1.1. A review of probability theory 29
Definition 1.1.23 (Disintegration). Let Y be a random variable with range
R. A disintegration (R , (μy)y∈R ) of the underlying sample space Ω with
respect to Y is a subset R of R of full measure in μY (thus Y R almost
surely), together with assignment of a probability measure P(|Y = y) on
the subspace Ωy := Ω : Y (ω) = y} of Ω for each y R, which is
measurable in the sense that the map y P(F |Y = y) is measurable for
every event F , and such that
P(F ) = EP(F |Y )
for all such events, where P(F |Y ) is the (almost surely defined) random
variable defined to equal P(F |Y = y) whenever Y = y.
Given such a disintegration, we can then condition to the event Y = y
for any y R by replacing Ω with the subspace Ωy (with the induced σ-
algebra), but replacing the underlying probability measure P with P(|Y =
y). We can thus condition (unconditional) events F and random variables X
to this event to create conditioned events (F |Y = y) and random variables
(X|Y = y) on the conditioned space, giving rise to conditional probabil-
ities P(F |Y = y) (which is consistent with the existing notation for this
expression) and conditional expectations E(X|Y = y) (assuming absolute
integrability in this conditioned space). We then set E(X|Y ) to be the (al-
most surely defined) random variable defined to equal E(X|Y = y) whenever
Y = y.
A disintegration is also known as a regular conditional probability in the
literature.
Example 1.1.24 (Discrete case). If Y is a discrete random variable, one can
set R to be the essential range of Y , which in the discrete case is the set of all
y R for which P(Y = y) 0. For each y R , we define P(|Y = y) to be
the conditional probability measure relative to the event Y = y, as defined
in Definition 1.1.18. It is easy to verify that this is indeed a disintegration;
thus the continuous notion of conditional probability generalises the discrete
one.
Example 1.1.25 (Independent case). Starting with an initial sample space
Ω, and a probability measure μ on a measurable space R, one can adjoin a
random variable Y taking values in R with distribution μ that is independent
of all previously existing random variables, by extending Ω to Ω × R as in
Lemma 1.1.7. One can then disintegrate Y by taking R := R and letting
μy be the probability measure on Ωy = Ω × {y} induced by the obvious
isomorphism between Ω×{y} and Ω; this is easily seen to be a disintegration.
Note that if X is any random variable from the original space Ω, then
(X|Y = y) has the same distribution as X for any y R.
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