Let $(\Omega, (\mathcal F_t)_{t\geq 0}, \mathcal F)$ be a filtered measurable space. Consider the deterministic process $X(t,\omega) = h(t)$ where $h: \mathbb R_+ \rightarrow \mathbb R$ is a function. Prove that $X$ (or just $h$) is a predictable process if and only if $h$ is Borel measurable.
In this link, @saz gave an answer using a result that says every Borel (measurable) function $h$ can be approximated by simple left continuous processes of the form
$$ s(t) = \sum_{i=1}^n c_i 1_{(a_i,b_i]}(t), \quad n\geq1,\quad (a_i,b_i]'s \ \text{are disjoint}. $$
However, it seems that this is wrong. If this is true, it means that $1_A(t)$, where $A$ is a Borel set, can also be approximated by these simple, left-continuous functions. But I cannot prove or disprove this. Any help is appreciated.
Also, I don't know how to start the part where $h$ is first assumed to be predictable then it implies $h$ is Borel measurable. Thank you for all your help!
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