Series of Positive Measurable Functions is Positive Measurable Function

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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 $\displaystyle \sum_{n \mathop \in \N} f_n$ is also a positive measurable function.


Proof


Sources