Absolute value and powers of a martingale are submartingales

Fix $p\ge 1$ and set $\varphi (t)=\mathrm{\mid }t{\mathrm{\mid }}^p$ on all of $\mathbb{R}$, using $\varphi (0)=0$. The power lemma supplies finiteness, Borel measurability and convexity, including real noninteger $p$ and the endpoint $p=1$. Since $\varphi (M_n)\ge 0$, the assumed finite pth moment is exactly its $L^1$ condition. The convex-transform theorem therefore gives $E[\mathrm{\mid }{M}_{n+1}{\mathrm{\mid }}^p\mid {\mathcal{F}}_n]\ge \mathrm{\mid }{M}_n{\mathrm{\mid }}^p$ a.s. for every $n$.

At $p=1$, $E\mathrm{\mid }{M}_n\mathrm{\mid }<\mathrm{\infty }$ already follows from the martingale definition, so the first assertion requires no extra moment hypothesis. AC is inherited from the convex-transform theorem and its conditional Jensen argument. For $p>1$ the moment assumption is retained; no assertion for $p<1$ is made.