathbb{R}^n} \left( \int_{\mathbb{R}^m} f(x,y) \, d\nu_x(y) \right) d\lambda_n(x),

where the scaling by |\det D\phi(u)| ensures equivalence to the original Lebesgue integral without the absolute value in the transformed coordinates. This formulation highlights how disintegration decomposes the integral into marginal and conditional components, with the Jacobian accounting for the geometric distortion along the fibers.^[12] A concrete illustration arises in polar coordinates on \mathbb{R}^2, where the Lebesgue measure \lambda_2 disintegrates with respect to the radial projection \rho: \mathbb{R}^2 \to [0, \infty), \rho(x) = \|x\|. By the coarea formula,

\int_{\mathbb{R}^2} f(x) \, d\lambda_2(x) = \int_0^\infty \left( \int_{\{ \|z\| = r \}} f(z) \, d\mathcal{H}^1(z) \right) dr,

for nonnegative integrable f: \mathbb{R}^2 \to [0, \infty). The inner integral uses 1-dimensional Hausdorff measure (arc length) on the circle of radius r, equivalent to \int_0^\infty \int_0^{2\pi} f(r \cos \theta, r \sin \theta) r \, d\theta \, dr, where the Jacobian factor r from the map (r, \theta) \mapsto (r \cos \theta, r \sin \theta) is incorporated. This decomposition exemplifies how disintegration captures the radial and angular components of the measure, facilitating computations in cylindrical symmetry.^[13] The coarea formula represents a variant of this disintegration for level sets of Lipschitz maps u: \mathbb{R}^n \to \mathbb{R}, disintegrating the Lebesgue measure along the submanifolds \{x : u(x) = t\}. For a Lipschitz function f: \mathbb{R}^n \to [0, \infty) and u Lipschitz with |\nabla u| > 0 almost everywhere,

\int_{\mathbb{R}^n} f(x) |\nabla u(x)| \, d\lambda_n(x) = \int_{-\infty}^\infty \left( \int_{\{u = t\}} f(x) \, d\mathcal{H}^{n-1}(x) \right) dt,

where \mathcal{H}^{n-1} is the (n-1)-dimensional Hausdorff measure. This arises as a disintegration of \lambda_n with respect to the pushforward u_* \lambda_n, with conditional measures \nu_t given by the normalized Hausdorff measures on the level sets, scaled by the coarea factor $1/|\nabla u|. Such formulas generalize the Jacobian adjustment to nonsmooth settings, linking directly to applications in geometric measure theory.^[12]

Extensions to optimal transport

In the context of Wasserstein spaces, the disintegration theorem provides a framework for decomposing optimal transport plans into conditional measures, representing joint measures \gamma \in \Pi(\mu, \nu) as \gamma = \int_X \nu_x \, d\mu(x), where \nu_x are probability measures on the target space Y uniquely determined up to \mu-negligible sets.^[14] This decomposition facilitates the analysis of transport costs in the space \mathcal{P}_2(X) of probability measures equipped with the 2-Wasserstein metric W_2, enabling the construction of geodesics between measures via optimal plans and supporting stability results in non-branching spaces.^[15] A key application arises in the Monge-Kantorovich problem, where optimal transport plans are viewed through disintegrations of couplings, leading to transport maps derived from the conditional measures \nu_x. For instance, the abstract Monge problem \inf_f \int_X \tilde{c}(x, f(x)) \, d\mu(x), with f: X \to \mathcal{P}(Y) and \tilde{c}(x, \lambda) = \int_Y c(x,y) \, d\lambda(y), equates to the constrained Kantorovich problem over transport classes [\gamma], allowing recovery of deterministic transport maps t(x) = \beta(f(x)) for non-atomic \mu under suitable conditions.^[14] In discrete settings, such as \mu = \frac{1}{3} \sum_{i=1}^3 \delta_{x_i} and \nu = \frac{1}{6} \delta_{y_1} + \frac{5}{6} \delta_{y_2}, disintegrations reveal how mass splitting affects optimal classes, illustrating the theorem's role in partitioning transport costs.^[14] Recent developments connect disintegration to geometric properties in optimal transport, particularly through fiber-wise metrics and regularity of disintegration maps in metric measure spaces. In locally compact separable metric spaces, transport plans disintegrate as \gamma = \mu \otimes \gamma_x, with the disintegration map exhibiting nearly weak continuity when the second marginal is absolutely continuous with respect to a reference volume, ensuring paths of measures link disintegrations weakly continuously and imposing rigidity: if one fiber measure is absolutely continuous, all are.^[16] For metric measure foliations, this yields isometry preservation under optimal transport, enhancing understanding of curvature and geodesic structures in Wasserstein spaces.^[16] The disintegration theorem extends to differential forms and currents in geometric measure theory, where it decomposes varifolds V \in V_n(U) as V(dx, dT) = \|V\|(dx) \otimes \mu_x(dT), with \mu_x probability measures on the Grassmannian G(n,m), separating mass and tangential directions to prove rectifiability.^[17] This applies to currents, such as rectifiable 1-currents T = \int_{\mathrm{Lip}\, R} \gamma \, d\pi(\gamma), enabling slicing, boundary analysis, and energy estimates like M_\alpha(T), with links to optimal transport via branched models where decompositions ensure cost efficiency for sub-measures in traffic paths.^[17] In anisotropic settings, it supports Allard's rectifiability theorem variants, showing varifolds with bounded first variation are rectifiable at density-positive points, aiding compactness for integral varifolds in transport-related minimizers.^[17]