Probability with Martingales:
About $(c)$
My understanding is that $(c)$ is equivalent to:
$$\sup E[A_n] < \infty \iff E[\lim A_n] < \infty$$
Why is that so?
2026-04-06 21:44:13.1775511853
Why is $M$ bounded in $\mathscr L^2$ iff $E[\lim A_n] < \infty$?
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$$E[\lim A_n] = \lim E[A_n]\ \text{by MCT}$$
$$= \sup E[A_n] \ \because \ P(A_n \le A_{n+1}) = 1 \ \forall \ n \ \because \ P(A_n \le A_{n+1} \ \forall \ n) = 1$$