Re: Measurable set in R and mean values

On Nov 1, 6:59 pm, jane <jane1...@xxxxxxxxxx> wrote:
Let E be a measurable set in R. Must there exist points x,y in E such that (x+y)/2 is also in E ?

Thanks.

No, the empty set is a counterexample. It is the only counterexample:
if E is any nonempty subset of R, then there exist points x,y in E
such that (x+y)/2 is also in E.

Perhaps you meant to ask a different question? E.g., if E is a set of
*positive* Lebesgue measure in R, must there exist *distinct* points
x,y in E wuch that (x+y)/2 is in E? Seems likely to me.

.