This comes from proposition 2.34 of Jacod/Shiryaev: limit theorems for stochastic process
For any stochastic process $X$, we denote $^P X$ as the predictable projection of $X$. Jacod then writes this: for a predictable time $S$ and optional random set $A$, \begin{equation} \mathbb{P}((^P 1_A)_S >0,\;\; S <\infty ) = \mathbb{P}((1_A)_S>0,\;\; S<\infty) \end{equation} $\textbf{I have no idea why this equality holds. Can anyone help?}$
$\textbf{Some definitions:}$
(i) A stochastic basis is a probability space $(\Omega, \mathscr{F}, P)$ equipped with a filtration $\mathbf{F}=\left(\mathscr{F}_t\right)_{t \in \mathbb{R}_{+}}$; here, filtration means increasing and right-continuous family of sub- $\sigma$-fields of $\mathscr{F}$ (in other words, $\mathscr{F}_s \subset \mathscr{F}_t$ for $s \leq t$ and $\mathscr{F}_t=\bigcap_{s>t} \mathscr{F}_s$ ). By convention, we set: $\mathscr{F}_{\infty}=\mathscr{F}$ and $\mathscr{F}_{\infty-}=\bigvee_{s \in \mathbb{R}_{+}} \mathscr{F}_s$.
(ii) An optional sigma-field (denoted by $\mathcal{O}$) is the $\sigma$-field on $\Omega\times \mathbb{R}_+$ that is generated by all cadlag adapted processes
(iii) A predictable sigma-field (denoted by $\mathscr{P}$) is the $\sigma$-field on $\Omega\times \mathbb{R}_+$ that is generated by all cag adapted processes
(iv) For any two stopping times $S$ and $T$, \begin{array}{l} [\![ S, T ]\!] :=\left\{(\omega, t): t \in \mathbb{R}_{+}, S(\omega) \leq t \leq T(\omega)\right\} \\ [\![ S, T [\![ :=\left\{(\omega, t): t \in \mathbb{R}_{+}, S(\omega) \leq t<T(\omega)\right\} \\ ]\!] S, T ]\!] :=\left\{(\omega, t): t \in \mathbb{R}_{+}, S(\omega)<t \leq T(\omega)\right\} \\ ]\!] S, T [\![ :=\left\{(\omega, t): t \in \mathbb{R}_{+}, S(\omega)<t<T(\omega)\right\} . \end{array}
(v) A predictable time is a mapping $T:\Omega\rightarrow \bar{\mathbb{R}}_+$ such that the stochastic interval $[\![ 0, T [\![ \in \mathscr{P}$
(vi) If T is a stopping time, we denote by $ \mathscr{F}_{T-}$ the $\sigma$-field generated by $\mathscr{F}_0$ and all the sets of the form $A \cap \{ t<T \}$, where $t \in \mathbb{R}_+$ and $A\in\mathscr{F}_t$
(vii) For any $\bar{\mathbb{R}}$-valued process $X$ which is $\mathscr{F}\otimes \mathcal{B}(\mathbb{R}_+) $ measurable, the predictable projection of $X$, denoted by $^P X$, is the process determined uniquely up to an evanescent set by the following two conditions:
$a)\quad ^P X \;is\;predictable $
$b)\quad (^PX)_T\cdot1_{T<\infty} = \mathbb{E}(X_T|\mathscr{F}_{T-})\cdot 1_{T<\infty}\quad for\;all\;predictable\;time\;T$
$\textbf{Remark:}$ I know that $(^P1_A)_S \cdot 1_{S<\infty} = \mathbb{E}((1_A)_S \cdot1_{S<\infty}|\mathcal{F}_{S-})$, which follows by the definition of predictable projection (see Theorem 2.28 of Jacod/Shiryaev)