Re: meger and null sets in R

From: Artur (artur_at_opendf.com.br)
Date: 07/13/04 

Date: 13 Jul 2004 06:57:33 -0700

David C. Ullrich <ullrich@math.okstate.edu> wrote in message news:<46b6f0lj2rhfov0g1891944tft2ndc71nc@4ax.com>...
> On 12 Jul 2004 16:18:19 -0700, artur@opendf.com.br (Artur) wrote:
>
> >Hi,
> >I'd like some hints to prove the following statement:
> >
> >If a subset of R has measure zero, then it is meager,
>
> Well, it seemed like this couldn't be so hard to prove; I
> felt really stupid being unable to prove it. So I looked
> it up, and then I felt _really_ stupid, not even having
> considered the possibility that it was false.
>
> The first thing I saw was an exercise saying that there
> exist residual subsets of [0,1] which have measure zero.
> (A residual set is the complement of a meager set, and
> so the Baire categormy theorem shows that in particular
> a residual set is not meager...)

Thank you all for your help. Sorry, I realy thought the statement was
true, because it was a guy at sci.math that made it in one of his
posts. And he seems to know a lot about Math.
Artur
>
> The exercise is easy, after the solution to the
> second question:
>
> >but the converse is not true.
>
> In fact there exists a closed nowhere-dense set
> of positive measure. The canonical example would
> be a "fat Cantor set": a set of positive measure
> homoemorphic to the middle-thirds Cantor set.
> (Modify the construction of the middle-thirds set,
> removing the middle r_n-th of each complementary
> interval at the n-th stage (so if r_n = 1/3 for
> all n you get the middle-thirds set). If you
> haven't seen this it's a good exercise to show
> that if you let r_n -> 0 fast enough you get a
> Cantor set of positive measure.
>
> So the converse is not true. You can use these
> Cantor sets to show that the first statement
> is not true either:
>
> Say K is a fat Cantor set in [0,1], of measure
> 1/2. Let K_0 = K, and let K_{n+1} consist of
> K_n plus copies of K, one for each interval
> in the complement of K_n, scaled and translated
> to fit the interval. The union of the first n
> K_j has measure 1 - 2^(-n) or something like
> that, so the union of all the K_n is a meager
> set of full measure. Hence its complement
> is a residual, hence non-meager, set of
> measure zero.
>
> It's curious that an elaboration of the construction
> used to disprove a statement serves to disprove the
> converse.
>
> >It's easy to see that null sets must have an empty interior, but I
> >couldn't prove they have to be meager. I know meager sets may have
> >positive measure, but I couldn't find an example.
> >Thank you.
> >Artur
>
>
> ************************
>
> David C. Ullrich

Relevant Pages
Quantcast