Re: Non-zero gaps between real numbers

On Mon, 03 Dec 2007 17:25:30 -0600, David C Ullrich wrote:
On Sun, 2 Dec 2007 22:00:22 -0800 (PST), Venkat Reddy
<vreddyp@xxxxxxxxx> wrote:

On Dec 3, 6:18 am, William Hughes <wpihug...@xxxxxxxxxxx> wrote:
On Dec 2, 7:51 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:

Proove that the sum of measures of all real numbers in [0, 1] is equal
to 1.

The sum of all measures of all the real numbers in [0,1] is zero.
The measure of the set of real numbers in [0,1] is one.
The measure of the set of real numbers in [0,1] is not the sum
of all measures of the real numbers in [0,1].

That helps.

I guess it follows from the definition for measure? If it is not sum,
then what is it?

The definition of the measure of the interval [a,b] is b-a.

No, that's not the definition. The definition involves first defining
outer measure and then demonstrating (using the Heine-Borel theorem) that
the outer measure of an interval is its length. Finally, one must shaw
that intervals are indeed measurable and therefore that the outer measure
is the measure.

Bottom line: it's a *theorem* that the measure of an interval [a,b] is
b-a. And nowhere in the proof of the theorem or in the definition of
outer measure is there anything about adding up the measures of an
uncountable number of points.


--
Dave Seaman
Oral Arguments in Mumia Abu-Jamal Case heard May 17
U.S. Court of Appeals, Third Circuit
<http://www.abu-jamal-news.com/>
.

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