For a paper I am writing, I need these two facts. The proofs are fairly short, but I would rather just cite them. This is for martingales index by natural numbers. Also, I call a martingale which converges to 0 "singular". I have also seen them called "potentials".
Is there a good reference for these two facts?
Do these decompositions have standard names?
Is there a standard term for a martingale which converges to 0?
Below, $\Vert M \Vert$ is the $L^1$-bound of the martingale $M_k$.
Decomposition 1. Let $(M_{k})$ be an $L^{1}$-bounded martingale with respect to the filtration $({\mathcal{F}}_{k})$. Then there are two nonnegative martingales $(P_k)$ and $(N_k)$ such that such that $M_{k}=P_k-N_k$ a.e. for all $k$, and $\left\Vert M\right\Vert =\left\Vert P\right\Vert +\left\Vert N\right\Vert = \Vert P_0 \Vert_1 + \Vert N_0 \Vert_1$. Further, this decomposition is a.e. unique; $(P_k)=\sup_{n\geq k}E[[M_{n}]^{+}\mid\mathcal{F}_{k}]$ a.e.; $N_k=\sup_{n \geq k}E[[M_{n}]^{-}\mid\mathcal{F}_{k}]$ a.e.; $\lim_{k\rightarrow\infty}P_k=[\lim_{k}M_{k}]^{+}$ a.e.; and $\lim_{k\rightarrow\infty}N_k=[\lim_{k}M_{k}]^{-} a.e.$
Decomposition 2. Let $(M_{k})$ be an $L^{1}$-bounded martingale with respect to the filtration $(\mathcal{F}_{k})$ and let $M_{\infty}=\lim_{n}M_{n}$. Then there is a uniformly integrable martingale $(U_k)$ and a singular martingale $(S_k)$ such that $M_{k}=U_k+S_k$ a.e. for all $k$. Further, this decomposition is a.e. unique; $U_k=E[M_{\infty}\mid\mathcal{F}_{k}]$ a.e.; $S_k=E[M_{k}-M_{\infty}\mid\mathcal{F}_{k}]$ a.e.; and $\left\Vert M\right\Vert =\left\Vert U\right\Vert +\left\Vert S\right\Vert $.
