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Let $(f_n:n\geq 0)$ be a sequence of measurable real-valued functions on some measurable space. Show $S =\{x: \sum_{n=0}^\infty f_n(x)\text{ is absolutely convergent}\}$ is measurable

My attempt: $\sum_{n=0}^\infty |f_n(x)|$ converges, then $\forall k\in\mathbb{N},\,\exists N\in\mathbb{N},\,\forall n\geq N$ $$\sum_{n\geq N}^\infty|f_n(x)|<\frac 1 k$$ Then set S can be rewritten as: $$S=\bigcap_{k\in\mathbb{N}}\bigcup_{N\in \mathbb{N}}\bigcap_{n\geq N}\{x:\sum_{n\geq N}^\infty|f_n(x)|<\frac 1 k\}$$

Am I on the right track? (Trying to show that the tail of the absolute series can be arbitrarily small)

Now I am stuck at the point to show that $\{x:\sum_{n\geq N}^\infty|f_n(x)|<\frac 1 k\}$ is measurable.

Or is there a better way to do this? Appreciate any help!

Felicks
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    Sure, note that the series is absolutely convergent if and only if the partial sums (with absolute values) are Cauchy; this should make the sets involved more obvious (no more tails of infinite series). Another way is if you know that supremums of (a sequence of) measurable functions is measurable, then the full infinite series is the supremum of the partial sums. Thus, the set of points where tue supremum is finite (i.e belongs to the Borel set $\Bbb{R}$) is measurable. Hence the points of absolute convergence is measurable. – peek-a-boo Dec 21 '21 at 17:12
  • @peek-a-boo, that helps a lot, thank you! I like your approach using the Cauchy criterion – Felicks Dec 21 '21 at 17:18
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    I'm glad that was helpful. Note however that the second approach using supremum is more direct/flexible, because from supremum we get analogous results for infimum, hence limit superior/inferior. Hence the set of points where limits exist and belong to a certain Borel set of the extended reals is also measurable in the domain. – peek-a-boo Dec 21 '21 at 17:21
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    I think this https://math.stackexchange.com/questions/1456350/the-supremum-and-infimum-of-a-sequence-of-measurable-functions-is-measurable entails that $\sum_{n=0}^\infty |f_n|$ is measurable. – Gribouillis Dec 21 '21 at 17:22
  • Thank you all for the help! – Felicks Dec 21 '21 at 17:43

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