measure theory - When is this bounded?

Suppose we have non-negative measurable functions $f_n$ which are square integrable on a finite measure space $\Omega$, i.e. $\mu(\Omega) < \infty$, where $\mu$ is the measure. We know

$$f:=\sum_{n\ge 1} f_n <\infty \hspace{8pt}\text{a.s.}$$

Under which assumption is this bounded, i.e.

$$\int_{\Omega} f \; d\mu <\infty$$

Thanks for your help

hulik