in the book "Diffusions, Markov Processes, and Martingales: Volume 1, Foundations" by L. C. G. Rogers,David Williams, Doob's $L^p$-inequality is stated as theorem 52.6
Let $p > 1$ and $q^{−1} + p^{-1} = 1$ . Let $Z$ be a non-negative sub-martingale bounded in $L^p$ , and define $Z^∗ ≡ \sup_{k\in N} Z_k$. Then $Z^∗ ∈ L^p$ and $||Z^*||_p \le q\cdot\sup_{n\in N}||Z_n||_p$. Also $Z_\infty = \lim_{n\to\infty}Z_n$ almost surely and in $L^p$ and $||Z_\infty||_P=\lim_{n\to\infty}||Z_n||_p=\sup_{n\in N}||Z_n||_p$.
If $Z$ is of the form $ Z = |M|$, where $M$ is a martingale bounded in $L^p$ , then $M_\infty = \lim_{n\to\infty} M_n$ exists almost surely and in $L^p$ , and $Z_\infty = |M_\infty|$ almost surely.
I am having trouble seeing why the second paragraph is true. Clearly $Z$ is a non-negative sub-martingale, therefore we can apply the first part and see that the limit $Z_\infty$ exists almost surely and in $L^p$. But why does that also imply that $M_\infty$ exists and that $Z_\infty = |M_\infty|$?
The existence of $M_\infty$ does not follow from the first paragraph, but from the martingale convergence theorem, because boundedness of $M$ in $L^p$ with $p>1$ implies $E|M_n|\le(E|M_n|^p)^{1/p}$ is bounded.