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Indicators of the rectanglestogether with $A\times \{0\}$ generate the predictable sigma-algebra. Their finite linear span is precisely the set of simple processes. The monotone-class theorem therefore makes this span dense among bounded predictable functions in measure. Since $\nu$ is finite, truncation followed by bounded approximation proves density in $L^2(\mathcal{P},\nu )$.

The predictable sigma-algebra $\mathcal{P}$ on $\Omega \times [0,\infty )$ is generated by the sets $A\times \{0\}$ with $A\in {\mathcal{F}}_0$ and $A\times (s,t]$ with $0\le s<t$ and $A\in {\mathcal{F}}_s$. A process is previsible precisely when it is $\mathcal{P}$-measurable.

In discrete time, a process $A_n$ is predictable or previsible when $A_n$ is ${\mathcal{F}}_{n-1}$-measurable for every $n\ge 1$. In continuous time, predictability means measurability with respect to the predictable sigma-algebra.