[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
----- Original Message -----
From: <fom-request at cs.nyu.edu>
To: <fom at cs.nyu.edu>
Sent: Sunday, July 20, 2003 12:00 PM
Subject: FOM Digest, Vol 7, Issue 30
> Send FOM mailing list submissions to
> fom at cs.nyu.edu
>
> To subscribe or unsubscribe via the World Wide Web, visit
> http://www.cs.nyu.edu/mailman/listinfo/fom
> or, via email, send a message with subject or body 'help' to
> fom-request at cs.nyu.edu
>
> You can reach the person managing the list at
> fom-owner at cs.nyu.edu
>
> When replying, please edit your Subject line so it is more specific
> than "Re: Contents of FOM digest..."
>
>
> Today's Topics:
>
> 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>
> Content-Type: text/plain;
> charset="us-ascii"
> MIME-Version: 1.0
> Content-Transfer-Encoding: 7bit
> Precedence: list
> Message: 1
>
> 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'
\supset
> 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
> ------------------------------
>
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>
>
> End of FOM Digest, Vol 7, Issue 30
> **********************************
More information about the FOM
mailing list