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I have the following Problem:

Let $f_n:\mathbb{R}\rightarrow \mathbb{R}$ be Borel-mesurable functions. Show that $\{x|\limsup\,f_n(x)=\infty\}$ is borel set.

In the lecture, we did a similar proof with $\lim f_n(x)$ instead of $\limsup$. There we rewrote the condition in terms of quantors (i.e., with $\forall, \exists$) and then interpreted this as unions and intersections. But somehow, I don't see it here. Could maybe someone help me?

Thanks!

Sumanta
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user123234
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  • \limsup f_n(x), \lim f_n(x) – Sumanta Oct 19 '21 at 18:40
  • Your approach sounds right. Are you having trouble expanding the definition of $\limsup$ to rewrite the condition $\limsup f_n(x)=\infty$ using quantifiers? – Karl Oct 19 '21 at 18:43
  • One way to start is to show that $g_n = \sup_{k \ge n} f_k$ is Borel-measurable. Then apply the result from your lecture to $\lim g_n$. – Umberto P. Oct 19 '21 at 18:44
  • yes I somehow have troubles to rewrite this with quantifiers. – user123234 Oct 19 '21 at 18:45
  • See here https://math.stackexchange.com/a/1327089/591889 – Sumanta Oct 19 '21 at 18:45
  • @UmbertoP. If I want to do it in your way, is it correct that I also rewrite sup with quantifiers? – user123234 Oct 19 '21 at 18:46
  • If ${a_n}$ is a sequence, then $\sup_n a_n > t$ if and only if there exists at least one index with $a_n > t$. Thus $${ x \mid g_n(x) > t} = \cup_{k=n}^\infty {x \mid f_k(x) > t}$$ and these are all Borel sets. – Umberto P. Oct 19 '21 at 18:53
  • @UmbertoP. Your way sounds interesting, but why do I don't care about $\infty$ so in the exercise the limes superior is $\infty$ – user123234 Oct 19 '21 at 18:54
  • First, try rewriting $\limsup$ using the definition of $\limsup$. Then, try changing the definition so that the quantifiers range only over a countable set rather than all of $\mathbb{R}$. From here, turn these quantifiers over countable sets into intersections and unions. – Mark Saving Oct 19 '21 at 18:54
  • @MarkSaving. Yes this was just my idea but I don't manage it to do it with lim sup, somehow I'm confused. Could you maybe help me more because the link from above wasn't very helpful – user123234 Oct 19 '21 at 19:08

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