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 1 For any borel measurable convex set , and for any probability measure on with
By utilizing the above theorem and regular conditional distribution, we can prove our previous claim.
Theorem 2 Suppose 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 ,
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 .