When defining martingales (for example p. 458 Billingsley) one explicitly requires that $X_n$ is measurable $\mathscr{F}_n$. But doesn't this follow by the requirement $E[X_{n+1} \vert \mathscr{F}_n] = X_n$ ?
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2026-03-26 22:17:31.1774563451
Why explicitly require that a martingale is measurable
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