Math 2.0

Measure and integration Product measures Theorem

Disintegration theorem

Builds on Polish spaceProbability measurePushforward measureMeasurable function

Statement

Let YY and XX be Polish spaces with their Borel σ-algebras. Let μ\mu be a Borel probability measure on YY, let π ⁣:Y→X\pi \colon Y \to X be a Borel measurable map, and let ν=π∗μ\nu = \pi_* \mu be the pushforward measure. Then there is a family (μx)x∈X(\mu_x)_{x \in X} of Borel probability measures on YY with the following properties.

  1. For every Borel set B⊆YB \subseteq Y, the map x↦μx(B)x \mapsto \mu_x(B) is Borel measurable.
  2. For ν\nu-almost every x∈Xx \in X, μx\mu_x is concentrated on the fibre: μx(Y∖π−1(x))=0\mu_x(Y \setminus \pi^{-1}(x)) = 0.
  3. For every Borel function f ⁣:Y→[0,∞]f \colon Y \to [0, \infty], ∫Yf dμ=∫X∫π−1(x)f(y) dμx(y) dν(x).\int_Y f \, d\mu = \int_X \int_{\pi^{-1}(x)} f(y) \, d\mu_x(y) \, d\nu(x).

The family is unique up to ν\nu-null sets: two such families agree for ν\nu-almost every xx.

Remarks

One way to prove it is through regular conditional probabilities. For a Borel set BB, the measure A↦μ(B∩π−1(A))A \mapsto \mu(B \cap \pi^{-1}(A)) is absolutely continuous with respect to ν\nu, and the Radon–Nikodym theorem gives a density x↦μx(B)x \mapsto \mu_x(B). The Polish assumption lets one choose these densities for a countable generating algebra of Borel sets at once and extend them to measures.

Example: let μ\mu be Lebesgue measure on Y=[0,1]2Y = [0, 1]^2 and π(x,y)=x\pi(x, y) = x. Then ν\nu is Lebesgue measure on [0,1][0, 1], and μx\mu_x is Lebesgue measure on the vertical segment {x}×[0,1]\set{x} \times [0, 1]. Property 3 is then Fubini's theorem.

The theorem is often stated for Radon spaces, which include Polish spaces. Without such a regularity assumption a disintegration may fail to exist.

Going deeper

For a student

Disintegration makes conditioning on events of probability zero rigorous: μx\mu_x is the conditional law of the point yy given π(y)=x\pi(y) = x. It is used in optimal transport to decompose transport plans, in ergodic theory for the ergodic decomposition, and in geometric measure theory together with the coarea formula. The measures μx\mu_x are only defined for almost every xx. So a conditional law given a single null event depends on the choice of the map π\pi, as the Borel–Kolmogorov paradox shows.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Measurable function, concept, no evidence levelMeasurable functionconceptPolish space, concept, no evidence levelPolish spaceconceptProbability measure, concept, no evidence levelProbability measureconceptPushforward measure, concept, no evidence levelPushforward measureconceptDisintegration theorem, statement, no evidence levelDisintegration theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Measurable function concept
  2. Polish space concept
  3. Probability measure concept
  4. Pushforward measure concept
  5. Disintegration theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Disintegration_theorem
About this page
TagN8NXNCMSK5
URIm2://N8NXNCMSK5
Kindstatement, theorem
WritingW0 stub
MSC28A50, 60A10
StatusThis copy of the site has no build data. Run m2 status N8NXNCMSK5.
AboutProbability measure, Regular conditional probability
UsesPolish space, Probability measure, Pushforward measure, Measurable function
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Changed2 November 2026

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