Localization of processes

Question

my notation follows the book Limit theorems of stochastic processes by Shyraev/Jacod: $\nu$ denotes the class of all càdlàg adapted processes with finite variation of every intervall $[0,t]$,$t>0$. Now $\mathcal{A}$ is the class of all $A\in\nu$, such that $E[Variation(A)_\infty]<\infty$. Now, there is a theorem which states that every process $M$ is a local martingale iff $M^{\tau_n}$ is a local martingale for each $n$ (see Th. 15.1 in Kallenberg) where $(\tau_n)$ is a localizing sequence. Is there a similar statement for processes in $\mathcal{A}_{loc}$?