Disintegration Map: Measure & Wasserstein Geometry

• Disintegration map is a measurable function that decomposes a measure into conditional measures on fibers, embodying regular conditional probabilities.
• It is treated as a geometric object in a Wasserstein space, capturing how conditional measures and their supports vary and align with metric measure foliations.
• The framework introduces derivative and energy functionals to classify disintegration maps, which aids in diagnosing geometric perturbations and transport classes in optimal transport.

A disintegration map is the map that assigns to each point in a base or quotient space the corresponding conditional measure arising from the disintegration of a measure along a measurable map. In contemporary work, it is treated not merely as a bookkeeping device for conditional probabilities, but as a geometric object valued in a Wasserstein space, capable of encoding how conditional measures vary, how their supports sit inside fibers, and when this variation realizes a metric measure foliation (Münch et al., 17 Sep 2025). Closely related formulations appear in optimal transport, constructive conditioning on manifolds, computable analysis, and categorical probability, where the same basic object links conditional measures, transport plans, regularity properties, and Bayesian inversion (Granieri et al., 2012, Costa et al., 1 Aug 2025, Ackerman et al., 2015, Cho et al., 2017).

1. Measure-theoretic definition

Given measurable spaces $X$ and $Y$, a measure $\mu$ on $X$, and a measurable map $\pi:X\to Y$, disintegration is the decomposition of $\mu$ into a family of conditional measures $\{\mu_y\}_{y\in Y}$ such that, for $\nu:=\pi_*\mu$, the assignment $y\mapsto\mu_y$ is Borel for $\nu$-almost every $Y$0, each $Y$1 is a probability measure on $Y$2 supported on the fiber $Y$3, and

$Y$4

for every Borel set $Y$5. The associated disintegration map is then

$Y$6

or, in a more general measure-theoretic formulation, a Borel map into $Y$7 (Münch et al., 17 Sep 2025, Possobon et al., 2022).

This formulation is a version of regular conditional probability. In the kernel language used in constructive and categorical treatments, a disintegration is a transition kernel $Y$8 for a measurable map $Y$9, supported on $\mu$0 and satisfying the law of total probability. The same object is therefore simultaneously a family of conditional measures, a measurable kernel, and a map into a space of measures (Costa et al., 1 Aug 2025, Cho et al., 2017).

Existence and uniqueness depend on the ambient category of spaces and on the regularity demanded of the disintegration. For locally compact, separable metric spaces and finite Radon measures, one has a Borel disintegration $\mu$1 concentrated on fibers and reconstructing the original measure (Possobon et al., 2022). For complete separable metric spaces, measurable disintegrations exist in the Radon setting, but uniqueness is only up to null sets; if a continuous version exists, then it is unique on the support of the pushforward measure (Ackerman et al., 2015). In the standard Borel setting, disintegration is available for joint states in the measure-theoretic probability category, whereas outside such well-behaved classes existence can fail (Cho et al., 2017).

A recurrent misconception is to treat disintegration as intrinsically unique pointwise. The literature instead distinguishes measurable existence, uniqueness up to $\mu$2-almost everywhere equivalence, and stronger forms such as unique continuous disintegration. The Tjur property provides a sharp criterion: for Radon measures on a complete separable metric space with continuous surjective $\mu$3, every point is Tjur if and only if there is a unique continuous disintegration along $\mu$4 (Ackerman et al., 2015).

2. Wasserstein-geometric formulation

When the conditional measures are regarded as points of a Wasserstein space, the disintegration map becomes a geometric morphism. For $\mu$5, $\mu$6 denotes the space of probability measures with finite $\mu$7-th moment, equipped with the Wasserstein distance

$\mu$8

This turns $\mu$9 into a map from the base to a metric space of measures (Münch et al., 17 Sep 2025).

The central setting in the 2025 classification paper is a foliation $X$0 of $X$1 by closed subsets, with quotient map $X$2. A metric foliation requires that for all leaves $X$3 and any $X$4,

$X$5

If $X$6 is a disintegration of $X$7 with respect to $X$8, then a $X$9-metric measure foliation is characterized by

$\pi:X\to Y$0

In that case the disintegration map is an isometry between the base $\pi:X\to Y$1, endowed with $\pi:X\to Y$2, and its image in Wasserstein space (Münch et al., 17 Sep 2025).

This perspective aligns with earlier work on the geometric properties of disintegration. If the quotient or fibration is a metric measure foliation, then the disintegration map is weakly continuous everywhere, and the Wasserstein geometry of the image reflects the metric structure of the base. Under weak continuity and path-connectedness of the base, one obtains a path of conditional measures $\pi:X\to Y$3, and in suitable Riemannian settings absolute continuity can propagate along that path as a rigidity phenomenon (Possobon et al., 2022).

The geometric significance is that the disintegration map compares two structures at once: the geometry of the fibers in $\pi:X\to Y$4 and the geometry of the corresponding conditional measures in $\pi:X\to Y$5. In the metric measure foliation case these structures are perfectly aligned; outside that case the mismatch is measurable in Wasserstein terms (Münch et al., 17 Sep 2025).

3. Classification via derivative and energy

A principal contribution of the classification framework is the introduction of a derivative and an energy functional for disintegration maps. The derivative at $\pi:X\to Y$6 is defined by

$\pi:X\to Y$7

and the energy functional is

$\pi:X\to Y$8

The paper states two basic facts: $\pi:X\to Y$9 always, and $\mu$0 if and only if the conditional measures satisfy the isometry condition

$\mu$1

for all $\mu$2, and have full support on leaves, $\mu$3 (Münch et al., 17 Sep 2025).

This gives a classification criterion: the conditional measures arise from a metric measure foliation exactly when the disintegration map is isometric in this sense and each conditional measure is fully supported on its leaf. The paper also emphasizes that isometry alone is not sufficient; if support fails, one may still obtain an isometric relation in Wasserstein space without obtaining a genuine metric measure foliation. This directly addresses a common confusion between metric matching of conditional laws and full geometric realization of the leaves (Münch et al., 17 Sep 2025).

The derivative is presented as a generalization of the metric derivative to the present setting, and the energy functional quantifies how closely the measure-theoretic decomposition aligns with the geometric foliation. The minimum value $\mu$4 selects perfectly parallel, or “rigid,” measure-theoretic foliations, while values $\mu$5 indicate deviation from the ideal metric-foliation configuration (Münch et al., 17 Sep 2025).

The paper’s illustrative example takes

$\mu$6

with $\mu$7 the uniform measure on $\mu$8, $\mu$9, and leaves $\{\mu_y\}_{y\in Y}$0. For $\{\mu_y\}_{y\in Y}$1, the disintegration gives uniform measures on circles, and

$\{\mu_y\}_{y\in Y}$2

A deformation to ellipses

$\{\mu_y\}_{y\in Y}$3

perturbs the Wasserstein relation according to

$\{\mu_y\}_{y\in Y}$4

where $\{\mu_y\}_{y\in Y}$5 is the perimeter of the ellipse, and the example is used to show that the energy functional is sensitive to geometric perturbations of the foliation (Münch et al., 17 Sep 2025). This suggests that the energy is not only a classification invariant for the rigid case but also a perturbative diagnostic.

4. Constructive disintegration on manifolds

A separate line of work studies how to construct disintegrations explicitly, especially when conditioning occurs on lower-dimensional fibers or on events of measure zero. In this setting, the disintegration is not obtained by simply restricting a density to the consistent set and renormalizing. Instead, for a $\{\mu_y\}_{y\in Y}$6 map $\{\mu_y\}_{y\in Y}$7 that is a submersion, and for regular values $\{\mu_y\}_{y\in Y}$8, the conditional measure on the fiber $\{\mu_y\}_{y\in Y}$9 has density with respect to the Riemannian volume on the fiber given, up to normalization, by

$\nu:=\pi_*\mu$0

and in $\nu:=\pi_*\mu$1,

$\nu:=\pi_*\mu$2

The inverse Jacobian term is the corrective factor that distinguishes disintegration density from restriction density (Costa et al., 1 Aug 2025).

The significance of this formula is conceptual as well as technical. The “restrict-then-normalize” construction ignores how fibers are packed in the ambient space and therefore need not preserve the law of total probability. The disintegration density corrects for the local transverse volume scaling. When $\nu:=\pi_*\mu$3 is linear, the Jacobian factor is constant along the fiber and the two constructions coincide; for nonlinear $\nu:=\pi_*\mu$4, they can differ substantially (Costa et al., 1 Aug 2025).

The explicit example in $\nu:=\pi_*\mu$5 uses the standard Gaussian measure and the observation map

$\nu:=\pi_*\mu$6

with $\nu:=\pi_*\mu$7 and $\nu:=\pi_*\mu$8, so that the fibers are ellipses. The disintegration density is proportional to the Gaussian density multiplied by $\nu:=\pi_*\mu$9, whereas the restricted measure only normalizes the ambient density along the ellipse. The paper reports that the mode of the restricted density and the mode of the disintegration density are at distinct locations, in the example maximally apart (Costa et al., 1 Aug 2025).

This leads to a second important distinction: the modes of a disintegration need not coincide with “conditional modes” defined by maximizing the restricted density on the fiber. The weak modes of the disintegration maximize a functional containing the Jacobian correction, while the restricted modes do not. A plausible implication is that any methodology that identifies conditioning with restriction is using a different geometric object from the regular conditional measure itself (Costa et al., 1 Aug 2025).

5. Optimal transport and transport classes

In optimal transport, the disintegration map is the fiberwise description of a transport plan. If $y\mapsto\mu_y$0 has first marginal $y\mapsto\mu_y$1, then the disintegration theorem yields a measurable map

$y\mapsto\mu_y$2

such that

$y\mapsto\mu_y$3

For a transport map $y\mapsto\mu_y$4, the corresponding disintegration map is $y\mapsto\mu_y$5 (Granieri et al., 2012).

Granieri and Maddalena use this representation to define transport classes. Two transport plans $y\mapsto\mu_y$6 and $y\mapsto\mu_y$7 are equivalent when

$y\mapsto\mu_y$8

and the equivalence class

$y\mapsto\mu_y$9

is the transport class of $\nu$0. Fixing a transport class prescribes the law of the mass-splitting pattern encoded by the disintegration map (Granieri et al., 2012).

This framework recasts the Monge problem as a constrained Kantorovich problem. When the class is supported on Dirac masses, one recovers the map-based Monge setting; more generally, minimization in a fixed transport class is equivalent to an abstract Monge problem over a Wasserstein space of probability measures, with cost

$\nu$1

The paper emphasizes that the Monge problem is, in this sense, a “lucky case”: solvability is comparatively favorable in the Dirac class, while for general non-Dirac transport classes existence may fail even in otherwise classical settings (Granieri et al., 2012).

This transport-theoretic usage aligns with the broader geometric view of disintegration maps. The map records not only the marginal decomposition of a plan, but also an equivalence class in a space of measure-valued assignments. In that sense, transport classes are the pushforward laws of disintegration maps (Granieri et al., 2012).

6. Regularity, computability, and terminological variation

Regularity questions concern how smoothly a disintegration map varies. If $\nu$2 is absolutely continuous with respect to volume on $\nu$3, then the disintegration map is nearly weakly continuous: for every $\nu$4 there exists a closed set $\nu$5 with $\nu$6 such that the restriction of the map to $\nu$7 is weakly continuous. If $\nu$8 is bijective and continuous, then the disintegration map is weakly continuous and the conditional measures are Dirac masses. If the partition is a metric measure foliation, the disintegration map is an isometry and hence weakly continuous (Possobon et al., 2022).

Under weak continuity and path-connectedness, the conditional measures can be organized as a path in Wasserstein space, and in smooth compact Riemannian settings with suitable invariance hypotheses, absolute continuity at one endpoint propagates along the path. This is a rigidity statement for the regularity of the conditional measures (Possobon et al., 2022). A plausible implication is that weak continuity of the disintegration map can transmit structural properties across fibers, rather than merely describe them pointwise.

From the viewpoint of computable analysis, the disintegration operator is nontrivial even in well-behaved spaces. For computable metric spaces $\nu$9, the operator

$Y$00

that assigns to a measure with unique continuous disintegration its disintegration map is strongly Weihrauch equivalent to the limit operator $Y$01. When unique continuous disintegration fails but a computable Vitali basis of continuity sets is available, the resulting pointwise disintegration remains strongly Weihrauch reducible to $Y$02. The paper also constructs explicit families of measures realizing the lower bound, showing that conditioning is exactly as hard as taking limits in this formal sense (Ackerman et al., 2015).

Categorical probability abstracts the same construction further. In the string-diagram setting, disintegration extracts a channel from a joint state, and Bayesian inversion reverses a channel relative to a prior state. In the measure-theoretic case these channels are unique only up to almost-everywhere equality, matching the standard uniqueness caveat for regular conditional probabilities (Cho et al., 2017).

The term also has a distinct, non-measure-theoretic use in the soft-matter literature. In the study of smectic C freely suspended films, a “disintegration map” denotes the set of spatially allowed defect-cluster breakup patterns selected by tangential anchoring and one-constant elastic superposition rules; there it is a map of pattern selection rather than a map into a space of measures (Stannarius et al., 2023). This terminological divergence does not alter the mathematical meaning established in measure theory, optimal transport, and Wasserstein geometry, but it does show that the phrase is context-sensitive across arXiv literatures.