Let $(\Omega, \mathscr{F}, P)$ be a probability space with filtration $(\mathscr{F}_t)$, and $(X,\mathscr{B})$ be a measurable space.
We consider a function $f:[0,\infty) \times X \times \Omega \to \mathbb{R} $ with following properties:
(i) for every $t>0$, $(x,\omega) \longmapsto f(t,x,\omega) $ is $ \mathscr{F}_t \otimes \mathscr{B}$-measurable,
(ii) for every $(x,\omega)$, $t \longmapsto f(t,x, \omega)$ is left continuous.
Then, I have a question. Can we express $f$ by a limit of functions of the following form?:
$$g(t,x,\omega) = \sum_{i=1}^{n} K_i(\omega) \mathbf{1}_{(a_i,b_i]}(t) \mathbf{1}_{A_i}(x),$$ where $K_i(\omega)$ is $\mathscr{F}_{a_i}$-measurable random variable and $A_i \in \mathscr{B} $ for each $i$.