I am trying to solve the following problem III.2.30 from Karatzas, Shreve book: assume that $M$ is a continuos local martingale and $X$ is a "proper" integrand, i.e. a measurable, adapted process, such that $\int_0^T X_t^2dt$ is almost surely finite for every $T$. Then for every $\mathcal{F}_s$-measurable random variable $Z$ we have $$ \int_s^t ZX_udM_u=Z\int_s^t X_udM_u $$ Obviously, first I tried to prove in case when $M$ is a continuous, square integrable martingale (for simplicity I assume further that $M$ is a Brownian motion $W$) and integrand $X$ belongs to more narrow class of measurable, adapted processes such that $[X]_T\triangleq \mathbb{E}\int_0^TX_t^2dt<\infty$ for every $T$.
In fact, when $Z$ is a bounded random variable and $X$ is a simple process, then above proposition is obvious. Then, approximating $X$ with a simple process, this proposition can be extended to measurable, adapted processes satisfying $[X]_T\triangleq \mathbb{E}\int_0^TX_t^2dt<\infty$.
How should this proposition be extended to $\mathcal{F}_s$-measurable random variable $Z$? Am I right to assume that there should be some constraints on $Z$ (e.g. integrability)?