Boundary of a Lebesgue measurable set E with $\lambda^n(E)≠0$ has Lebesgue Measure zero?

Question

I could not think of any counter examples, so I am wondering whether this statement would be true, and ideas on how one could possible prove/disprove the statement?

The answer here may also be of interest: http://math.stackexchange.com/questions/1421934/set-e-subset-mathbbrn-of-positive-lebesgue-measure-such-that-the-lebesgue — , Dec 04 '16 at 14:22

score 1 · Answer 1 · edited Apr 13 '17 at 12:21

1

See this question and its answers. The gist is that such examples exist in $\mathbb{R}^2$ as the interiors of space-filling Jordan curves.

edited Apr 13 '17 at 12:21

Community

1

answered Dec 04 '16 at 14:19

Neal

32,659

score 0 · Answer 2 · edited Oct 31 '21 at 01:31

0

Well okay that was A stupid question I just realized. It is enough to look at $\mathbb R$ \ $\mathbb Q$ as a counterexample in $\mathbb R$, which obviously has infinite Lebesgue measure but is composed solely of boundary points.

edited Oct 31 '21 at 01:31

KCd

46,062

answered Dec 04 '16 at 14:24

El Duderino

335

Boundary of a Lebesgue measurable set E with $\lambda^n(E)≠0$ has Lebesgue Measure zero?

2 Answers2