Why aren't these two sets of stochastic processes equal?

Question

I'm learning about stochastic integrals now, and I don't understand the following:

If $S$ and $L$ are two classes of processes where:

$S=\{f(s,\omega) |f $ is progressively measurable and $E(\int_{0}^{t}f^2ds)<\infty \}$

$L=\{f(s,\omega) |f $ is progressively measurable and $P(\int_{0}^{t}f^2ds<\infty )=1\}$

I don't understand why $S\subset L.$ Why aren't they equal?