Reference for conditional expectation with respect to a sigma algebra

Question

I am having problem understanding the idea of conditional expectation with respect to a sigma algebra. Is there any reference which explains the concept in detail. I will prefer something which does not go to geometric intuitions such as orthogonal projections.

Answer 1

You can think of $E(X|\mathcal{G})$ as an approximation to a random variable $X$ that is $\mathcal{G}$-measurable. Eventually, it turns out to be the best possible $\mathcal{G}$-measurable approximation according to the least squares criterion. But you came here for references:

1. The wikipedia page is fine to see the evolution of the concept, and it is necessary to fully understand the following references.
2. Rick Durrett's book explains the modern definition in detail.
3. This document clarifies the relationships between conditional expectation and conditional distributions, which, I think, is the core of the concept.