 Solution

Put $C={\sup }_p\mathbb{E}|X_p|<\infty$. Since $x\mapsto x^{+}$ is an increasing convex function, the conditional Jensen inequality and the submartingale property giveFor fixed $n$ and $p\ge n$, the tower property of conditional expectation therefore impliesChoose versions for this countable family so that all these inequalities hold outside one null set. Its nonnegative increasing limit $M_n$ is ${\mathcal{F}}_n$-measurable. The monotone convergence theorem shows thatIn particular the limit is finite almost surely. Thus the increasing conditional means converge to an integrable nonnegative $M_n$, rather than merely to a possibly infinite extended value. This is the construction of the positive martingale majorant of an L1-bounded submartingale.