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.

Motivation

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

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

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

Statement of the theorem

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

• Let ${\displaystyle Y}$ and ${\displaystyle X}$ 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 ${\displaystyle \mu \in {\mathcal {P}}(Y)}$.
• Let ${\displaystyle \pi :Y\to X}$ be a Borel-measurable function. Here one should think of ${\displaystyle \pi }$ as a function to "disintegrate" ${\displaystyle Y}$, in the sense of partitioning ${\displaystyle Y}$ into ${\displaystyle \{\pi ^{-1}(x)\ |\ x\in X\}}$. For example, for the motivating example above, one can define ${\displaystyle \pi ((a,b))=a}$, ${\displaystyle (a,b)\in [0,1]\times [0,1]}$, which gives that ${\displaystyle \pi ^{-1}(a)=a\times [0,1]}$, a slice we want to capture.
• Let ${\displaystyle \nu \in {\mathcal {P}}(X)}$ be the pushforward measure ${\displaystyle \nu =\pi _{*}(\mu )=\mu \circ \pi ^{-1}}$. This measure provides the distribution of ${\displaystyle x}$ (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 ${\displaystyle \{\mu _{x}\}_{x\in X}\subseteq {\mathcal {P}}(Y)}$, which provides a "disintegration" of ${\displaystyle \mu }$ into ${\displaystyle \{\mu _{x}\}_{x\in X}}$, such that:

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

Applications

Product spaces

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

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

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

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 ${\displaystyle \Sigma \subset \mathbb {R} ^{3}}$, it is implicit that the "correct" measure on ${\displaystyle \Sigma }$ is the disintegration of three-dimensional Lebesgue measure ${\displaystyle \lambda ^{3}}$ on ${\displaystyle \Sigma }$, and that the disintegration of this measure on ∂Σ is the same as the disintegration of ${\displaystyle \lambda ^{3}}$ on ${\displaystyle \partial \Sigma }$.^[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] The theorem is related to the Borel–Kolmogorov paradox, for example.

See also

• 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

References

1. Dellacherie, C.; Meyer, P.-A. (1978). Probabilities and Potential. North-Holland Mathematics Studies. Amsterdam: North-Holland. ISBN 0-7204-0701-X.
2. 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.
3. 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.