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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

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?

>>
>>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

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 ?

add some more Cantor sets, one in each interval
the first set misses. Repeat until done...

>
>quasi

Date Subject Author
3/23/13 William Elliot
3/23/13 Mike Terry
3/23/13 William Elliot
3/24/13 quasi
3/24/13 Butch Malahide
3/24/13 quasi
3/24/13 quasi
3/24/13 David C. Ullrich
3/24/13 Butch Malahide
3/24/13 David C. Ullrich
3/24/13 Butch Malahide
3/25/13 David C. Ullrich
3/25/13 JT
3/25/13 JT
3/25/13 JT