Square-integrable martingale-difference array and variance clock

Assume AC on a fixed probability space $(\mathrm{\Omega },\mathcal{F},P)$. A rowwise square-integrable martingale-difference array consists of families $({\mathcal{F}}_{n,k}{)}_{n\ge 1,k\ge 0}$ and $(Z_{n,k}{)}_{n,k\ge 1}$ such that for each $n$ the ${\mathcal{F}}_{n,k}$ form an increasing filtration of sub-$\sigma$-algebras of $\mathcal{F}$, $Z_{n,k}\in L^2$ is ${\mathcal{F}}_{n,k}$-measurable, and $\mathbb{E}[Z_{n,k}\mid {\mathcal{F}}_{n,k-1}]=0\text{a.s.}$ Define $M_{n,m}={\sum }_{k=1}^m{Z}_{n,k},v_{n,k}=\mathbb{E}[Z_{n,k}^2\mid {\mathcal{F}}_{n,k-1}],{\mathrm{\Gamma }}_{n,m}={\sum }_{k=1}^m{v}_{n,k}\mathrm{.}$ Each $v_{n,k}$ is nonnegative, integrable, and ${\mathcal{F}}_{n,k-1}$-measurable; hence ${\mathrm{\Gamma }}_{n,m}$ is an increasing predictable variance clock, while $M_{n,m}$ is a rowwise $L^2$ martingale. A finite triangular row is included by putting $Z_{n,k}=0$ and holding the filtration fixed after its last column. This preserves both limiting sums. AC is used only through conditional-moment existence and countable representative selection.