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Topic: Half Measure
Replies: 14   Last Post: Mar 25, 2013 5:03 PM

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David C. Ullrich

Posts: 21,553
Registered: 12/6/04
Re: Half Measure and Correction
Posted: Mar 24, 2013 11:53 AM
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On Sun, 24 Mar 2013 00:48:48 -0500, quasi <quasi@null.set> 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?

>> 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...


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