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Let $A\subset [0,1]$ be countable. Is there a set $B\subset [0,1]$ such that $A\subset B$ and $B$ is uncountable with Lebesgue measure $0$ ?

Thank you.

1 Answers 1

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What about $B=C\cup A$, where $C$ is the Cantor set?

Recall that:

  • $C$ is uncountable
  • Lebesgue measure of $C$ is 0.
  • Lebesgue measure of $A$ is 0, since it is countable.
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    Martin, thank you very much. The quote "A stitch in time saves nine" resonates in me as "Asking questions on Math.SE saves time."2011-10-30