[FOM] Re: L-measurable sets

[Ali Enayat]

Yu Liang has asked whether it is possible to arrange a model M, and an
extension M' of M, both of ZFC, so that the following holds in M':

"Both the reals of M, and the reals in M'\M, form sets of positive Lebesgue
measure".

The answer is in the negative because:

(1) The reals of M must be closed under subtraction.

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

(3) Since the reals of M form a subgroup of the additive group of reals, we
conclude, by translation invariance of the Lebesgue measure, that M' thinks
that either the reals of M have measure 0, or the reals of M have full
measure.

To summarise: the answer is in the negative because M' thinks (A) the reals
of M form a group under +, and (B) every measurable subgroup of the additive
group of reals has either null, or full measure.


Ali Enayat