Integral of Series of Positive Measurable Functions

Theorem
Let $\struct {X, \Sigma}$ be a measurable space.

Let $\sequence {f_n}_{n \mathop \in \N} \in \MM_{\overline \R}^+$, $f_n: X \to \overline \R$ be a sequence of positive measurable functions.

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

Then:


 * $\ds \int \sum_{n \mathop \in \N} f_n \rd \mu = \sum_{n \mathop \in \N} \int f_n \rd \mu$

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