If two sets $A$ and $B$ defined on bounded intervals have the same cardinality and $ A \bigcap B $ is non empty and the Lebesgue outer measure of A is greater than zero. Is it then true that the Lebesgue outer measure of B is also greater than zero.
2026-05-05 15:45:43.1777995943
Interesting relationship between cardinality and Lebesgue outer measure
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The Cantor set has measure $0$ and has the same cardinality as $[0,1]$, which has measure $1$.