Re: Cardinality of equivalence classes and measure

eugene wrote:
Butch Malahide wrote:
eugene wrote:
Let's say two sets E_1 and E_2 in R are equivalent iff m( (E_1-E_2) V
(E_2-E_1)) = 0.
a). Prove that the cardinality of set of equivalence classes of
measurable sets is continuum.
b). What is the cardinality of set of equivalence classes of all sets ?

I know how to do the first question, it is clear. But i have no idea
about the second one, i think that it is should probably be 2^c but
still have no idea how to prove it, or it might even be wrong.

Construct a family of c pairwise disjoint nonmeasurable sets, and then
consider unions of subfamilies of that family; there are 2^c such
unions, no two equivalent.

Could you please explain why no two of them are equivalent.

Not the question I was expecting. I thought you were going to ask me
how to construct c disjoint nonmeasurable sets.

To answer the question you asked: they are not equivalent because their
symmetric difference contains a nonmeasurable set, and so is not a set
of measure zero. (We *are* talking about Lebesgue measure here, right?)

.

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