I'm struggling to understand the proof of Lemma 2 in these notes. Am I correct in interpreting $E(Y_N; A)$ as $E(Y_N 1_A)$? It seems to me that, rewriting either end of the chain of inequalities as integrals, all we've proven here is that for $A\in F_T$:
$$\int_A E(Y_N|F_T)dP\geq \int_A Y_T dP$$
But an inequality for all integrals over measurable sets doesn't imply the inequality is true almost everywhere.