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 $S = [0, 1] \times [0, 1]$ in the Euclidean plane $ℝ^{2}$ . Consider the probability measure $μ$ defined on $S$ by the restriction of two-dimensional Lebesgue measure $λ^{2}$ to $S$ . That is, the probability of an event $E \subseteq S$ 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 $L_{x} = {x} \times [0, 1]$ . $L_{x}$ has $μ$ -measure zero; every subset of $L_{x}$ is a $μ$ -null set; since the Lebesgue measure space is a complete measure space, $E \subseteq L_{x} ⟹ μ (E) = 0.$

While true, this is somewhat unsatisfying. It would be nice to say that $μ$ "restricted to" $L_{x}$ is the one-dimensional Lebesgue measure $λ^{1}$ , 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" $E \cap L_{x}$ : more formally, if $μ_{x}$ denotes one-dimensional Lebesgue measure on $L_{x}$ , then $μ (E) = \int_{[0, 1]} μ_{x} (E \cap L_{x}) d x$ for any "nice" $E \subseteq S$ . The disintegration theorem makes this argument rigorous in the context of measures on metric spaces.

Statement of the theorem

(Hereafter, $𝒫 (X)$ 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 Polish spaces (i.e. separably completely metrizable spaces).
Let $μ \in 𝒫 (Y)$ .
Let $π : Y \to X$ be a Borel-measurable function. Here one should think of $π$ as a function to "disintegrate" $Y$ , in the sense of partitioning $Y$ into ${π^{- 1} (x) | x \in X}$ . For example, for the motivating example above, one can define $π ((a, b)) = a$ , $(a, b) \in [0, 1] \times [0, 1]$ , which gives that $π^{- 1} (a) = a \times [0, 1]$ , a slice we want to capture.
Let $ν \in 𝒫 (X)$ be the pushforward measure $ν = π_{*} (μ) = μ \circ π^{- 1}$ . This measure provides the distribution of $x$ (which corresponds to the events $π^{- 1} (x)$ ).

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

the function $x \mapsto μ_{x}$ is Borel measurable, in the sense that $x \mapsto μ_{x} (B)$ is a Borel-measurable function for each Borel-measurable set $B \subseteq Y$ ;
$μ_{x}$ "lives on" the fiber $π^{- 1} (x)$ : for $ν$ -almost all $x \in X$ , $μ_{x} (Y ∖ π^{- 1} (x)) = 0,$ and so $μ_{x} (E) = μ_{x} (E \cap π^{- 1} (x))$ ;
for every Borel-measurable function $f : Y \to [0, \infty]$ , $\int_{Y} f (y) d μ (y) = \int_{X} \int_{π^{- 1} (x)} f (y) d μ_{x} (y) d ν (x) .$ In particular, for any event $E \subseteq Y$ , taking $f$ to be the indicator function of $E$ , $μ (E) = \int_{X} μ_{x} (E) d ν (x),$ which shows that the family ${μ_{x}}_{x \in X}$ is a regular conditional probability.

Applications

Product spaces

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 $Y = X_{1} \times X_{2}$ and $π_{i} : Y \to X_{i}$ is the natural projection, then each fibre $π_{1}^{- 1} (x_{1})$ can be canonically identified with $X_{2}$ and there exists a Borel family of probability measures ${μ_{x_{1}}}_{x_{1} \in X_{1}}$ in $𝒫 (X_{2})$ (which is $(π_{1})_{*} (μ)$ -almost everywhere uniquely determined) such that $μ = \int_{X_{1}} μ_{x_{1}} μ (π_{1}^{- 1} (d x_{1})) = \int_{X_{1}} μ_{x_{1}} d (π_{1})_{*} (μ) (x_{1}),$ which is in particular^{[clarification needed]} $\int_{X_{1} \times X_{2}} f (x_{1}, x_{2}) μ (d x_{1}, d x_{2}) = \int_{X_{1}} (\int_{X_{2}} f (x_{1}, x_{2}) μ (d x_{2} ∣ x_{1})) μ (π_{1}^{- 1} (d x_{1}))$ and $μ (A \times B) = \int_{A} μ (B ∣ x_{1}) μ (π_{1}^{- 1} (d x_{1})) .$

The relation to conditional expectation is given by the identities $E (f ∣ π_{1}) (x_{1}) = \int_{X_{2}} f (x_{1}, x_{2}) μ (d x_{2} ∣ x_{1}),$ $μ (A \times B ∣ π_{1}) (x_{1}) = 1_{A} (x_{1}) \cdot μ (B ∣ x_{1}) .$

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

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

References

↑ Bogachev 2007, §10.4, §10.6
↑ Dellacherie, C.; Meyer, P.-A. (1978). Probabilities and Potential. North-Holland Mathematics Studies. Amsterdam: North-Holland. ISBN 0-7204-0701-X.
↑ 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.
↑ Chang, J.T.; Pollard, D. (1997). "Conditioning as disintegration" (PDF). Statistica Neerlandica. 51 (3): 287. doi:10.1111/1467-9574.00056. S2CID 16749932.

Bogachev, Vladimir Igorevich (2007). Measure theory. Springer Berlin, Heidelberg. ISBN 978-3-540-34513-8.

[1] Bogachev 2007, §10.4, §10.6

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

[Ambrosio_Gigli_Savare-3] 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.

[Chang_Pollard-4] Chang, J.T.; Pollard, D. (1997). "Conditioning as disintegration" (PDF). Statistica Neerlandica. 51 (3): 287. doi:10.1111/1467-9574.00056. S2CID 16749932.