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. 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. Consider the unit square in the Euclidean plane R2, S = × . 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} × . Lx has μ-measure zero; every subset of Lx is a μ-null set; since the Lebesgue measure space is a complete measure space, 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 for any 'nice' E ⊆ S. The disintegration theorem makes this argument rigorous in the context of measures on metric spaces. (Hereafter, P(X) will denote the collection of Borel probability measures on a metric space (X, d).) Let Y and X be two Radon spaces (i.e. 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, and let ν {displaystyle u } ∈ P(X) be the pushforward measure ν {displaystyle u }  = π∗(μ) = μ ∘ π−1. Then there exists a ν {displaystyle u } -almost everywhere uniquely determined family of probability measures {μx}x∈X ⊆ P(Y) such that