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 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 in the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
R
2, . Consider the
probability 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''. The difference between a probability measure and the more ge ...
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
In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space (''X'', Σ, ''μ'') is comp ...
,
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'' ∩ ''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
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''. The difference between a probability measure and the more ge ...
on ''M'' is
inner regular 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 s ...
).
* Let μ ∈ ''P''(''Y'').
* Let π : ''Y'' → ''X'' be a Borel-
measurable function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in d ...
. 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