In
mathematics, the disintegration theorem is a result in
measure theory and
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
. It rigorously defines the idea of a non-trivial "restriction" of a
measure to a
measure zero
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null ...
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 In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of tw ...
.
Motivation
Consider the unit square 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'' ⊆ ''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'' = ×
, 1
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
''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,
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
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
of the one-dimensional probabilities of the vertical "slices" ''E'' ∩ ''L''
''x'': more formally, if μ
''x'' denotes one-dimensional Lebesgue measure on ''L''
''x'', then
for any "nice" ''E'' ⊆ ''S''. The disintegration theorem makes this argument rigorous in the context of measures on
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s.
Statement of the theorem
(Hereafter, ''P''(''X'') will denote the collection of
Borel probability measures on a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
(''X'', ''T'').)
The assumptions of the theorem are as follows:
* Let ''Y'' and ''X'' be two
Radon spaces (i.e. a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
such that every
Borel probability measure on ''M'' is
inner regular
In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets.
Definition
Let (''X'', ''T'') be a Hausdorff topological space and let Σ be a σ-algebra on ''X'' tha ...
e.g.
separable metric spaces on which every probability measure is a
Radon measure
In mathematics (specifically in measure theory), 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 finite on all compact sets, outer regular on all Borel ...
).
* Let μ ∈ ''P''(''Y'').
* Let π : ''Y'' → ''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
. For example, for the motivating example above, one can define