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.

Inhalt 1 Motivation 2 Aussage des Theorems 3 Anwendungen 3.1 Product spaces 3.2 Vector calculus 3.3 Conditional distributions 4 Siehe auch 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. Das ist, 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, dann {zeige ihn an (E)=int _{[0,1]}in _{x}(Ecap L_{x}),Mathrm {d} x} für alle "nice" E ⊆ S. The disintegration theorem makes this argument rigorous in the context of measures on metric spaces.

Aussage des Theorems (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 (d.h. 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 {Anzeigestil {Pi ^{-1}(x) | xin X}} . Zum Beispiel, for the motivating example above, kann man definieren {Anzeigestil pi ((a,b))= ein,(a,b)in [0,1]mal [0,1]} , which gives that {displaystyle pi ^{-1}(a)=atimes [0,1]} , a slice we want to capture. Lassen {Anzeigestil Nr } ∈ P(X) be the pushforward measure ν = π∗(m) = 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 {Anzeigestil Nr } -almost everywhere uniquely determined family of probability measures {μx}x∈X ⊆ P(Y), which provides a "disintegration" von {zeige ihn an } hinein {Anzeigestil {in _{x}}_{xin X}} , so dass: die Funktion {displaystyle xmapsto mu _{x}} is Borel measurable, in dem Sinne, dass {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): zum {Anzeigestil Nr } -almost all x ∈ X, {displaystyle ihn _{x}links(Ysetminus pi ^{-1}(x)Rechts)=0,} and so μx(E) = μx(E ∩ π−1(x)); for every Borel-measurable function f : Y → [0, ∞], {Anzeigestil int _{Y}f(j),Mathrm {d} in (j)=int _{X}int _{Pi ^{-1}(x)}f(j),Mathrm {d} in _{x}(j)Mathrm {d} nicht (x).} Im Speziellen, for any event E ⊆ Y, taking f to be the indicator function of E,[1] {zeige ihn an (E)=int _{X}in _{x}links(Eright),Mathrm {d} nicht (x).} Applications Product spaces This section needs additional citations for verification. Bitte helfen Sie mit, diesen Artikel zu verbessern, indem Sie zuverlässige Quellen zitieren. Nicht bezogenes Material kann angefochten und entfernt werden. (Kann 2022) (Erfahren Sie, wie und wann Sie diese Vorlagennachricht entfernen können) 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 {Anzeigestil {in _{x_{1}}}_{x_{1}in X_{1}}} in P(X2) (welches ist (p1)(m)-almost everywhere uniquely determined) so dass {displaystyle mu =int _{X_{1}}in _{x_{1}},mu left(Pi _{1}^{-1}(Mathrm {d} x_{1})Rechts)=int _{X_{1}}in _{x_{1}},Mathrm {d} (Pi _{1})_{*}(in )(x_{1}),} which is in particular[Klärung nötig] {Anzeigestil int _{X_{1}times X_{2}}f(x_{1},x_{2}),in (Mathrm {d} x_{1},Mathrm {d} x_{2})=int _{X_{1}}links(int _{X_{2}}f(x_{1},x_{2})in (Mathrm {d} x_{2}|x_{1})Rechts)mu left(Pi _{1}^{-1}(Mathrm {d} x_{1})Rechts)} und {zeige ihn an (Zeit B)=int _{EIN}mu left(B|x_{1}Rechts),mu left(Pi _{1}^{-1}(Mathrm {d} x_{1})Rechts).} The relation to conditional expectation is given by the identities {Anzeigestil Betreibername {E} (f|Pi _{1})(x_{1})=int _{X_{2}}f(x_{1},x_{2})in (Mathrm {d} x_{2}|x_{1}),} {zeige ihn an (Zeit B|Pi _{1})(x_{1})=1_{EIN}(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. Zum Beispiel, 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.; Meier, P.-A. (1978). Probabilities and Potential. Nordholländisches Mathematikstudium. Amsterdam: Nordholland. 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 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: Sätze in der MaßtheorieWahrscheinlichkeitssätze

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