I am trying to understand example of process which is adapted and measurable but not progressively measure.
I am analyzing Example 0.3 from this paper: https://projecteuclid.org/journals/electronic-communications-in-probability/volume-18/issue-none/On-existence-of-progressively-measurable-modifications/10.1214/ECP.v18-2548.full
And the question is: why $\mathbb{1}_{\Delta}$ is a pointwise limit of functions of the form $$\sum_{k=1}^{N} c_{k} \mathbb{1}_{B_{k} \times C_{k}}$$ where $N \in \mathbb{N}$, $c_{k} \in \mathbb{R}$, $B_{k} \in \mathcal{B}([0, \frac{1}{2}])$, $C_{k} \in \mathcal{A}$???
I completely don't see it...