Purely Atomic Structures Supporting Undominated and Nonuniformly Integrable Martingales

Let $(F_n)_{n=1,2,\cdots}$ be a sequence of sigma-fields on a set $\Omega$, each $F_n$ purely atomic with respect to a measure $P$. Let $C$ denote a nested sequence of sets $C_n$, where $C_n$ is a $P$-atom of $F_n$ for each $n$. Define $S(C) = \Sigma_n(P(C_n - C_{n+1})/P(C_n))$. Then every $L^1$-bounded martingale relative to $(F_n)_{n=1,2,\cdots}$ and $P$ is uniformly integrable if and only if $S$ is finite-valued, and every such martingale is dominated if and only if $S$ is uniformly bounded.