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Use conditional monotone convergence and the tower property of conditional expectation in the preceding construction:This proves the martingale property. The construction gives $M_n\ge 0$ and ${\sup }_n\mathbb{E}{M}_n\le C$, so $M$ is bounded in $L^1$. Taking $p=n$ in the increasing family also gives $M_n\ge X_n^{+}\ge X_n$. Define $Y_n=M_n-X_n$. Then $Y$ is nonnegative and adapted, andThus $Y$ is a supermartingale, with $\mathbb{E}|Y_n|\le \mathbb{E}{M}_n+\mathbb{E}|X_n|\le 2C$. The requested conclusion isThis positive martingale majorant of an L1-bounded submartingale is not asserted to have uniform integrability; boundedness in $L^1$ alone does not imply that stronger property.

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.