I'm in the process of learning about conditional expectation in the framework of modern probability theory. The sudden change brought about by the notion of conditional expectation being a function on the sample space instead of just a number is a little jarring and difficult for me to understand intuitively at this moment.
I've searched for examples to clarify my understanding but haven't been able to find any so far, so I'd be grateful if anyone could point out references containing such examples.
I'm looking for simple examples that we deal with in basic probability (like calculating cond. exp. for successive coin toss or 3-coin toss, etc.) that are explained in the modern probability theoretic framework, i.e., how to find cond. exp. for successive coin toss using Radon-Nikodym derivative, how we should be able to define the sub sigma-algebras used in these cases, etc.
Basically anything that will help me get a better intuitive feel for this concept.
Thanks in advance
David Williams, Probability with martingales (Cambridge Mathematical Textbooks).