Fix a finite time horizon $T>0$ and consider a probability space $(\Omega,\mathcal{F},(\mathcal{F}_t),\mathbb{P})$ where the filtration is assumed to satisfy the usual conditions, and an integrable random variable $V$. For any adapted process $X$, we define a martingale by $$ Y_t = \mathbb{E}(V|\mathcal{F}_t^X), \quad 0\leq t\leq T. $$ We can without loss of generality assume that it is right continuous (by taking the right continuous modification, for example). I am interested in knowing what condition on $X$ can guarantee continuity of $Y$. I know if $X$ is left continuous, then the filtration generated by it, $(\mathcal{F}_t^X)$, will be left continuous. In this case, we obtain continuity of $Y$ through simple martingale convergence theorem.
Are there any conditions weaker than left-continuity that will work? I think predictability will not be sufficient...Thanks!