Let X and Y be random variables with on some probability space. In fact, let us suppose that.

By the Doob–Dynkin Lemma (see for example, Oksendal, 2010, Lemma 2.1.2)

for some Borel function g : if Y takes values in.

If Y takes countably many distinct values, then g can be described in more detail. The atoms of σ(Y), the σ-algebra generated by Y, are the disjoint sets

This means that any element of σ(Y) is either ∅ or a countable union of atoms. The function g is characterized as follows: g is constant on each atom, and

for all A ∈ σ(Y). In fact (D.1) is equivalent to the result holding for atoms A. This leads to

where. If P, then

But if, then (D.2) becomes

as. This means that gk can be given any finite value. If we set, then

So we can say then when, then.

We could also note that

and we can set on sets when and obtain a random variable equal to the conditional expected value with probability 1. As many authors use this convention, we shall also adopt it.

Now let W be another random variable which takes countably many distinct values. If, then Then

with probability 1. This is because

with probability 1, since

We could apply these ideas with, and and. If, then

with probability 1. In the application above, if

implies

We could also discuss what is meant by P(A|B) when P(B) = 0. It should be the value of

on the set B. Here.With our convention, we would set.