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For the converse in the characterization of a martingale by stopped expectations, fix $n\ge 0$ and $A\in {\mathcal{F}}_n$. Define the bounded stopping timeIndeed, its only nontrivial sublevel event is $\{T_A\le n\}=A^c\in {\mathcal{F}}_n$. Applying the assumed stopped-expectation equality to $T_A$ and to the deterministic stopping time $n$ givesThis holds for every $A\in {\mathcal{F}}_n$. Since $M_n$ is adapted and both variables are integrable, the defining test-event property of conditional expectation yieldsTogether with the given adaptation and integrability, this is exactly the martingale property.

Assume (iii). Fix $n\ge 0$ and $A\in {\mathcal{F}}_n$, and defineThis is a stopping time: $\{T\le n\}=A\in {\mathcal{F}}_n$. Applying (iii) at time $n+1$ first to $T$ and then to the deterministic stopping time $n+1$ givesSubtracting yieldsBy the defining characterization of conditional expectation, this says $\mathbb{E}[M_{n+1}\mid {\mathcal{F}}_n]=M_n$. Thus $M$ is a martingale, proving (i). Together the three implications establish the characterization of a martingale by stopped expectations.