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". 



> 
> 1. Is there a good reference for these
>  two facts?
> 2. Do these decompositions have standard
>  names?
> 3. 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 $.



Reference request: Martingale decompositions (positive/negative and u.i./singular)
These are the *Krickeberg* and *Riesz* decompositions, respectively. A good reference is section 4 of Chapter V in **Probabilities and Potential B** by Claude Dellacherie and Paul-Andre Meyer. 


