Conditional Expectation of $M_n$ given $F_n$

Question

Why is that the Conditional Expectation of $M_n$ given $F_n$ is equal to $M_n$ where $M_n$ is collection of the random variable $X_i$ where i runs from 0 to n and $F_n$ is the filtration at time n?

Answer 1

This question lacks the machinery/terminology required to capture what you are (probably) trying to express, but I imagine what you meant to say is that the process $X$ is adapted to $\left(F_n\right)_{n\in \mathbb{N} }$ and that $M_n \equiv f\left(X_1,X_2,\ldots,X_n\right)$ and $f$ is "nice enough". If you want more than a heuristic explanation, I would start by reading a text on measure-theoretic probability.

A link describing what measure-theoretic probability is https://en.wikipedia.org/wiki/Probability_theory#Measure-theoretic_probability_theory

Good books are found at "Best measure theoretic probability theory book?"