I want to prove the above lemma. How can I show (c)? There is a hint: for $f:I\to \mathbb R$ convex and increasing, there exist $a \in \mathbb R$ and $b \in \mathbb R_+$ such that $a+bx\leq f(x)$. I showed that the hint is correct, but how can I apply it in this context?
The negative part $Y_t^-$ of $Y_t$ is integrable because $f$ is bounded below by an affine function, per the hint. Meanwhile, by Jensen, if $0\le s<t$, then $E[Y_t^+|\mathcal F_s]\ge f(E[M_t|\mathcal F_s])^+=f(M_s)^+=Y_s^+$ (the positive part of $Y_s$), even on $\{E[Y_t|\mathcal F_s]=+\infty\}$. Consequently, $E[Y^+_t]\ge E[Y_s^+]$, so the integrability of $Y_t$ implies that of $Y_s$ for $0\le s<t$.