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.

Consider the unit square ${\displaystyle S=[0,1]\times [0,1]}${\displaystyle S=[0,1]\times [0,1]} in the Euclidean plane ${\displaystyle \mathbb {R} ^{2}}${\displaystyle \mathbb {R} ^{2}}. Consider the probability measure ${\displaystyle \mu }${\displaystyle \mu } defined on ${\displaystyle S}${\displaystyle S} by the restriction of two-dimensional Lebesgue measure ${\displaystyle \lambda ^{2}}${\displaystyle \lambda ^{2}} to ${\displaystyle S}${\displaystyle S}. That is, the probability of an event ${\displaystyle E\subseteq S}${\displaystyle E\subseteq S} is simply the area of ${\displaystyle E}${\displaystyle E}. We assume ${\displaystyle E}${\displaystyle E} is a measurable subset of ${\displaystyle S}${\displaystyle S}.

Consider a one-dimensional subset of ${\displaystyle S}${\displaystyle S} such as the line segment ${\displaystyle L_{x}=\{x\}\times [0,1]}${\displaystyle L_{x}=\{x\}\times [0,1]}. ${\displaystyle L_{x}}${\displaystyle L_{x}} has ${\displaystyle \mu }${\displaystyle \mu }-measure zero; every subset of ${\displaystyle L_{x}}${\displaystyle L_{x}} is a ${\displaystyle \mu }${\displaystyle \mu }-null set; since the Lebesgue measure space is a complete measure space, $${\displaystyle E\subseteq L_{x}\implies \mu (E)=0.}$${\displaystyle E\subseteq L_{x}\implies \mu (E)=0.}

While true, this is somewhat unsatisfying. It would be nice to say that ${\displaystyle \mu }${\displaystyle \mu } "restricted to" ${\displaystyle L_{x}}${\displaystyle L_{x}} is the one-dimensional Lebesgue measure ${\displaystyle \lambda ^{1}}${\displaystyle \lambda ^{1}}, rather than the zero measure. The probability of a "two-dimensional" event ${\displaystyle E}${\displaystyle E} could then be obtained as an integral of the one-dimensional probabilities of the vertical "slices" ${\displaystyle E\cap L_{x}}${\displaystyle E\cap L_{x}}: more formally, if ${\displaystyle \mu _{x}}${\displaystyle \mu _{x}} denotes one-dimensional Lebesgue measure on ${\displaystyle L_{x}}${\displaystyle L_{x}}, then $${\displaystyle \mu (E)=\int _{[0,1]}\mu _{x}(E\cap L_{x}),円\mathrm {d} x}$${\displaystyle \mu (E)=\int _{[0,1]}\mu _{x}(E\cap L_{x}),円\mathrm {d} x} for any "nice" ${\displaystyle E\subseteq S}${\displaystyle E\subseteq S}. The disintegration theorem makes this argument rigorous in the context of measures on metric spaces.

(Hereafter, ${\displaystyle {\mathcal {P}}(X)}${\displaystyle {\mathcal {P}}(X)} will denote the collection of Borel probability measures on a topological space ${\displaystyle (X,T)}${\displaystyle (X,T)}.) The assumptions of the theorem are as follows:

• Let ${\displaystyle Y}${\displaystyle Y} and ${\displaystyle X}${\displaystyle X} be two Radon spaces (i.e. a topological space such that every Borel probability measure on it is inner regular, e.g. separably metrizable spaces; in particular, every probability measure on it is outright a Radon measure).
• Let ${\displaystyle \mu \in {\mathcal {P}}(Y)}${\displaystyle \mu \in {\mathcal {P}}(Y)}.
• Let ${\displaystyle \pi :Y\to X}${\displaystyle \pi :Y\to X} be a Borel-measurable function. Here one should think of ${\displaystyle \pi }${\displaystyle \pi } as a function to "disintegrate" ${\displaystyle Y}${\displaystyle Y}, in the sense of partitioning ${\displaystyle Y}${\displaystyle Y} into ${\displaystyle \{\pi ^{-1}(x)\ |\ x\in X\}}${\displaystyle \{\pi ^{-1}(x)\ |\ x\in X\}}. For example, for the motivating example above, one can define ${\displaystyle \pi ((a,b))=a}${\displaystyle \pi ((a,b))=a}, ${\displaystyle (a,b)\in [0,1]\times [0,1]}${\displaystyle (a,b)\in [0,1]\times [0,1]}, which gives that ${\displaystyle \pi ^{-1}(a)=a\times [0,1]}${\displaystyle \pi ^{-1}(a)=a\times [0,1]}, a slice we want to capture.
• Let ${\displaystyle \nu \in {\mathcal {P}}(X)}${\displaystyle \nu \in {\mathcal {P}}(X)} be the pushforward measure ${\displaystyle \nu =\pi _{*}(\mu )=\mu \circ \pi ^{-1}}${\displaystyle \nu =\pi _{*}(\mu )=\mu \circ \pi ^{-1}}. This measure provides the distribution of ${\displaystyle x}${\displaystyle x} (which corresponds to the events ${\displaystyle \pi ^{-1}(x)}${\displaystyle \pi ^{-1}(x)}).

The conclusion of the theorem: There exists a ${\displaystyle \nu }${\displaystyle \nu }-almost everywhere uniquely determined family of probability measures ${\displaystyle \{\mu _{x}\}_{x\in X}\subseteq {\mathcal {P}}(Y)}${\displaystyle \{\mu _{x}\}_{x\in X}\subseteq {\mathcal {P}}(Y)}, which provides a "disintegration" of ${\displaystyle \mu }${\displaystyle \mu } into ${\displaystyle \{\mu _{x}\}_{x\in X}}${\displaystyle \{\mu _{x}\}_{x\in X}}, such that:

• the function ${\displaystyle x\mapsto \mu _{x}}${\displaystyle x\mapsto \mu _{x}} is Borel measurable, in the sense that ${\displaystyle x\mapsto \mu _{x}(B)}${\displaystyle x\mapsto \mu _{x}(B)} is a Borel-measurable function for each Borel-measurable set ${\displaystyle B\subseteq Y}${\displaystyle B\subseteq Y};
• ${\displaystyle \mu _{x}}${\displaystyle \mu _{x}} "lives on" the fiber ${\displaystyle \pi ^{-1}(x)}${\displaystyle \pi ^{-1}(x)}: for ${\displaystyle \nu }${\displaystyle \nu }-almost all ${\displaystyle x\in X}${\displaystyle x\in X}, $${\displaystyle \mu _{x}\left(Y\setminus \pi ^{-1}(x)\right)=0,}$${\displaystyle \mu _{x}\left(Y\setminus \pi ^{-1}(x)\right)=0,} and so ${\displaystyle \mu _{x}(E)=\mu _{x}(E\cap \pi ^{-1}(x))}${\displaystyle \mu _{x}(E)=\mu _{x}(E\cap \pi ^{-1}(x))};
• for every Borel-measurable function ${\displaystyle f:Y\to [0,\infty ]}${\displaystyle f:Y\to [0,\infty ]}, $${\displaystyle \int _{Y}f(y),円\mathrm {d} \mu (y)=\int _{X}\int _{\pi ^{-1}(x)}f(y),円\mathrm {d} \mu _{x}(y),円\mathrm {d} \nu (x).}$${\displaystyle \int _{Y}f(y),円\mathrm {d} \mu (y)=\int _{X}\int _{\pi ^{-1}(x)}f(y),円\mathrm {d} \mu _{x}(y),円\mathrm {d} \nu (x).} In particular, for any event ${\displaystyle E\subseteq Y}${\displaystyle E\subseteq Y}, taking ${\displaystyle f}${\displaystyle f} to be the indicator function of ${\displaystyle E}${\displaystyle E},^[1] $${\displaystyle \mu (E)=\int _{X}\mu _{x}(E),円\mathrm {d} \nu (x).}$${\displaystyle \mu (E)=\int _{X}\mu _{x}(E),円\mathrm {d} \nu (x).}

The original example was a special case of the problem of product spaces, to which the disintegration theorem applies.

When ${\displaystyle Y}${\displaystyle Y} is written as a Cartesian product ${\displaystyle Y=X_{1}\times X_{2}}${\displaystyle Y=X_{1}\times X_{2}} and ${\displaystyle \pi _{i}:Y\to X_{i}}${\displaystyle \pi _{i}:Y\to X_{i}} is the natural projection, then each fibre ${\displaystyle \pi _{1}^{-1}(x_{1})}${\displaystyle \pi _{1}^{-1}(x_{1})} can be canonically identified with ${\displaystyle X_{2}}${\displaystyle X_{2}} and there exists a Borel family of probability measures ${\displaystyle \{\mu _{x_{1}}\}_{x_{1}\in X_{1}}}${\displaystyle \{\mu _{x_{1}}\}_{x_{1}\in X_{1}}} in ${\displaystyle {\mathcal {P}}(X_{2})}${\displaystyle {\mathcal {P}}(X_{2})} (which is ${\displaystyle (\pi _{1})_{*}(\mu )}${\displaystyle (\pi _{1})_{*}(\mu )}-almost everywhere uniquely determined) such that $${\displaystyle \mu =\int _{X_{1}}\mu _{x_{1}},円\mu \left(\pi _{1}^{-1}(\mathrm {d} x_{1})\right)=\int _{X_{1}}\mu _{x_{1}},円\mathrm {d} (\pi _{1})_{*}(\mu )(x_{1}),}$${\displaystyle \mu =\int _{X_{1}}\mu _{x_{1}},円\mu \left(\pi _{1}^{-1}(\mathrm {d} x_{1})\right)=\int _{X_{1}}\mu _{x_{1}},円\mathrm {d} (\pi _{1})_{*}(\mu )(x_{1}),} which is in particular^{[clarification needed ]} $${\displaystyle \int _{X_{1}\times X_{2}}f(x_{1},x_{2}),円\mu (\mathrm {d} x_{1},\mathrm {d} x_{2})=\int _{X_{1}}\left(\int _{X_{2}}f(x_{1},x_{2})\mu (\mathrm {d} x_{2}\mid x_{1})\right)\mu \left(\pi _{1}^{-1}(\mathrm {d} x_{1})\right)}$${\displaystyle \int _{X_{1}\times X_{2}}f(x_{1},x_{2}),円\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}\mid x_{1})\right)\mu \left(\pi _{1}^{-1}(\mathrm {d} x_{1})\right)} and $${\displaystyle \mu (A\times B)=\int _{A}\mu \left(B\mid x_{1}\right),円\mu \left(\pi _{1}^{-1}(\mathrm {d} x_{1})\right).}$${\displaystyle \mu (A\times B)=\int _{A}\mu \left(B\mid x_{1}\right),円\mu \left(\pi _{1}^{-1}(\mathrm {d} x_{1})\right).}

The relation to conditional expectation is given by the identities $${\displaystyle \operatorname {E} (f\mid \pi _{1})(x_{1})=\int _{X_{2}}f(x_{1},x_{2})\mu (\mathrm {d} x_{2}\mid x_{1}),}$${\displaystyle \operatorname {E} (f\mid \pi _{1})(x_{1})=\int _{X_{2}}f(x_{1},x_{2})\mu (\mathrm {d} x_{2}\mid x_{1}),} $${\displaystyle \mu (A\times B\mid \pi _{1})(x_{1})=1_{A}(x_{1})\cdot \mu (B\mid x_{1}).}$${\displaystyle \mu (A\times B\mid \pi _{1})(x_{1})=1_{A}(x_{1})\cdot \mu (B\mid x_{1}).}

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 ${\displaystyle \Sigma \subset \mathbb {R} ^{3}}${\displaystyle \Sigma \subset \mathbb {R} ^{3}}, it is implicit that the "correct" measure on ${\displaystyle \Sigma }${\displaystyle \Sigma } is the disintegration of three-dimensional Lebesgue measure ${\displaystyle \lambda ^{3}}${\displaystyle \lambda ^{3}} on ${\displaystyle \Sigma }${\displaystyle \Sigma }, and that the disintegration of this measure on ∂Σ is the same as the disintegration of ${\displaystyle \lambda ^{3}}${\displaystyle \lambda ^{3}} on ${\displaystyle \partial \Sigma }${\displaystyle \partial \Sigma }.^[2]

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] The theorem is related to the Borel–Kolmogorov paradox, for example.

• Ionescu-Tulcea theorem – Probability theorem
• Joint probability distribution – Type of probability distribution
• Copula (statistics) – Statistical distribution for dependence between random variables
• Conditional expectation – Expected value of a random variable given that certain conditions are known to occur
• Borel–Kolmogorov paradox – Conditional probability paradox
• Regular conditional probability
• Lifting theory

• Theorems in measure theory
• Theorems in probability theory

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