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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

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    One assumption is $\sum_n \int f_n < \infty$. Of course you do need some assumption: the general nonnegative function finite a.s. can be written as a sum of square-integrable functions.2012-03-06

1 Answers 1