= Solution
For fixed $u>0$, define the <left-limit sigma-algebra> by $\mathcal F_{u-}=\sigma(\bigcup_{s<u}\mathcal F_s)$. The section at $u$ of a generating rectangle $A\times(s,t]$ of the <predictable sigma-algebra> is $A$ when $s<u\leq t$, and empty otherwise. In the first case $A\in\mathcal F_s\subseteq\mathcal F_{u-}$. Sections of the time-zero generators are empty for $u>0$.
The collection of subsets of sample-time space whose section at $u$ belongs to $\mathcal F_{u-}$ is a <sigma-algebra>: sections commute with complements and countable unions. It therefore contains all of $\mathcal P$. Apply this to $\{(\omega,t):H_t(\omega)\in E\}$ for every <Borel set> $E$. Thus \b[<predictable sections at deterministic times> satisfy]
$$
\boxed{H_u\text{ is }\mathcal F_{u-}\text{-measurable}.}
$$
At $u=0$ the same argument gives $\mathcal F_0$-<measurability>, using the convention $\mathcal F_{0-}=\mathcal F_0$. The PDF asks for information strictly before $u$; the TeX aid drops the minus sign.
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