Theorem

Let $X\in\Lambda^{2}(\mathscr{P},M)$, let $0\le s<t$, let $Z$ be $\mathcal{F}_{s}$-measurable and bounded. Then: 1. $\mathbb{1}_{(s,t]}Z$ is $\mathscr{P}$-measurable (i.e. predictable) 2. $\mathbb{1}_{(s,t]}ZX\in L^{2}(\mathbb{R}^{+}\times\Omega,\mathscr{P},\mu_{M})$ 3. $$\int\limits \mathbb{1}_{(s,t]}ZX \, dM =Z\int\limits \mathbb{1}_{(s,t]}X \, dM \text{ a.s.}$$