Conditional Expectation

The conditional expectation of an integrable random variable on a probability space given a sigma-algebra is a -measurable random variable, denoted , such that

for every . It is unique up to sets of probability zero.

For a nonnegative random variable , the definition extends to conditional expectations that may take the value by the almost-sure increasing limit . Here, denotes the minimum.

When conditioning on a random variable , the notation means conditioning on the sigma-algebra generated by . Conditional expectation is the probability-theoretic analog of averaging while retaining specified information.