All examples of a dense and co-dense set I have seen are either of full Lebesgue measure or of measure zero. For instance, in restriction to the unit interval I = [ 0 , , 1 ], we could have respectively I ∩ Q or I ∖ Q . What I am looking for is a dense and co-dense subset A ⊂ I such that m ⁡ ( A ) = m ⁡ ( I ∖ A ) = 1 2 .I have attempted this task sequentially by, ever more finely, nibbling holes out of subintervals of I and partially back-filling the previously created holes. It's easy to approach half measure at each step, but I can't see how to to get convergence.

Question

All examples of a dense and co-dense set I have seen are either of full Lebesgue measure or of measure zero. For instance, in restriction to the unit interval       I    =  [  0              ,              ,      1  ], we could have respectively       I    ∩      Q   or       I    ∖      Q  . What I am looking for is a dense and co-dense subset   A  ⊂      I   such that  m  ⁡  (  A  )  =  m  ⁡  (      I    ∖  A  )  =            1      2        .I have attempted this task sequentially by, ever more finely, nibbling holes out of subintervals of       I   and partially back-filling the previously created holes. It&#039;s easy to approach half measure at each step, but I can&#039;t see how to to get convergence.

conveneau71 · Accepted Answer

You can take   A  =      [    0    ,          1      2        ]    ∪      (          [              1        2            ,      1      ]        ∩          Q        )

Ciara Mcdaniel · Accepted Answer

Let   C be a fat Cantor set with measure 1/2. Set   A  =  C  ∪      Q    ∩  [  0  ,  1  ] and you&#039;re done.