Doob martingale

Example

Let $(\Omega, \mathcal{F}, P)$ be a probability space with a filtration $(\mathcal{F}_n)_{n \ge 0}$, and let $X$ be a real random variable with $E[\abs{X}] < \infty$. For $n \ge 0$ let $$M_n = E[X \mid \mathcal{F}_n],$$ the conditional expectation of $X$ given $\mathcal{F}_n$. Then $(M_n)_{n \ge 0}$ is a martingale, the Doob martingale of $X$.

Check of the three clauses:

1. $M_n$ is $\mathcal{F}_n$-measurable by the definition of conditional expectation.
2. $E[\abs{M_n}] \le E\bigl[E[\abs{X} \mid \mathcal{F}_n]\bigr] = E[\abs{X}] < \infty$.
3. Since $\mathcal{F}_n \subseteq \mathcal{F}_{n+1}$, the tower property gives $E[M_{n+1} \mid \mathcal{F}_n] = E\bigl[E[X \mid \mathcal{F}_{n+1}] \mid \mathcal{F}_n\bigr] = E[X \mid \mathcal{F}_n] = M_n$.