Filtration: a sequence of embedding sigma-algebra

Martingale: w.r.t. A filtration is a stochastic process where (1) finite expectations (2) each r.v. is measurable w.r.t. respective sigma-algebra in the filtration (3) the expectation of next r.v. conditioned on current sigma-algebra is the current r.v.

Conditional expectation of a r.v. on a sigma-algebra is another r.v. which: (1) is measurable w.r.t. That sigma-algebra (2) for any element (event) from that sigma-algebra, on which those 2 r.v. have the same conditional expectation.

That is the reason why the conditional expectation is that r.v. itself if it is measurable. We call this “conditional const”.

Levy’s upward and downward theorem: a.s. And L1 convergence of martingales

Useful preliminary links:

Sigma-algebra: https://en.wikipedia.org/wiki/%CE%A3-algebra

Measure: https://en.wikipedia.org/wiki/Measure_(mathematics)

Probability space: https://en.wikipedia.org/wiki/Probability_space

Borel set: https://en.wikipedia.org/wiki/Borel_set