Theorem in measure theory
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
in the
Euclidean plane
. Consider the
probability measure
defined on
by the restriction of two-dimensional
Lebesgue measure
to
. That is, the probability of an event
is simply the area of
. We assume
is a measurable subset of
.
Consider a one-dimensional subset of
such as the line segment
.
has
-measure zero; every subset of
is a
-
null set; since the Lebesgue measure space is a
complete measure space,
![{\displaystyle E\subseteq L_{x}\implies \mu (E)=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e77e61e0972b4a20df26847ef37776a7788462ef)
While true, this is somewhat unsatisfying. It would be nice to say that
"restricted to"
is the one-dimensional Lebesgue measure
, rather than the
zero measure. The probability of a "two-dimensional" event
could then be obtained as an
integral of the one-dimensional probabilities of the vertical "slices"
: more formally, if
denotes one-dimensional Lebesgue measure on
, then
![{\displaystyle \mu (E)=\int _{[0,1]}\mu _{x}(E\cap L_{x})\,\mathrm {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f47a1870d36eef3db833a3d8c274afb6a9176f7f)
for any "nice"
![{\displaystyle E\subseteq S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0929e015999da74b44c676c8208f9d820edebfd8)
. The disintegration theorem makes this argument rigorous in the context of measures on
metric spaces.
Statement of the theorem
(Hereafter,
will denote the collection of
Borel probability measures on a
topological space
.)
The assumptions of the theorem are as follows:
- Let
and
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
.
- Let
be a Borel-
measurable function. Here one should think of
as a function to "disintegrate"
, in the sense of partitioning
into
. For example, for the motivating example above, one can define
,
, which gives that
, a slice we want to capture.
- Let
be the
pushforward measure
. This measure provides the distribution of
(which corresponds to the events
).
The conclusion of the theorem: There exists a
-
almost everywhere uniquely determined family of probability measures
, which provides a "disintegration" of
into
, such that:
- the function
is Borel measurable, in the sense that
is a Borel-measurable function for each Borel-measurable set
;
"lives on" the
fiber
: for
-
almost all
, ![{\displaystyle \mu _{x}\left(Y\setminus \pi ^{-1}(x)\right)=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50af4974046e6ddd045924866ed784d8a2420350)
and so
;
- for every Borel-measurable function
, ![{\displaystyle \int _{Y}f(y)\,\mathrm {d} \mu (y)=\int _{X}\int _{\pi ^{-1}(x)}f(y)\,\mathrm {d} \mu _{x}(y)\,\mathrm {d} \nu (x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4819c19cd6a319975dec5d6fddd4c3ce18c50e23)
In particular, for any event
, taking
to be the
indicator function of
,
[1] ![{\displaystyle \mu (E)=\int _{X}\mu _{x}(E)\,\mathrm {d} \nu (x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1078febfc3b707df16e8ab616384f53be1f4638)
Applications
Product spaces
The original example was a special case of the problem of product spaces, to which the disintegration theorem applies.
When
is written as a
Cartesian product
and
is the natural
projection, then each fibre
can be
canonically identified with
and there exists a Borel family of probability measures
in
(which is
-almost everywhere uniquely determined) such that
![{\displaystyle \mu =\int _{X_{1}}\mu _{x_{1}}\,\mu \left(\pi _{1}^{-1}(\mathrm {d} x_{1})\right)=\int _{X_{1}}\mu _{x_{1}}\,\mathrm {d} (\pi _{1})_{*}(\mu )(x_{1}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a974806e823388d483fe42a773bfd41598c5260)
which is in particular
[
clarification needed]
![{\displaystyle \int _{X_{1}\times X_{2}}f(x_{1},x_{2})\,\mu (\mathrm {d} x_{1},\mathrm {d} x_{2})=\int _{X_{1}}\left(\int _{X_{2}}f(x_{1},x_{2})\mu (\mathrm {d} x_{2}\mid x_{1})\right)\mu \left(\pi _{1}^{-1}(\mathrm {d} x_{1})\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0cae02433aed6b3ca788b8d7e884415d275e5cd)
and
![{\displaystyle \mu (A\times B)=\int _{A}\mu \left(B\mid x_{1}\right)\,\mu \left(\pi _{1}^{-1}(\mathrm {d} x_{1})\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03eade33b8f754c2cbddaa8a8369d5b0f3be3809)
The relation to
conditional expectation is given by the identities
![{\displaystyle \operatorname {E} (f\mid \pi _{1})(x_{1})=\int _{X_{2}}f(x_{1},x_{2})\mu (\mathrm {d} x_{2}\mid x_{1}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e4b2342a27125afbb1e87a610136c24f07ce966)
![{\displaystyle \mu (A\times B\mid \pi _{1})(x_{1})=1_{A}(x_{1})\cdot \mu (B\mid x_{1}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a7967b0ae21f5aa864e7b96513e1630770c405)
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
, it is implicit that the "correct" measure on
is the disintegration of three-dimensional Lebesgue measure
on
, and that the disintegration of this measure on ∂Σ is the same as the disintegration of
on
.
[2]
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.
[3]
See also
References