Disintegration theorem

Consider the unit square $S\; =\; [0,1]\backslash times[0,1]$ in the Euclidean plane $\backslash mathbb\{R\}^2$. Consider the probability measure $\backslash mu$ defined on $S$ by the restriction of two-dimensional Lebesgue measure $\backslash lambda^2$ to $S$. That is, the probability of an event $E\backslash subseteq\; S$ 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 $L\_x\; =\; \backslash \{x\backslash \}\backslash times[0,\; 1]$. $L\_x$ has $\backslash mu$-measure zero; every subset of $L\_x$ is a $\backslash mu$-null set; since the Lebesgue measure space is a complete measure space,

$$E\; \backslash subseteq\; L\_\{x\}\; \backslash implies\; \backslash mu\; (E)\; =\; 0.$$

While true, this is somewhat unsatisfying. It would be nice to say that $\backslash mu$ "restricted to" $L\_x$ is the one-dimensional Lebesgue measure $\backslash lambda^1$, 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" $E\backslash cap\; L\_x$: more formally, if $\backslash mu\_x$ denotes one-dimensional Lebesgue measure on $L\_x$, then

$$\backslash mu\; (E)\; =\; \backslash int\_\{[0,\; 1]\}\; \backslash mu\_\{x\}\; (E\; \backslash cap\; L\_\{x\})\; \backslash ,\; \backslash mathrm\{d\}\; x$$

for any "nice" $E\backslash subseteq\; S$. The disintegration theorem makes this argument rigorous in the context of measures on metric spaces.

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

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

• the function $x\; \backslash mapsto\; \backslash mu\_\{x\}$ is Borel measurable, in the sense that $x\; \backslash mapsto\; \backslash mu\_\{x\}\; (B)$ is a Borel-measurable function for each Borel-measurable set $B\backslash subseteq\; Y$;
• $\backslash mu\_x$ "lives on" the fiber $\backslash pi^\{-1\}(x)$: for $\backslash nu$-almost all $x\backslash in\; X$, $$\backslash mu\_\{x\}\; \backslash left(\; Y\; \backslash setminus\; \backslash pi^\{-1\}\; (x)\; \backslash right)\; =\; 0,$$ and so $\backslash mu\_x(E)\; =\backslash mu\_x(E\backslash cap\backslash pi^\{-1\}(x))$;
• for every Borel-measurable function $f\; :\; Y\; \backslash to\; [0,\backslash infty]$, $$\backslash int\_\{Y\}\; f(y)\; \backslash ,\; \backslash mathrm\{d\}\; \backslash mu\; (y)\; =\; \backslash int\_\{X\}\; \backslash int\_\{\backslash pi^\{-1\}\; (x)\}\; f(y)\; \backslash ,\; \backslash mathrm\{d\}\; \backslash mu\_x\; (y)\; \backslash ,\; \backslash mathrm\{d\}\; \backslash nu\; (x).$$ In particular, for any event $E\backslash subseteq\; Y$, taking $f$ to be the indicator function of $E$,{{cite book |author1=Dellacherie, C. |author2=Meyer, P.-A. |title=Probabilities and Potential |series=North-Holland Mathematics Studies |publisher=North-Holland |location=Amsterdam |year=1978 |isbn=0-7204-0701-X }} $$\backslash mu\; (E)\; =\; \backslash int\_X\; \backslash mu\_x\; (E)\; \backslash ,\; \backslash mathrm\{d\}\; \backslash nu\; (x).$$

{{More citations needed section|date=May 2022}}

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 $Y\; =\; X\_1\backslash times\; X\_2$ and $\backslash pi\_i\; :\; Y\backslash to\; X\_i$ is the natural projection, then each fibre $\backslash pi\_1^\{-1\}(x\_1)$ can be canonically identified with $X\_2$ and there exists a Borel family of probability measures $\backslash \{\; \backslash mu\_\{x\_\{1\}\}\; \backslash \}\_\{x\_\{1\}\; \backslash in\; X\_\{1\}\}$ in $\backslash mathcal\{P\}(X\_2)$ (which is $(\backslash pi\_1)\_*(\backslash mu)$-almost everywhere uniquely determined) such that

$$\backslash mu\; =\; \backslash int\_\{X\_\{1\}\}\; \backslash mu\_\{x\_\{1\}\}\; \backslash ,\; \backslash mu\; \backslash left(\backslash pi\_1^\{-1\}(\backslash mathrm\; d\; x\_1)\; \backslash right)=\; \backslash int\_\{X\_\{1\}\}\; \backslash mu\_\{x\_\{1\}\}\; \backslash ,\; \backslash mathrm\{d\}\; (\backslash pi\_\{1\})\_\{*\}\; (\backslash mu)\; (x\_\{1\}),$$

which is in particular{{Clarify|date=May 2022|reason=Notation "\mu(d x_2[pipe]x_1)" has not been defined}}

$$\backslash int\_\{X\_1\backslash times\; X\_2\}\; f(x\_1,x\_2)\backslash ,\; \backslash mu(\backslash mathrm\; d\; x\_1,\backslash mathrm\; d\; x\_2)\; =\; \backslash int\_\{X\_1\}\backslash left(\; \backslash int\_\{X\_2\}\; f(x\_1,x\_2)\; \backslash mu(\backslash mathrm\; d\; x\_2\backslash mid\; x\_1)\; \backslash right)\; \backslash mu\backslash left(\; \backslash pi\_1^\{-1\}(\backslash mathrm\{d\}\; x\_\{1\})\backslash right)$$

and

$$\backslash mu(A\; \backslash times\; B)\; =\; \backslash int\_A\; \backslash mu\backslash left(B\backslash mid\; x\_1\backslash right)\; \backslash ,\; \backslash mu\backslash left(\; \backslash pi\_1^\{-1\}(\backslash mathrm\{d\}\; x\_\{1\})\backslash right).$$

The relation to conditional expectation is given by the identities

$$\backslash operatorname\; E(f\backslash mid\; \backslash pi\_1)(x\_1)=\; \backslash int\_\{X\_2\}\; f(x\_1,x\_2)\; \backslash mu(\backslash mathrm\; d\; x\_2\backslash mid\; x\_1),$$

$$\backslash mu(A\backslash times\; B\backslash mid\; \backslash pi\_1)(x\_1)=\; 1\_A(x\_1)\; \backslash cdot\; \backslash mu(B\backslash mid\; x\_1).$$

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 {{nowrap|$\backslash Sigma\; \backslash subset\; \backslash mathbb\{R\}^3$}}, it is implicit that the "correct" measure on $\backslash Sigma$ is the disintegration of three-dimensional Lebesgue measure $\backslash lambda^3$ on $\backslash Sigma$, and that the disintegration of this measure on ∂Σ is the same as the disintegration of $\backslash lambda^3$ on $\backslash partial\backslash Sigma$.{{cite book |author1=Ambrosio, L. |author2=Gigli, N. |author3=Savaré, G. |title=Gradient Flows in Metric Spaces and in the Space of Probability Measures |publisher=ETH Zürich, Birkhäuser Verlag, Basel |year=2005 |isbn=978-3-7643-2428-5 }}

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.{{cite journal |last=Chang |first=J.T. |author2=Pollard, D. |title=Conditioning as disintegration |journal=Statistica Neerlandica |year=1997 |volume=51 |issue=3 |url=http://www.stat.yale.edu/~jtc5/papers/ConditioningAsDisintegration.pdf |doi=10.1111/1467-9574.00056 |page=287 |citeseerx=10.1.1.55.7544 |s2cid=16749932 }} The theorem is related to the Borel–Kolmogorov paradox, for example.

• {{annotated link|Ionescu-Tulcea theorem}}
• {{annotated link|Joint probability distribution}}
• {{annotated link|Copula (statistics)}}
• {{annotated link|Conditional expectation}}
• {{annotated link|Borel–Kolmogorov paradox}}
• Regular conditional probability

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{{Measure theory}}

Category:Theorems in measure theory

Category:Theorems in probability theory