Optional stopping requires a passage-to-the-limit hypothesis

Assume AC. Bounded optional sampling proves $\mathbb{E}{M}_{\tau \wedge n}=\mathbb{E}{M}_0$; almost-sure convergence $M_{\tau \wedge n}\to M_{\tau }$ does not by itself permit expectations to pass to the limit. Almost-sure finiteness of $\tau$ alone is insufficient, and even $\mathbb{E}\tau <\mathrm{\infty }$ is insufficient when the martingale increments are unrestricted.

Three valid mechanisms established here are distinct:

• Optional stopping under uniform integrability supplies uniform integrability of the stopped family;
• Optional stopping with integrable time and bounded increments supplies the $L^1$ dominator $\mathrm{\mid }{M}_0\mathrm{\mid }+C\tau$ for the stopped family (and $C\tau$ for its difference from $M_{\tau }$);
• Optional stopping with a dominating integrable variable assumes a direct integrable dominator.

The companion counterexamples witness both advertised failures. This remark adds no new optional-stopping theorem. AC is exactly the conditional-expectation dependence inherited from the three cited results.