Consider $R$, with the lebesgue measure defined on its borel set. If $E$ is a real subset of $R$ and is measurable and is not open then does that mean that it has measure zero? If not please provide a counterexample.
2026-04-04 03:41:46.1775274106
Real measurable not open subsets of R have measure zero?
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$\lambda([0,1])=1$ and $[0,1]$ is not open... The irrationals are neither open nr closed and have $+\infty$ Lebesgue measure. If $E$ is Borel and contains a non-empty open set (so its interior is non-empty) we do have $\lambda(E) > 0$, but the reverse is very far from true.