Re: A set containing a nonempty open interval

"Amanda" <sca18@xxxxxxxxxxx> writes in article <1125067955.450481.88600@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> dated 26 Aug 2005 07:52:35 -0700:
>
>I'd like some hints on how to prove that, if a set S has positive
>Lebesgue measure, then (A + A)/2 = {(x + y)/2 | x and y are in A}
>contains a non-empty open interval.

Unless you specify x <> y, you might get all closed intervals.

For example,
A = [0,1] U [10,11]
gives
(A+A)/2 = [0,1] U [5,6] U [10,11]
which contains no open intervals.

But if you stipulate x<>y you get
(A+A)/2 = (0,1) U [5,6] U (10,11)
which does contain open intervals.

U is the union operator.

--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
.

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