Re: Size of equivalence class of Cauchy sequences
- From: "Dave L. Renfro" <renfr1dl@xxxxxxxxx>
- Date: 11 Jan 2007 13:26:35 -0800
Michael Press wrote:
Given a rational number, r, how many Cauchy sequences
converge to r?
This reminds me of a question I thought of back in
1984 or 1985: What is the cardinality of the Lebesgue
spaces L^p? The thing is, I knew there were 2^c
p-integrable *functions*, but how many almost-everywhere
*equivalence classes* of functions do we have? [The usual
interpretation is L^p as a metric space whose elements
are these equivalence classes, although you can also
stick with the functions themselves and view L^p as
a pseudo-metric space.] Now if there were fewer than
2^c many functions in each of the equivalence classes,
then it would follow that the cardinality of L^p was 2^c.
However, each equivalence class has 2^c functions,
since you can add the characteristic function of any
subset of the Cantor middle thirds set to a function
and not leave the equivalence class it belongs to.
Thus, since there are 2^c functions and the equivalence
classes each have cardinality 2^c, it's not automatic
how many equivalence classes there will be. The cardinality
of the collection of equivalence classes of functions
(modulo almost everywhere equality) could, a prior, be
any cardinal number b <= 2^c, since b*2^c = 2^c for each
cardinal number b such that b <= 2^c.
After thinking about it some, I realized that since the
L^p spaces are separable, the cardinality can't be more
than (aleph_0)^(alpeh_0) = c, and of course it was at
least c (distinct constant functions obviously belong
to different almost-everywhere equal equivalence classes),
so the cardinality was exactly c.
I didn't think this was all that well known (I asked
a few professors where I was at and no one off-hand
seemed to know the answer), so I sent it in to be an
American Mathematical Monthly problem. It was rejected
as being perhaps too easy (which I agree, especially
now), and I think the letter I got outlined two additional
methods that were different from the one I submitted.
I think I later came up with still another method
a year or two later, but I don't remember what any
of the other ways are right now (besides separability).
Hummm...Just before sending this I realized L^oo isn't
covered by the separability argument. I'm sure L^oo has
cardinality c, and probably using one of the other ways
of showing L^p for 1 < p < oo has cardinality c will
do it, but I can't think of anything now.
Dave L. Renfro
.
- References:
- Size of equivalence class of Cauchy sequences
- From: Michael Press
- Size of equivalence class of Cauchy sequences
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