Is there an example of progressively measurable process that is not predictable?

This question is motivated by Revuz-Yor, Continuous Martingales and Brownian Motion http://www.springer.com/gb/book/9783540643258, where 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 than the one we get with the predictability imposition, or if this yields the same result in this case.