Re: A set containing a nonempty open interval

On 26 Aug 2005 11:33:14 -0700, "Amanda" <sca18@xxxxxxxxxxx> wrote:

>Oh, there's a typo, of course i mean, if a set A has positive
>Lebesgue measure, then (A + A)/2 = {(x + y)/2 | x and y are in A}
>contains a non-empty open interval.

To expand a bit on one of RI's suggestions:

First show that you can assume that the measure of A is finite
(consider a subset with finite measure). Now let f be the
characteristic function of A and let g be the convolution
f*f (definition somewhere in the book). Now f is integrable,
and it follows that g is continuous (this may be a theorem
or exercise in the book, if not it follows from something
that's sure to be in the book, although it may not be
obvious which fact in the book does it - it's not just
dominated convergence, for example). And something in
the book shows that the integral of g is m(A)^2 (it
shouldn't be too hard to see what theorem in the book
is used here).

So g is continuous and not identically zero, hence
g is non-zero on some open interval, and the result
you want follows, because ___.


************************

David C. Ullrich
.

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