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fom
Posts:
1,037
Registered:
12/4/12
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Re: How WM is cheating - fat Cantor set measure
Posted:
Jan 3, 2013 4:52 AM
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On 12/31/2012 10:16 AM, Ross A. Finlayson wrote:
> On Dec 30, 12:17 am, fom <fomJ...@nyms.net> wrote:
>> ...
>> Unlike many other measures, Lebesgue
>> measure has an invariance property
>> that permits its product measures
>> to be defined without the general
>> theory of product measures. To
>> see why, consider the binary
>> expansions on the interval
>>
>> 0<=y<1
>>
>> taking the eventually constant
>> sequences ending in constant 0
>> as the representation for rational
>> numbers. ...
>
> Not all rationals as binary expansions end with zeros, only multiples
> of inverse powers of two, for any finite string of zeros and ones
> there are expansions of rationals that end with those repeating.
I probably need to rethink this, then. The basic point
remains, however, if the ones I overlooked comprise a set
of measure zero.
The point is that ancient Greeks used to think of numbers
differently with respect to lines, squares, and cubes. Although
it was Vieta's work in algebra that started to break these
distinctions, the idea here is that the invariance of the Lebesgue
measure actually implements the "unity" of number.
WM is cheating since this invariance with respect to Lebesgue
measure depends in the completed infinity of the sigma algebra
of Borel sets.
His "number" is not the same as what everyone else intends.
>
> "Thus, the claim that the non-zero
> measure of fat Cantor sets arises
> from relation with an atomic
> measurable space whose atoms are
> not singletons has been verified. "
>
> Having measurable, countable, atoms, comprising units of measure:
> what would be their character as sets of reals?
Nice question. I do not think that the atoms are measurable.
Rather, at each iteration of the fat Cantor set intermediaries
should be viewed as having the measure for the interval distributed
among the compact intervals. This would then be a content on the
intervals in the sense of content from which a Borel measure might
be obtained.
One might imagine this content as the area under a normal
distribution vanishing outside of the intervals.
In the limit, however, the measure on the interval would have to be
taken as non-measurable in the components (singletons) since the
additivity conditions for content functions would be violated.
Actually, one might not even think of them as singletons. Recently,
I have been thinking quite a bit about double-origin topologies.
You can find mention of one
http://en.wikipedia.org/wiki/Connected_space
"An example of a space which is path-connected but not arc-connected is
provided by adding a second copy 0' of 0 to the nonnegative real numbers
[0, ?). One endows this set with a partial order by specifying that 0'<a
for any positive number a, but leaving 0 and 0' incomparable. One then
endows this set with the order topology, that is one takes the open
intervals (a, b) = {x | a < x < b} and the half-open intervals [0, a) =
{x | 0 ? x < a}, [0', a) = {x | 0' ? x < a} as a base for the topology.
The resulting space is a T1 space but not a Hausdorff space. Clearly 0
and 0' can be connected by a path but not by an arc in this space."
So, it may be reasonable to think of the non-measurable content as
a double-origin segment.
That there is non-measurable content associated with the atoms at all,
however, would seem to reside with the fact that the points have
zero measure relative to the product only if the sequences of zeros
and ones are infinite. The invariance of the Lebesgue measure
depends on double-indexed infinity. Once for the underlying
point set and once for the Borel sigma algebra on that point
set.
After writing this, I have started to ask myself if
there might not be a measure for the interval that would
distribute across lengths near the Planck length that
would have the Planck energy. Maybe one day...
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