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In Karatzas and Shreve, the integral for Bounded Progressively measurable processes is defined first. Then, for Bounded measurable and adapted processes ($f(t,\omega)$), the authors say that there exists a progressively measurable modification and show how to define an integral in this case.

However, for a bounded, measurable and adapted process the progressively measurable modification is also bounded (this can be seen from Theorem IV T46 in P.A. Meyer's book Probability and Potentials, Blaisdell Publishing 1966). 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.

(I have asked this question in stackexchange but did not get any answers there).

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