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
 A: In a general context, we can say this: If $(f_n)$ are non-negative and measurable functions and $f=\sum_{n\geq 1}f_n$ then, by the Monotone Convergence Theorem we have that $$\int fd\mu=\sum_{n\geq 1}\int f_nd\mu$$
so $\int fd\mu<+\infty$ if and only if the series on the right-hand side is finite.
As I see it, it really doesn't matter if we have some integrability condition on the $(f_n)$ or the measure is finite. Of course, with additional assumptions (like the ones you are stating) you could derive more specialized conditions under wich the series converges, but any usual rule will suffice. For example, in your setting one criterion could be that $$\sum_{n\geq 1}\|f_n\|_{L^2}<+\infty.$$
This is because if we are in finite measure, then by Hölder's Inequality one has that $\|g\|_{L^1}\leq\|g\|_{L^2}$ for all square integrable $g$. Then, noting that $$\sum_{n\geq 1}\int f_nd\mu=\sum_{n\geq 1}\|f_n\|_{L^1}\leq\sum_{n\geq 1}\|f_n\|_{L^2}<+\infty$$ you get the result.
Edit: I just realized the following. Since $L^2$ is a Banach space, the condition I gave implies that $f\in L^2$, and hence finite a.s.
