3

A subset $S$ of $\Bbb R^n$ is Jordan measurable if and only if the measure of the set of its boundary points is zero. Basically I tried to imagine a set which has boundary points of nonzero measure, but I couldn't. Can you give me some examples? In my mind every boundary set of points of any set has zero measure.

J.-E. Pin
  • 40,163
dato nefaridze
  • 321
  • 5
    Does this answer your question? Domain whose boundry has non zero volume. – Isaac Ren May 08 '20 at 17:47
  • 1
    i thinks no because volume and measure are different. – dato nefaridze May 08 '20 at 17:49
  • @datonefaridze in fact Jordan measure (whenever it's defined) can indeed be thought of as "volume" – Ben Grossmann May 08 '20 at 17:50
  • @datonefaridze It's not quite correct to say that the boundary of a non-measureable set has non-zero measure. In particular, it is possible that the boundary itself fails to be Jordan measurable. This is what happens in the case of $S = \Bbb Q \cap [0,1]$, for instance. Note that $S$ is equal to its own boundary. – Ben Grossmann May 08 '20 at 17:51
  • 1
    what is the volume?how do you define volume? definition of volume has many interpretation, for example lebesgue measure, 'measures' volume differently, jordan measure measures volume also differently.so saying it is volume incorrect, you have to specify according to which definition. – dato nefaridze May 08 '20 at 17:53
  • @datonefaridze If you're specifically looking for a set whose boundary is Jordan measurable with non-zero measure, then that's a much trickier question than asking for a set that fails to be Jordan measurable. You should clarify exactly what it is you're asking for. – Ben Grossmann May 08 '20 at 17:54
  • @datonefaridze Rejecting an answer as irrelevant for the sole reason that it uses the term "volume" instead of "measure" is awfully pedantic. That said, it is notable that the answer in the linked post gives a set whose boundary is not Jordan measurable which, depending on the question that you're actually trying to ask, might not answer your question. – Ben Grossmann May 08 '20 at 17:58

1 Answers1

6

The set points with rational coordinates.

cha21
  • 912