= Solution
Use a <filtered probability space> $(\Omega,\mathcal F,(\mathcal F_t),\mathbb P)$, and put $\mathcal F_{0-}=\mathcal F_0$. The \b[<predictable sigma-algebra>] on $\Omega\times[0,\infty)$ is
$$
\boxed{\mathcal P=\sigma\{A\times\{0\},\ A\in\mathcal F_0;\ A\times(s,t],\ A\in\mathcal F_s,\ 0\leq s<t<\infty\}.}
$$
Equivalently it is generated by left-continuous <adapted processes>. A real-valued \b[<previsible process>] is a function $(\omega,t)\mapsto H_t(\omega)$ that is <measurable> for $\mathcal P$ and the <Borel sigma-algebra> of $\mathbb R$.
A \b[<simple predictable process>] has the form
$$
H=h_{-1}\mathbf1_{\{0\}}+\sum_{i=0}^{n-1}h_i\mathbf1_{(t_i,t_{i+1}]},\qquad 0=t_0<t_1<\cdots<t_n<\infty,
$$
where $h_{-1}$ is bounded and $\mathcal F_0$-<measurable>, and each $h_i$ is bounded and $\mathcal F_{t_i}$-<measurable>. Set $H=0$ after $t_n$; allowing a final infinite interval gives an equivalent dense class. The values may be restricted to finitely many values if desired, by approximating bounded <measurable functions> by simple functions. The initial-time term matters for arbitrary <finite measures> on $\mathcal P$: a measure can charge time zero, even though the <stochastic integral> against a continuous <martingale> does not.
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