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Topic: From addition preservation to linearity
Replies: 3   Last Post: Sep 24, 2012 1:14 PM

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Rupert

Posts: 3,809
Registered: 12/6/04
Re: From addition preservation to linearity
Posted: Sep 24, 2012 4:41 AM
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On Sep 24, 1:14 am, José Carlos Santos <jcsan...@fc.up.pt> wrote:
> Hi all,
>
> Can someone please tell me where can I find a reference for the fact
> that any Lebesgue-measurable map from R^n into R^m which preserves sums
> is necessarily a linear map?
>
> Best regards,
>
> Jose Carlos Santos


I don't know a reference, but one of the crucial points is that if A
is a subset of R^n of positive measure then A+A has nonempty interior.
David Ullrich showed me a proof of this a while back but he didn't
know a reference in the literature.

I think this might be relevant

@article{Kestelman51,
author="Hyman Kestelman",
title="Automorphisms of the field of complex
numbers",
journal="Proc. London Math. Soc.",
issue=2,
number=53,
pages="1-12",
year=1951}



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