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A standard result in real analysis says that if $f : (\mathbb{R}, +) \rightarrow (\mathbb{C}^*, *)$ is a Lebesgue measurable (group) homomorphism with $|f| = 1$, then $$(\exists ~\xi \in \mathbb{R})~ f(x) = e^{\large 2\pi{}i\langle \xi, x\rangle},~ \forall x \in \mathbb{R}.$$ The only proof I know uses the Lebesgue Differentiation Theorem to establish that there exists an $a > 0$ such that $$\int_0^a f(x)dx \neq 0.$$ I'd like to see an alternative proof of this statement, that such an $a$ exists, using a more elementary argument.

For instance, if $f$ is continuous, then this is very easy to prove. Thus, following one of Littlewood's three principles of analysis (namely, that every measurable function is almost continuous), one should be able to extend this to measurable functions, say, via Lusin's Theorem, but so far I haven't been able to do so.

Any suggestions?

student
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  • There is probably something that you can do with showing that the kernel is measurable, and conclude it has to be of measure zero or $\Bbb R^n$. If the kernel is everything then this is the zero homomorphism; otherwise you can probably show that the function is actually continuous, and finish the proof. – Asaf Karagila Jul 13 '13 at 20:39

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By Luzin's theorem, the restriction of $f$ is continuous on a positive measure compact set $K$. So $f$ is also continuous on $K - K$ (Why?) which contains an interval. Hence $f$ is everywhere continuous.

I do not much care about Littlewood slogans but let me know if you think either one of "Luzin's theorem" and "$K - K$ has interior" is not elementary.

hot_queen
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  • That second theorem is so useful! Huzzah! – Asaf Karagila Jul 13 '13 at 23:35
  • Here's a fun problem in similar spirit: It is easy to see (on real line) that every positive measure set contains a similar copy (scale and translate) of ${1, 1/2, 1/4, 1/8, \dots, 1/2^{n-1}}$. What happens if we replace $n$ by $\omega$? – hot_queen Jul 13 '13 at 23:49
  • So the question whether or not every set of positive measure contains a [strictly increasing] sequence with its limit? – Asaf Karagila Jul 13 '13 at 23:51
  • Sorry if I wasn't clear but the question is: Does every positive measure set of reals contain a similar copy of ${1, 1/2, 1/4, \dots }$? I think this problem (due to Erdos) is still open. – hot_queen Jul 13 '13 at 23:55
  • Ah. I see. Very different than what I thought you were asking. Good question. – Asaf Karagila Jul 13 '13 at 23:57
  • Here's a survey: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.rae/1214571347 – hot_queen Jul 13 '13 at 23:58
  • You should remind me in a week. I will either be finished with grading papers and exams, or the department administration will hang me upside down and flay me alive. In either way, I'll have time to read that only a week from now. ;-) – Asaf Karagila Jul 14 '13 at 00:00