Nonnegative or absolutely $L^1$-summable series may be integrated term by term.Suggest a slogan
Let \((X,\mcm,\mu)\) be a measure space.
If \(f_n\in\) $L^+$ for \(n\geq1\), then \(\int_X\sum_n f_n\dmu=\sum_n\int_X f_n\dmu\) in \([0,\infty]\) [@Fol13].
If \(f_n\in L^1(\mu)\) for \(n\geq1\) and \(\sum_n\int_X\abs{f_n}\dmu<\infty\), then \(\sum_n f_n\) converges almost everywhere to a function in \(L^1(\mu)\) and \(\int_X\sum_n f_n\dmu=\sum_n\int_X f_n\dmu\) [@Fol13].
By (a) applied to \(\abs{f_n}\), the hypothesis \(\sum_n\int_X\abs{f_n}\dmu<\infty\) in (b) is the same as \(\int_X\sum_n\abs{f_n}\dmu<\infty\).