Integral of Series of Positive Measurable Functions

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Theorem

Let $\left({X, \Sigma}\right)$ be a measurable space.

Let $\left({f_n}\right)_{n \in \N} \in \mathcal{M}_{\overline{\R}}^+$, $f_n: X \to \overline{\R}$ be a sequence of positive measurable functions.

Let $\displaystyle \sum_{n \mathop \in \N} f_n: X \to \overline{\R}$ be the pointwise series of the $f_n$.


Then:

$\displaystyle \int \sum_{n \mathop \in \N} f_n \, \mathrm d \mu = \sum_{n \mathop \in \N} \int f_n \, \mathrm d \mu$

where the integral sign denotes $\mu$-integration.


Proof


Sources