Conditional Expectation and an Introduction to Martingales

In this chapter, the conditional expectation operator will be defined and then used in the study of martingales. To begin, one considers the simplest cases of conditional expectation, which are closely related to conditional probability. Then, one proves the Riesz representation theorem for continuous linear functionals on Hilbert space as a tool for defining conditional expectation for square integrable random variables. Given this, it is easy to then define the conditional expectation of integrable random variables.