If there exists a sequence $(\phi_n)$ of step functions such that $\phi_n \longrightarrow f$ almost everywhere on $[a,b]$, can we prove that $f\in L$?

Question

I know, if $f\in L$ (the set of all Lebesgue integrable functions), then there exists a sequence $(\phi_n)$ of step functions such that $\phi_n \longrightarrow f$ almost everywhere on $[a,b]$ and $\lim\int_a^b\phi_n = \int_a^bf$. But the question is: the converse is also true? If yes, please give me a proof. If no, give me a counterexample, please.

Thanks a lot.