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 ∈ 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.