All examples of a dense and co-dense set I have seen are either of full...

Wisniewool

Wisniewool

Answered

2022-07-03

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.

Answer & Explanation

conveneau71

conveneau71

Expert

2022-07-04Added 17 answers

You can take A = [ 0 , 1 2 ] ( [ 1 2 , 1 ] Q )
Ciara Mcdaniel

Ciara Mcdaniel

Expert

2022-07-05Added 4 answers

Let C be a fat Cantor set with measure 1/2. Set A = C Q [ 0 , 1 ] and you're done.

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