Re: Uncountable many reals without Cantor

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 11/30/04 

Date: Tue, 30 Nov 2004 10:10:14 -0600

On Tue, 30 Nov 2004 14:56:30 +0000 (UTC), Dave Seaman
<dseaman@no.such.host> wrote:

>On Tue, 30 Nov 2004 08:42:39 -0600, David C Ullrich wrote:
>> On Tue, 30 Nov 2004 12:57:04 +0000 (UTC), Dave Seaman
>><dseaman@no.such.host> wrote:
>
>>>On Tue, 30 Nov 2004 08:09:18 GMT, Jürgen R. wrote:
>>>> David C. Ullrich <ullrich@math.okstate.edu> wrote:
>>>
>>>>>This argument _is_ much more complicated, if you include
>>>>>the missing details. In particular you need a _proof_ of
>>>>>the intuitively reasonable fact that if [0,1] is contained
>>>>>in the union of countably many intervals I_n then
>>>>>
>>>>>(*) sum length(I_n) >= 1.
>>>>>
>>>>>How do you _prove_ that?
>>>
>>>> Assume the opposite, put the intervals end to end etc. This kind of
>>>> thing is proven in the beginning of any Real Variables text, e.g.
>>>> Royden. Where do you see a problem?
>>>
>>>Sounds like you want to use induction on the number of intervals.
>>>Problem is, induction works only if the number of intervals is finite.
>>>
>>>Hint: that's where compactness comes in.
>
>> Amusing technicality:
>
>> In various places for various reasons we need to talk about
>> coverings by half-open intervals instead of open intervals.
>> Say [0,1) is the union of disjoint intervals [a_n, b_n).
>> Then sum(b_n - a_n) = 1. You can't prove that (directly)
>> by compactness, but you _can_ prove it by a simple
>> transfinite induction!
>
>That's not quite the approach I had in mind. I'm sure you are aware of
>this argument, but I wonder why we "need" to talk about half-open
>intervals to show that m([0,1]) = 1. What I meant was to show the outer
>measure of [0,1] is >= 1 by considering open covers, extracting finite
>subcovers, and using ordinary induction over the number of intervals in
>the subcover. Then you can show the outer measure is less than 1+epsilon
>for any epsilon > 0.

I understood all that - I wasn't disputing anything you said,
just pointing out a _similar_ result where it's curious that
one actually can use transfinite induction but not compactness
(as opposed to the previous result where one can use compactness
but not induction.)

************************

David C. Ullrich

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