Conditional-expectation martingale

For an integrable random variable $Z$ and an increasing filtration $({\mathcal{F}}_n)$, the process $M_n=\mathbb{E}[Z\mid {\mathcal{F}}_n]$ is a martingale. The tower property of conditional expectation gives the martingale identity.

Expose the independent edge indicators of a binomial random graph in a fixed order, and set $M_i=\mathbb{E}[Z\mid \text{the first }i\text{ indicators}]$ for an integrable statistic $Z$. This is a martingale from $\mathbb{E}Z$ to $Z$. If changing one edge changes $Z$ by at most $c$, coupling the unexposed indicators gives $|M_i-M_{i-1}|\le c$, permitting the Azuma-Hoeffding inequality.