Disintegration theorem

Disintegration theorem In mathematics, the disintegration theorem is a result in measure theory and probability theory. It rigorously defines the idea of a non-trivial "restriction" of a measure to a measure zero subset of the measure space in question. It is related to the existence of conditional probability measures. In a sense, "disintegration" is the opposite process to the construction of a product measure.

Contents 1 Motivation 2 Statement of the theorem 3 Applications 3.1 Product spaces 3.2 Vector calculus 3.3 Conditional distributions 4 See also 5 References Motivation Consider the unit square in the Euclidean plane R2, S = [0, 1] × [0, 1]. Consider the probability measure μ defined on S by the restriction of two-dimensional Lebesgue measure λ2 to S. That is, the probability of an event E ⊆ S is simply the area of E. We assume E is a measurable subset of S.

Consider a one-dimensional subset of S such as the line segment Lx = {x} × [0, 1]. Lx has μ-measure zero; every subset of Lx is a μ-null set; since the Lebesgue measure space is a complete measure space, {displaystyle Esubseteq L_{x}implies mu (E)=0.} While true, this is somewhat unsatisfying. It would be nice to say that μ "restricted to" Lx is the one-dimensional Lebesgue measure λ1, rather than the zero measure. The probability of a "two-dimensional" event E could then be obtained as an integral of the one-dimensional probabilities of the vertical "slices" E ∩ Lx: more formally, if μx denotes one-dimensional Lebesgue measure on Lx, then {displaystyle mu (E)=int _{[0,1]}mu _{x}(Ecap L_{x}),mathrm {d} x} for any "nice" E ⊆ S. The disintegration theorem makes this argument rigorous in the context of measures on metric spaces.

Statement of the theorem (Hereafter, P(X) will denote the collection of Borel probability measures on a topological space (X, T).) The assumptions of the theorem are as follows: Let Y and X be two Radon spaces (i.e. a topological space such that every Borel probability measure on M is inner regular e.g. separable metric spaces on which every probability measure is a Radon measure). Let μ ∈ P(Y). Let π : Y → X be a Borel-measurable function. Here one should think of π as a function to "disintegrate" Y, in the sense of partitioning Y into {displaystyle {pi ^{-1}(x) | xin X}} . For example, for the motivating example above, one can define {displaystyle pi ((a,b))=a,(a,b)in [0,1]times [0,1]} , which gives that {displaystyle pi ^{-1}(a)=atimes [0,1]} , a slice we want to capture. Let {displaystyle nu } ∈ P(X) be the pushforward measure ν = π∗(μ) = μ ∘ π−1. This measure provides the distribution of x (which corresponds to the events {displaystyle pi ^{-1}(x)} ).

The conclusion of the theorem: There exists a {displaystyle nu } -almost everywhere uniquely determined family of probability measures {μx}x∈X ⊆ P(Y), which provides a "disintegration" of {displaystyle mu } into {displaystyle {mu _{x}}_{xin X}} , such that: the function {displaystyle xmapsto mu _{x}} is Borel measurable, in the sense that {displaystyle xmapsto mu _{x}(B)} is a Borel-measurable function for each Borel-measurable set B ⊆ Y; μx "lives on" the fiber π−1(x): for {displaystyle nu } -almost all x ∈ X, {displaystyle mu _{x}left(Ysetminus pi ^{-1}(x)right)=0,} and so μx(E) = μx(E ∩ π−1(x)); for every Borel-measurable function f : Y → [0, ∞], {displaystyle int _{Y}f(y),mathrm {d} mu (y)=int _{X}int _{pi ^{-1}(x)}f(y),mathrm {d} mu _{x}(y)mathrm {d} nu (x).} In particular, for any event E ⊆ Y, taking f to be the indicator function of E,[1] {displaystyle mu (E)=int _{X}mu _{x}left(Eright),mathrm {d} nu (x).} Applications Product spaces This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (May 2022) (Learn how and when to remove this template message) The original example was a special case of the problem of product spaces, to which the disintegration theorem applies.

When Y is written as a Cartesian product Y = X1 × X2 and πi : Y → Xi is the natural projection, then each fibre π1−1(x1) can be canonically identified with X2 and there exists a Borel family of probability measures {displaystyle {mu _{x_{1}}}_{x_{1}in X_{1}}} in P(X2) (which is (π1)∗(μ)-almost everywhere uniquely determined) such that {displaystyle mu =int _{X_{1}}mu _{x_{1}},mu left(pi _{1}^{-1}(mathrm {d} x_{1})right)=int _{X_{1}}mu _{x_{1}},mathrm {d} (pi _{1})_{*}(mu )(x_{1}),} which is in particular[clarification needed] {displaystyle int _{X_{1}times X_{2}}f(x_{1},x_{2}),mu (mathrm {d} x_{1},mathrm {d} x_{2})=int _{X_{1}}left(int _{X_{2}}f(x_{1},x_{2})mu (mathrm {d} x_{2}|x_{1})right)mu left(pi _{1}^{-1}(mathrm {d} x_{1})right)} and {displaystyle mu (Atimes B)=int _{A}mu left(B|x_{1}right),mu left(pi _{1}^{-1}(mathrm {d} x_{1})right).} The relation to conditional expectation is given by the identities {displaystyle operatorname {E} (f|pi _{1})(x_{1})=int _{X_{2}}f(x_{1},x_{2})mu (mathrm {d} x_{2}|x_{1}),} {displaystyle mu (Atimes B|pi _{1})(x_{1})=1_{A}(x_{1})cdot mu (B|x_{1}).} Vector calculus The disintegration theorem can also be seen as justifying the use of a "restricted" measure in vector calculus. For instance, in Stokes' theorem as applied to a vector field flowing through a compact surface Σ ⊂ R3, it is implicit that the "correct" measure on Σ is the disintegration of three-dimensional Lebesgue measure λ3 on Σ, and that the disintegration of this measure on ∂Σ is the same as the disintegration of λ3 on ∂Σ.[2] Conditional distributions The disintegration theorem can be applied to give a rigorous treatment of conditional probability distributions in statistics, while avoiding purely abstract formulations of conditional probability.[3] See also Ionescu-Tulcea theorem Joint probability distribution – Type of probability distribution Copula (statistics) Conditional expectation – Expected value of a random variable given that certain conditions are known to occur Regular conditional probability References ^ Dellacherie, C.; Meyer, P.-A. (1978). Probabilities and Potential. North-Holland Mathematics Studies. Amsterdam: North-Holland. ISBN 0-7204-0701-X. ^ Ambrosio, L., Gigli, N. & Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures. ETH Zürich, Birkhäuser Verlag, Basel. ISBN 978-3-7643-2428-5. ^ Chang, J.T.; Pollard, D. (1997). "Conditioning as disintegration" (PDF). Statistica Neerlandica. 51 (3): 287. CiteSeerX 10.1.1.55.7544. doi:10.1111/1467-9574.00056. S2CID 16749932. hide vte Measure theory Basic concepts Absolute continuityLebesgue integrationLp spacesMeasureMeasure space Probability spaceMeasurable space/function Sets Almost everywhereBorel setCarathéodory's criterionConvergence in measure -systemEssential range infimum/supremumLocally measurableπ-systemσ-algebraNon-measurable set Vitali setNull setSupportTransverse measure Types of Measures BaireBanachBesovBorelComplexCompleteContent(Logarithmically) ConvexDiscreteFiniteInner(Quasi-) InvariantLocally finiteMaximisingMetric outerOuterPerfectPre-measure(Sub-) ProbabilityProjection-valuedRadonRandomRegular Borel regularInner regularOuter regularSaturatedSet functionσ-finites-finiteSignedSingularSpectralStrictly positiveTightVector Particular measures CountingDiracEulerGaussianHaarHarmonicHausdorffIntensityLebesgueLogarithmicProductPushforwardSpherical measureTangentTrivialYoung Main results Carathéodory's extension theoremConvergence theorems DominatedMonotoneVitaliDecomposition theorems HahnJordanEgorov'sFatou's lemmaFubini'sHölder's inequalityMinkowski inequalityRadon–NikodymRiesz–Markov–Kakutani representation theorem Other results Disintegration theoremLebesgue's density theoremLebesgue differentiation theoremSard's theorem Applications Probability theoryReal analysisSpectral theory Categories: Theorems in measure theoryProbability theorems