Approximation of $L^{1}$-bounded martingales by martingales of bounded variation

Abstract:

If $f = ({f_1},{f_2}, \ldots )$ is a real ${L^1}$-bounded martingale and $\varepsilon > 0$, then there is a martingale g of bounded variation satisfying ${\left \| {f - g} \right \|_1} < \varepsilon$. The same result holds for X-valued martingales, where X is a Banach space, provided X has the Radon-Nikodým property. In fact, this characterizes Banach spaces having the Radon-Nikodým property. Theorem 1 identifies, for an arbitrary Banach space, the class of ${L^1}$-bounded martingales that converge almost everywhere.