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authorAdrian Kummerlaender2017-03-17 20:29:05 +0100
committerAdrian Kummerlaender2017-03-17 20:29:05 +0100
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Add section on sums of measurable functions
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@@ -295,6 +295,12 @@ Sei $(X, \A, \mu)$ ein Maßraum, $f_n : X \to [0,\infty]$ $\A$-$\overline\B_+$-m
Dies gilt nicht ohne Monotonie oder für eine fallende Folge $(f_n)_{n \in \N}$.
+\subsubsection*{Summen messbarer Funktionen}
+
+Seien $f_j : X \to [0,\infty]$ $\A$-$\overline\B_+$-mb. für $\forall j \in \N$. Dann ist auch $\sum_{j=1}^\infty f_j$ $\A$-$\overline\B_+$-messbar und:
+
+$$\int_X \sum_{j=1}^\infty f_j(x) d\mu(x) = \sum_{j=1}^\infty \int_X f_j(x) d\mu(x)$$
+
\subsection*{Integral für $\overline\R$-wertige Funktionen}
Sei $f : X \to \overline\R$ eine $\A$-$\overline\B_1$-mb. Funktion. Dann sind auch $f_+$ und $f_-$ mb. $f$ ist Lebesgue-integrierbar, wenn: