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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. ==Motivation== Consider the unit square in the Euclidean plane R2, ''S'' = (1 ) × (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 ''L''''x'' = × (1 ). ''L''''x'' has μ-measure zero; every subset of ''L''''x'' 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" ''L''''x'' 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'' ∩ ''L''''x'': more formally, if μ''x'' denotes one-dimensional Lebesgue measure on ''L''''x'', then : for any "nice" ''E'' ⊆ ''S''. The disintegration theorem makes this argument rigorous in the context of measures on metric spaces. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Disintegration theorem」の詳細全文を読む スポンサード リンク
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