A CHARACTERIZATION OF BANACH FUNCTION SPACES ASSOCIATED WITH MARTINGALES

Abstract

Let $(\Omega,\, \Sigma,\,\Prob)$ be a nonatomic probability space and let $\F=(\F_n)_{n{\in}\Z}$ be a filtration. If $f=(\,f_n)_{n{\in}\Z}$ is a uniformly integrable $\F$-martingale, let $\A_{\F}f=(\A_{\F}f_n)_{n{\in}\Z}$ denote the martingale defined by $\A_{\F}f_n =\E[|\,f_{\infty}||\F_n]\; (n \,{\in}\, \Z)$, where $f_{\infty}=\lim_n f_n$ a.s. Let $X$ be a Banach function space over $\Omega$. We give a necessary and sufficient condition for $X$ to have the property that $S(\,\hspace*{.2pt}f\hspace*{.3pt}) \,{\in}\, X$ if and only if $S(\A_{\F}f) \,{\in}\, X$, where $S(\,\hspace*{.2pt}f\hspace*{.3pt})$ stands for the square function of $f=(\,f_n)$.