Disintegration theorem
Encyclopedia
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
.
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 Lx = {x} × [0, 1]. Lx has μ-measure zero; every subset of Lx 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" Lx 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 ∩ Lx: more formally, if μx denotes one-dimensional Lebesgue measure on Lx, then
for any "nice" E ⊆ S. The disintegration theorem makes this argument rigorous in the context of measures on metric space
s.
(X, d).)
Let Y and X be two Radon space
s (i.e. separable metric spaces on which every probability measure is a Radon measure
). Let μ ∈ P(Y), let π : Y → X be a Borel-measurable function
, and let ν ∈ P(X) be the pushforward measure
ν = π∗(μ) = μ ∘ π−1. Then there exists a ν-almost everywhere
uniquely determined family of probability measures {μx}x∈X ⊆ P(Y) such that
When Y is written as a Cartesian product
Y = X1 × X2 and πi : Y → Xi is the natural projection
, then each fibre π1−1(x1) can be canonically identified with X2 and there exists a Borel family of probability measures
in P(X2) (which is (π1)∗(μ)-almost everywhere uniquely determined) such that
which is in particular
and
The relation to conditional expectation
is given by the identities
as applied to a vector field
flowing through a compact
surface
Σ ⊂ R³, it is implicit that the "correct" measure on Σ is the disintegration of three-dimensional Lebesgue measure λ³ on Σ, and that the disintegration of this measure on ∂Σ is the same as the disintegration of λ³ on ∂Σ.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the disintegration theorem is a result in measure theory and probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
. It rigorously defines the idea of a non-trivial "restriction" of a measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
to a measure zero subset of the measure space in question. It is related to the existence of conditional probability measures
Conditioning (probability)
Beliefs depend on the available information. This idea is formalized in probability theory by conditioning. Conditional probabilities, conditional expectations and conditional distributions are treated on three levels: discrete probabilities, probability density functions, and measure theory...
. In a sense, "disintegration" is the opposite process to the construction of a product measure
Product measure
In mathematics, given two measurable spaces and measures on them, one can obtain the product measurable space and the product measure on that space...
.
Motivation
Consider the unit square in the Euclidean plane R², S = [0, 1] × [0, 1]. Consider the probability measureProbability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
μ defined on S by the restriction of two-dimensional Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
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 Lx = {x} × [0, 1]. Lx has μ-measure zero; every subset of Lx is a μ-null set
Null set
In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...
; since the Lebesgue measure space is a complete measure space
Complete measure
In mathematics, a complete measure is a measure space in which every subset of every null set is measurable...
,
While true, this is somewhat unsatisfying. It would be nice to say that μ "restricted to" Lx is the one-dimensional Lebesgue measure λ1, rather than the zero measure
Trivial measure
In mathematics, specifically in measure theory, the trivial measure on any measurable space is the measure μ which assigns zero measure to every measurable set: μ = 0 for all A in Σ.-Properties of the trivial measure:...
. The probability of a "two-dimensional" event E could then be obtained as an integral
Lebesgue integration
In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...
of the one-dimensional probabilities of the vertical "slices" E ∩ Lx: more formally, if μx denotes one-dimensional Lebesgue measure on Lx, then
for any "nice" E ⊆ S. The disintegration theorem makes this argument rigorous in the context of measures on metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s.
Statement of the theorem
(Hereafter, P(X) will denote the collection of Borel probability measures on a metric spaceMetric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
(X, d).)
Let Y and X be two Radon space
Radon space
In mathematics, a Radon space, named after Johann Radon, is a separable metric space such that every Borel probability measure on M is inner regular. Since a probability measure is globally finite, and hence a locally finite measure, every probability measure on a Radon space is also a Radon...
s (i.e. separable metric spaces on which every probability measure is a Radon measure
Radon measure
In mathematics , a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.-Motivation:...
). Let μ ∈ P(Y), let π : Y → X be a Borel-measurable function
Measurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
, and let ν ∈ P(X) be the pushforward measure
Pushforward measure
In measure theory, a pushforward measure is obtained by transferring a measure from one measurable space to another using a measurable function.-Definition:...
ν = π∗(μ) = μ ∘ π−1. Then there exists a ν-almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
uniquely determined family of probability measures {μx}x∈X ⊆ P(Y) such that
- the function
is Borel measurable, in the sense that 
is a Borel-measurable function for each Borel-measurable set B ⊆ Y; - μx "lives on" the fiber π−1(x): for ν-almost allAlmost allIn mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....
x ∈ X,
- and so μx(E) = μx(E ∩ π−1(x));
- for every Borel-measurable function f : Y → [0, +∞],
- In particular, for any event E ⊆ Y, taking f to be the indicator function of E,
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
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
Y = X1 × X2 and πi : Y → Xi is the natural projection
Projection (mathematics)
Generally speaking, in mathematics, a projection is a mapping of a set which is idempotent, which means that a projection is equal to its composition with itself. A projection may also refer to a mapping which has a left inverse. Bot notions are strongly related, as follows...
, then each fibre π1−1(x1) can be canonically identified with X2 and there exists a Borel family of probability measures
in P(X2) (which is (π1)∗(μ)-almost everywhere uniquely determined) such thatwhich is in particular
and
The relation to conditional expectation
Conditional expectation
In probability theory, a conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution....
is given by the identities
Vector calculus
The disintegration theorem can also be seen as justifying the use of a "restricted" measure in vector calculus. For instance, in Stokes' theoremStokes' theorem
In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
as applied to a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
flowing through a compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
Σ ⊂ R³, it is implicit that the "correct" measure on Σ is the disintegration of three-dimensional Lebesgue measure λ³ on Σ, and that the disintegration of this measure on ∂Σ is the same as the disintegration of λ³ on ∂Σ.
Conditional distributions
The disintegration theorem can be applied to give a rigorous treatment of conditioning probability distributions in statistics, while avoiding purely abstract formulations of conditional probability.See also
- Joint probability distribution
- Copula (statistics)Copula (statistics)In probability theory and statistics, a copula can be used to describe the dependence between random variables. Copulas derive their name from linguistics....
- Conditional expectationConditional expectationIn probability theory, a conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution....