I am now at the end of Chapter 1 in with four sections to go on random walks, martingales, and markov chains. I am going to skip the section on random walks for now and come back to it later. I haven’t seen martingales before so I’ll start with that. Since martingales makes use of expectations with respect to decompositions heavily I want to get a little more comfortable with it.

Suppose that are two decompositions of the sample space where is *finer* than . Finer means that .

Let be a random variable. First, recall the expectation of a random variable with respect to a decomposition .

Note the special case when (i.e., when is -measurable).

Next, recall the generalized total probability formula

Suppose we took a conditional expection instead

This gets simplified if is a finer decomposition than because is now decomposed by

Therefore if

And in general if

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