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fom
Posts:
1,969
Registered:
12/4/12


Re: How WM is cheating  fat Cantor set measure
Posted:
Jan 3, 2013 4:52 AM


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 nonzero > 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 nonmeasurable 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 doubleorigin topologies. You can find mention of one
http://en.wikipedia.org/wiki/Connected_space
"An example of a space which is pathconnected but not arcconnected 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 halfopen 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 nonmeasurable content as a doubleorigin segment.
That there is nonmeasurable 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 doubleindexed 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...



