Suppose $f(t,\omega)$ is separable stochastic process which is adapted to the filtration $F_t, 0<t<T$, is integral $\int_0^tf(s,\omega)ds$ measurable with respect to $F_t$? We assume that integral exists almost sure. Definition: A stochastic process $(X(t), t\in I)$ is called separable if there exists a countable sequence $(t_j)$ that is a dense subset of $I$ and a subset N of $\Omega$ with $P(N)=0$ such that, if $\omega \notin N$, then

$$\{X(t, w) \in F\ for\ all\ t\in J\} = \{X(t_j, w) \in F\ for\ all\ t_j\in J\}$$ for any open subset J of I and for any closed subset F of R