measurability assuming failure of the Continuum Hypothesis
quasi
If an uncountable set has cardinality less than that of R
and is measurable. must it have measure 0?
quasi
FredJeffries
I believe this was asked recently:
http://groups.google.com/group/sci.math/msg/c09ca0e41328fc48
smn
If the uncountable set is Borel measurable and uncountable there is a
bijection with R which is Borel measurable (inverse images of Borel
sets are Borel sets) with Borel measurable inverse.There is a Proof in
Dudley,Richard -Real Analysis and Probability (Cambridge Press) PS
Also the only cardinalities possible for a Borel subset of R are
countable and the cardinality c of R.This theorem does not use the
continuum hypothesis or its negation.Regards,smn
Butch Malahide
>
> If an uncountable set has cardinality less than that of R
> and is measurable. must it have measure 0?
I think so. I seem to recall (from a course in measure theory that I
took 50 years ago) a theorem to the effect that, if A is a Lebesgue
measurable set of real numbers, and if A has positive measure, then
the difference set D(A) = {x - y: x, y in A} contains an interval. If
A had cardinality less than the continuum, then D(A) couldn't contain
an interval, at least if the axiom of choice is assumed.
quasi
<fredje...@gmail.com> wrote:
>On Feb 12, 11:19 am, quasi <qu...@null.set> wrote:
>> Question:
>>
>> If an uncountable set has cardinality less than that of R
>> and is measurable, must it have measure 0?
>
>I believe this was asked recently:
>http://groups.google.com/group/sci.math/msg/c09ca0e41328fc48
Ah -- I must have missed it.
Apparently I asked the exact same question!
The question relates naturally to some questions I asked in
another thread.
Thanks for the link.
quasi
David C. Ullrich
<fred....@gmail.com> wrote:
>On Feb 12, 1:19 pm, quasi <qu...@null.set> wrote:
>>
>> If an uncountable set has cardinality less than that of R
>> and is measurable. must it have measure 0?
>
>I think so. I seem to recall (from a course in measure theory that I
>took 50 years ago) a theorem to the effect that, if A is a Lebesgue
>measurable set of real numbers, and if A has positive measure, then
>the difference set D(A) = {x - y: x, y in A} contains an interval.
You recall correctly. (Easy proof: Assume that the measure of A
is finite. Let f be the characteristic function of A. Consider the
convolution f*f; note that f*f is continuous and has positive
integral, hence must be positive on some interval).
>If
>A had cardinality less than the continuum, then D(A) couldn't contain
>an interval, at least if the axiom of choice is assumed.
Very clever.
Butch Malahide
> On Sat, 12 Feb 2011 12:48:09 -0800 (PST), FredJeffries
>
> >On Feb 12, 11:19 am, quasi <qu...@null.set> wrote:
> >> Question:
>
> >> If an uncountable set has cardinality less than that of R
> >> and is measurable, must it have measure 0?
>
> >I believe this was asked recently:
> >http://groups.google.com/group/sci.math/msg/c09ca0e41328fc48
>
> Ah -- I must have missed it.
>
> Apparently I asked the exact same question!
Looks like two different questions to me.
Must a measurable set of cardinality less than c have measure zero?
Yes.
Must a set of cardinality less than c have measure zero? Not
necessarily; there are models of ZFC in which there are nonmeasurable
sets of cardinality less than c.
quasi
Ah -- I see that I read the linked thread too quickly.
Thanks for the clarification.
quasi
Dave L. Renfro
>> If an uncountable set has cardinality less than that of R
>> and is measurable. must it have measure 0?
Butch Malahide wrote:
> I think so. I seem to recall (from a course in measure theory that I
> took 50 years ago) a theorem to the effect that, if A is a Lebesgue
> measurable set of real numbers, and if A has positive measure, then
> the difference set D(A) = {x - y: x, y in A} contains an interval. If
> A had cardinality less than the continuum, then D(A) couldn't contain
> an interval, at least if the axiom of choice is assumed.
Unless I'm missing something, doesn't "must have measure 0"
follow immediately from the more elementary result that every
measurable set of positive measure contains a closed set of
positive measure (and the fact that uncountable closed sets
have cardinality c)? [I'm in the middle of something at work,
so my concentration isn't great right now, so I might be
overlooking something ...]
Dave L. Renfro
David C. Ullrich
<renf...@cmich.edu> wrote:
>quasi wrote:
>
>>> If an uncountable set has cardinality less than that of R
>>> and is measurable. must it have measure 0?
>
>Butch Malahide wrote:
>
>> I think so. I seem to recall (from a course in measure theory that I
>> took 50 years ago) a theorem to the effect that, if A is a Lebesgue
>> measurable set of real numbers, and if A has positive measure, then
>> the difference set D(A) = {x - y: x, y in A} contains an interval. If
>> A had cardinality less than the continuum, then D(A) couldn't contain
>> an interval, at least if the axiom of choice is assumed.
>
>Unless I'm missing something, doesn't "must have measure 0"
>follow immediately from the more elementary result that every
>measurable set of positive measure contains a closed set of
>positive measure (and the fact that uncountable closed sets
>have cardinality c)?
It certainly follows from those facts - it's not at all clear to
me that this is "more elementary".
Of course "elementary" is not well defined...