Suppose we have a random vector and we know that only takes value in a convex set , i.e.,

Previously, we showed in this post that as long as is convex, we will have

if exists. It is then natural to ask how about conditional expectation. Is it true for any reasonable sigma-algebra that

The answer is affirmative. Let us first recall our previous theorem.

Theorem 1For any borel measurable convex set , and for any probability measure on with

we have

By utilizing the above theorem and regular conditional distribution, we can prove our previous claim.

Theorem 2Suppose is a random vector from a probability space to where is the usual Borel sigma algebra on . If is Borel measurable and

then we have

for any sigma algebra .

*Proof:* Since takes value in , we know there is a family of conditional distribution such that for almost all , we have for any

and for any real valued borel function with domain and , we have

The above two actually comes from the existence of regular conditional distribution which is an important result in measure-theoretic probability theory.

Now take and , we have for almost all ,

and

But since , we know that for almost all ,

Thus for almost all we have a probability distribution on with mean equals to and . This is exactly the situation of Theorem 1. Thus by applying Theorem 1 to the probability measure , we find that

for almost all .