Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 01/25/05 

Date: 24 Jan 2005 20:28:45 -0800

Perhaps this is mistaken: in measure theory the only concerns of
transfinite cardinality are countability vis-a-vis the cardinality of
the continuum. Probability's use of transfinite cardinality is based
upon measure theory.

This leads some to that you can select a random real number from a
uniform or pseudouniform distribution over an interval of the reals,
but not an integer. That's because of issues dealing with "zero
probability", or infinitesimal probability.

One consideration of how to sample a real number from the unit interval
is to flip coins, at each step narrowing the resultant interval towards
a binary expansion of the real number.

A consideration with that is that each sample from {0,1} is as well the
beginning of the sample of another real number, and that any other
permutation of the sequence is another sample. Then, if there's an
iterative process, a consideration is whether an infinite number of
ones and zeros is reached at the same time, where the average value is
1/2 as the average of 1/3 and 2/3, or an infinite number of zeros is
reached before infinitely many ones, leading to an average value of
zero and many rationals, or an infinite number of ones and finitely
many zeros, with an average value of one and many rationals. The
integral of a function symmetric about the origin is zero.

That is with the notion that if you have any infinite sequence of
zeros, however you reorder the sequence it represents zero, if you have
a sequence with infinitely many ones and zero you can generate most
irrationals, and half of the possible sequences.

That consideration of sampling real numbers is digression from the
point about measure theory and probability: that the utility is
primarily about the cardinality of the continuum and continua, and thus
the basically geometric nature of the continuum instead of its
cardinality, and that it could be explained that way.

So, aside from that digression, if I want to support measure
theoretical results or provide alternate mechanisms for correct results
using my little theory where infinite sets are equivalent, then it
would lead to some retrofitted underpinnings of measure theory as
necessary, so I wonder: is there any use in meaure of transfinite
cardinality besides the cardinality of the continuum? Second: does
probability theory use transfinite cardinals besides using measure
theory?

Consider "complex measure theory".

Regards,

Ross Finlayson

Relevant Pages

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