If $g:[a,b]\to[0,\infty]$ is lebesgue integrable on the compact interval $[a,b]$, is $g$ bounded $\textit{almost everywhere}$?

Question

I know that $g\in L^1[a,b]$ implies that $g<\infty$ almost everywhere. It seems like it should follow that $g$ is bounded almost everywhere, but I can't think of a proof or a counterexample.