I'm trying to determine conditions on a filtration $\{ \mathcal{F}_t \}_t$ such that the identity
$\lim\limits_{\Delta \longrightarrow 0+} E\big[ f(t,\Delta) | \mathcal{F}_{(t-\Delta)-} \big] = \lim\limits_{\Delta \longrightarrow 0+} E\big[ f(t,\Delta) | \mathcal{F}_{t-} \big] $
holds, when, for instance $f \in L^1$ uniformly in $\Delta$.
It is stated in Protter (Stochastic Integration and Differential Equations, V2.1, p.191):
"Note that if $T$ is predictable, and if $T_n$ is an announcing sequence of stopping times for $T$, and if $X \in L^1$, then $\lim_{n\longrightarrow \infty}E\{ X | \mathcal{F}_{T_n}\} = E \{ X | \mathcal{F}_{T-} \}$."
I think this would imply my desired result if $f$ was not a function of $\Delta$, but as stated it is unclear what is required of $\{ \mathcal{F}_t \}_t$. Moreover, I'm not sure how to deal with $f$ being a function of $\Delta$.