Re: Cardinality of equivalence classes and measure


Dave L. Renfro wrote:
eugene wrote:

Could you please explain how are you going to choose
c disjoint non-measurable sets.

I think there are several ways that make use of nonlinear
additive functions and/or nonmeasurable Hamel bases
(for the reals over the rationals), Bernstein sets, etc.
Celestyn Burstin may have been the first to do this
(he might have done it for "uncountable" instead of "c")
around 1915, but Lusin/Sierpinski published a paper in
1917 (see below; note that a .pdf file for the paper is
available) where they showed the unit interval [0,1]
can be decomposed into c many pairwise disjoint sets,
each of which has outer Lebesgue equal to 1. [Of course,
this gives c many nonmeasurable sets, since at most one
of these sets can be measurable.]

Some rather strong results along these lines can be
found in the Erdös/Marcus paper cited below.

Nikolai N. Lusin and Waclaw Sierpinski, "Sur une décomposition
d'un intervalle en une infinité non dénombrable d'ensembles
non mesurables", Comptes Rendus Académie des Sciences (Paris)
165 (1917), 422-424. [JFM 46.0294.01]
http://www.emis.de/cgi-bin/JFM-item?46.0294.01
ftp://ftp.bnf.fr/000/N0003118_PDF_422_424.pdf

Thank you very much for the reference. I'd like to figure out in this
paper, but i have some questions from this paper.

So, here how it goes as far as i understand it:

Let Omega_0 be the least ordinal which corresponds to the set of
cardinality c, such that all

ordinals < Omega_0 correspond to sets of cardinality <= c.

Then, since every set can be well-ordered and [0,1] has cardinality c
we can well-order

x_1, x_2, x_3, ..., x_omega,... , x_alpha , ... (alpha < Omega_0) (1)

all the reals from [0,1]. And (1) is of order type Omega_0.

Then since the card(perfect sets) = c in the same way we have
well-ordering of perfect sets in [0,1]

P_1, P_2, P_3, ..., P_omega,... , P_alpha , ... (alpha < Omega_0) (2)

Now choose p_1 from P_1, then p_2 from P_2 different from p_1 and so on
so that we get

p_1, p_2, p_3, ..., p_omega,... , p_alpha , ... (alpha < Omega_0) (3)


- this is also of order type Omega_0.

(We can do this since card(P) = c and card(p_1, p_2,..., p_alpha)) < c
for all alpha. )

Then consider some perfect set P in [0,1] and as it is claimed in the
paper,

* There are continuum many indices alpha in (2) such that P_alpha = P
).

This * is exactly the place where i got in troubles, could you please
explain why is * true ?

After understanding this * i understand what is said above in that
paper and one

can construct the set X_t which Butch Malahide mentioned and we would
be done.



And also i'd be very grateful if someone could explain it to me,
regarding that i'm far from

well familiar person with such that things.


Thanks


Paul Erdös and Solomon Marcus, "Sur la décomposition de
l'espace euclidien en ensembles homogènes", Acta Mathematica
Academiae Scientiarum Hungaricae 8 (1957), 443-452.
[MR 20 #1958; Zbl 79.07802]
http://www.emis.de:80/cgi-bin/zmen/ZMATH?type=html&an=0079.07802

Dave L. Renfro

.

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