Measure Theory & the Continuum Hypothesis

[Math1723]

We know in Measure Theory that sets on R of countable cardinality
(finite or Aleph_0) have measure 0, but sets on R of uncountable
cardinality may have measures of 0, a finite positive number, or even
non-measurable. In the case of the Continuum Hypothesis, There are
only two cardinalities for which infinite subsets of R can have:
Aleph_0 or c, the size of the continuum.

Now assume a model of Set Theory for which CH is false, and thus there
are three or more infinite cardinalities for which subsets of R may
hold. Is it true that all such subsets of R of smaller cardinality
than c but be of measure 0? It seems obvious to me that any
measurable set of size less than c must be of measure 0. But can
there be non-measurable sets smaller than c?

[Dave L. Renfro]

Math1723 wrote (in part):

> Now assume a model of Set Theory for which CH is false,
> and thus there are three or more infinite cardinalities
> for which subsets of R may hold. Is it true that all such
> subsets of R of smaller cardinality than c but be of measure 0?

No, this does not have to be true, but it can be, depending
on the model.

There is a cardinal number called non(L) that is defined
to be the minimum cardinality possible for a set of real
numbers that does not have Lebesgue measure zero. (We don't
have to say "infinimum", because every set of cardinal
numbers has a minimal element -- recall that sets of cardinal
numbers are well-ordered.) Thus, every set of reals with
cardinality less than non(L) has measure 0. There are very
few a prior restrictions on what values non(L) can have, besides
being greater than aleph_0 and not greater than 2^(aleph_0),
but its relationship to many other so-called "cardinal
invariants of the real line" has been studied extensively.

http://en.wikipedia.org/wiki/Cicho%C5%84's_diagram

http://en.wikipedia.org/wiki/Set_theory_of_the_real_line

Dave L. Renfro

[Butch Malahide]

On Jan 27, 3:23 pm, "Dave L. Renfro" <renfr...@cmich.edu> wrote:
> Math1723 wrote (in part):
>
> > Now assume a model of Set Theory for which CH is false,
> > and thus there are three or more infinite cardinalities
> > for which subsets of R may hold. Is it true that all such
> > subsets of R of smaller cardinality than c but be of measure 0?
>
> No, this does not have to be true, but it can be, depending
> on the model.
>
> There is a cardinal number called non(L) that is defined
> to be the minimum cardinality possible for a set of real
> numbers that does not have Lebesgue measure zero. (We don't
> have to say "infinimum", because every set of cardinal
> numbers has a minimal element -- recall that sets of cardinal
> numbers are well-ordered.) Thus, every set of reals with
> cardinality less than non(L) has measure 0. There are very
> few a prior restrictions on what values non(L) can have, besides
> being greater than aleph_0 and not greater than 2^(aleph_0),
> but its relationship to many other so-called "cardinal
> invariants of the real line" has been studied extensively.

One a priori restriction (assuming AC) is that non(L) has uncountable
cofinality. Is that all?

[Dave L. Renfro]

Butch Malahide wrote:

> One a priori restriction (assuming AC) is that non(L) has
> uncountable cofinality. Is that all?

I have no idea, and I wasn't sure I could easily find
out, so I purposely worded my reply in a way that I
hoped would be safe (i.e. basically correct, if not
all that informative).

Dave L. Renfro