On Sun, 24 Mar 2013 00:48:48 -0500, quasi <firstname.lastname@example.org> wrote:
>quasi wrote: >>William Elliot wrote: >> >>>Question: Is there a dense D, subset of R with >>>empty interior and measure D /\ [0,1] = 1/2? >> >>Yes. >> >>Let >> >> A = (0,1/2)\Q >> >> B = Q\A > >I meant: > > A = [0,1/2]\Q > > B = Q\[0,1/2] > >> D = A U B
Of course this answers the question as asked. But it's sort of a cheat. If we're doing measure theory then we want to ignore sets of measure zero...
Hmm. One might put it this way. _If_ we're talking about measurable sets _modulo_ null sets then Q doesn't count as dense because Q is the same as the empty set; similarly your D doesn't really count as having empty interior since A is the same as [0,1/2].
So the more interesting version of the question, in any case less trivial, amounts to this: Is there a measurable set D such that
0 < m(D intersect I) < m(I)
for every open interval I, and such that m(D intersect [0,1]) = 1/2 ?
The answer is yes, by the way. Think about "fat Cantor sets". Start with a Cantor set, then add some more Cantor sets, one in each interval the first set misses. Repeat until done...