= Solution
Write the <predictable sigma-algebra> on positive times as
$$
\mathcal P=\sigma\{B\times(s,t]:0\leq s<t<\infty,\ B\in\mathcal F_s\}.
$$
It is equivalently the smallest <sigma-algebra> making all left-continuous <adapted processes> measurable. Indeed, the indicator of each generating rectangle is left-continuous and adapted, and approximating any left-continuous <adapted process> by its values at the preceding points of deterministic partitions gives the reverse inclusion. \b[Previsibility records information available immediately before observation.] If time zero is included, add $B\times\{0\}$ with $B\in\mathcal F_0$. The displayed description in part (b) uses positive times, as indicated by its product space.
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