## Invariant Measure on (Q,+)

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• Can we say that there is no countably additive invariant measure on the additive group of rational numbers with the subspace topology from the real line?
Message 1 of 2 , May 1, 2009
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Can we say that there is no countably additive invariant measure on the additive group of rational numbers with the subspace topology from the real line?
S.Srinivas Rau
• Any subset of rational numbers is a countable union of points. As all points are congruent, you can have either m(p)=0, and then the whole measure is 0. Or you
Message 2 of 2 , May 1, 2009
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Any subset of rational numbers is a countable union of points. As all
points are congruent, you can have either m(p)=0, and then the whole
measure is 0. Or you can have a non-zero point mass a and then m(E)=a(#E),
i.e. measure is finite only for finite sets.

> Can we say that there is no countably additive invariant measure on the
> additive group of rational numbers with the subspace topology from the
> real line?
> S.Srinivas Rau
>
>
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