In Kartazas and Shreve, the integral for Bounded Progressively measurable processes is defined first. Then, for Bounded measurable processes ($f(t,\omega)$), the authors say that there exists a progressively measurable modification and show how to define an integral in this case.
For a bounded, measurable and adapted process the progressively measurable modification is also bounded (I found this from P.A. Meyer's book Probability and Potentials). Therefore, if $g(t,\omega)$ is the modification we know that there exist elementary $g_n(t,\omega)$ such that $E\int_S^T(g(t,\omega)-g_n(t,\omega))^2dt \to 0$ as $n \to \infty$. Then, it seems that $E\int_S^T(f(t,\omega)-g_n(t,\omega))^2dt \to 0$. I do not understand why the authors proceed in a more complicated manner (see the page which contains equation 2.8 in section 3.2) to define the integral.