Uniform convergence a.e. does not imply convergence in mean

Question

According to my notes: \begin{align} f_n \rightarrow f \text{ u.a.e} \nRightarrow f_n \rightarrow f \text{ in }L^1 \end{align} but if $ \mu(\Omega)<\infty$ and $ f \in L^1(\Omega)$, then the implication holds. I'm not sure why it is not sufficient $ \mu(\Omega)<\infty $, since I was not able to find an example in which $ f_n \rightarrow f \text{ u.a.e} \text{ but } f_n \nrightarrow f \text{ in }L^1$ in this case. I should find an example in which $ f_n \rightarrow f $ u.a.e. but $ f \notin L^1. $ The only way I've thought to find an example is to find a series whose uniform limit is not measurable, so I was thinking to use Vitali's set but I could not find anything.