Doob Decomposition Theorem in Williams is working backward? Unique modulo indistinguishability? Probability with Martingales * * * . 4. The proof that $M = X - X_0 - A$ is a martingale null at 0. 5. The details of unique modulo indistinguishability Is that right? How can we prove #5?

For 5.: The key is this: A previsible martingale is constant in time. In more detail, if $X$ admits a second decomposition $X_n=X_0+M'_n+A'_n$ then by subtracting you find that $M_n-M'_n$($=A'_n-A_n$) is both a martingale and previsible, hence constant in time a.s., whence $M_n-M'_n=M_0-M'_0=0-0=0$ for all $n$, a.s.