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What properties can be said for a subset of the reals which has a Lebesgue measure strictly greater than zero?

I tried googling but there weren't that many and if there was they were poorly explained

dahaka5
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1 Answers1

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If the measure of $E$ is greater than $0$ then $E$ has the cardinality of $\mathbb{R}$. This is because the set $E+E$ contains an interval and $|E+E| = |E|$. This answer uses the axiom of choice.

Jay
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  • An easier way of showing this is to assume for the sake of contradiction that $E$ is countable, and prove from there that $E$ has measure 0. – B. Mehta Nov 29 '17 at 15:59
  • Suppose that $|\mathbb{R}| > { \aleph}{1}$ and $|E| = {\aleph}{1}$? – Jay Nov 29 '17 at 16:07
  • Fair point. Instead my comment shows $E$ is uncountable while your answer proves a stronger statement. – B. Mehta Nov 29 '17 at 16:21