The files which appear first below are concerned with background material for Mathematical Finance. It includes some integration theory and some important (but tedious?) material about Stochastic Processes.
The files, "Families of Sigma Fields" give an introduction to Conditional Expectation and Stochastics Bases.
Notes on Condex , Part 1 of the notes on sigma fields , Part 2 of the notes on sigma fields, Part 3 of the notes on sigma fields.This file takes the discussion of Measurable Functions a little further.
This file looks at the conditional expectation on $L^{1}$. Conditional Expectation
The files Doob_Meyer_1 and Doob_Meyer_2 give a treatment of the Doob_Meyer Decomposition for a submartingale. Along the way the idea of a Natural process is introduced. In discrete time, Natural is equivalent to Predictable in the strongest sense. In continuous time you modify your idea of equivalent for the result to remain true.
Notes on the Hahn-Banach Theorem