Could someone give me such an example?

The question that originally motivated this is actually the following: In Revuz-Yor, "Continuous martingales and Brownian Motion" http://www.springer.com/gb/book/9783540643258 they define the space of integrands with respect to a continuous $L^2$-bounded martingale $M$ as the progressively measurable processes $\phi$ such that $\int_0^\infty \phi^2d<M><\infty$. So they ask $\phi$ to be progressively measurable instead of the very usual, stronger, hypothesis of $\phi$ being predictable. I wonder whether this space is actually bigger to the one we get with the predictability imposition or if this yields the same result in this case.