Infinite Series of Measurable Functions is Measurable

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

Let $\sequence {f_n}_{n \mathop \in \N}$ be a sequence of $\Sigma$-measurable functions $f_n : X \to \overline \R$ such that:


 * for each $N \in \N$ and $x \in X$, the summation $\ds \sum_{n \mathop = 1}^N \map {f_n} x$ is well-defined.

Then:


 * the series $\ds \sum_{n \mathop = 1}^\infty f_n$ is $\Sigma$-measurable.

Proof
From Pointwise Sum of Measurable Functions is Measurable: General Result, we have:


 * $\ds \sum_{n \mathop = 1}^N f_n$ is $\Sigma$-measurable for each $N \in \N$.

Then, from Pointwise Limit of Measurable Functions is Measurable, we have:


 * $\ds \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N f_n$ is $\Sigma$-measurable.

From the definition of infinite series, we have:


 * $\ds \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N f_n = \sum_{n \mathop = 1}^\infty f_n$

so that:


 * $\ds \sum_{n \mathop = 1}^\infty f_n$ is $\Sigma$-measurable.