= Solution
Use the <predictable sigma-algebra> generated by $A\times\{0\}$, $A\in\mathcal F_0$, and $A\times(s,t]$, $A\in\mathcal F_s$. A <previsible process> is a measurable function for this sigma-algebra. Set $J_t=\mathbf1_{\{S<t\leq T\}}$, with $J_0=0$. The events $\{S<t\}$ and $\{T<t\}$ are in $\mathcal F_t$, because they are countable unions of events $\{S\leq q\}$ and $\{T\leq q\}$ for rational $q<t$. Thus $J$ is an <adapted process>. Each path is left-continuous, including at the two endpoints, and is zero if $S=T$.
For a direct measurability argument, let
$$
J_t^{(n)}=\sum_{k\geq0}J_{k2^{-n}}\mathbf1_{(k2^{-n},(k+1)2^{-n}]}(t).
$$
Each summand is measurable for the <predictable sigma-algebra>, since its coefficient is $\mathcal F_{k2^{-n}}$-measurable. At every $t>0$ the sampling time approaches $t$ strictly from below, so left-continuity gives $J_t^{(n)}\to J_t$. Therefore
$$
\boxed{\mathbf1_{(S,T]}\text{ is previsible}.}
$$
The original PDF has a right-closed interval. The open right endpoint in the converted TeX is a transcription error; replacing $(S,T]$ by $(S,T)$ is not valid for arbitrary <stopping times>.
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