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

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Posts: 3,810
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 <> 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

author="Hyman Kestelman",
title="Automorphisms of the field of complex
journal="Proc. London Math. Soc.",

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