[FOM] Re: L-measurable sets

Ali Enayat enayat at american.edu
Sun Jul 20 22:01:03 EDT 2003

In an earlier posting, I answered a question of Yu Liang in the negative,
and asserted that the negative answer is a corollary of the fact that every
measurable subgroup of the additive of reals has either null, or full
measure.  Here I wish to make the "fact" more precise, and perspicuous.

Theorem (Surely Classical) Every measurable proper subgroup S of the the
additive group of
reals has measure 0.

Proof:  It suffices to show that if S is an additive subgroup of reals R,
and S is of
positive measure, then S = R.

(1) By a classical theorem of measure theory, if a set X of reals has
positive L-measure, then the difference set D(X) consisting of reals of the
form x-y, where x and y are in X, contains an interval of the form [0,s],
for s>0. Therefore, S contains an interval [0,s], for s>0.

(2) Since S is closed under addition and subtraction, S contains translation
of [0,s] by elements of the form ns, where n is an arbitrary integer.  By
the Archimedean property of R, such translations cover the real line, so the
proof is complete.

Corollary: If M is an inner model of a model M' of ZFC, then the
reals of M are either (A) of measure 0, or (B) not measurable.

Question: Clearly (A) in the corollary above can be arranged by forcing the
continuum of M to become countable in a generic extension M[G]=M'.  What
about (B)?

This certainly should be known to the cognoscenti; hopefully one will
enlighten us.

Ali Enayat

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Subject: FOM Digest, Vol 7, Issue 30

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>    1. a question about L-measurable set (Yu Liang)
> ----------------------------------------------------------------------
> Date: Sun, 20 Jul 2003 11:28:49 +1200
> From: Yu Liang <Yu.Liang at mcs.vuw.ac.nz>
> To: fom at cs.nyu.edu
> Subject: [FOM] a question about L-measurable set
> Message-ID: <200307201128.50051.Yu.Liang at mcs.vuw.ac.nz>
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> Here R denotes the real set in V.
> Is there a model of ZFC say M so that there is  a model of ZFC say M'
> M  so that M' \models u(R  \cap M)>0 and M' \models u(R  \cap (M'\M))>0?
> Is it possible  L=M?
> Thanks
> Liang Yu
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