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Invariant Measure on (Q,+)

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  • rauindia
    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
    • Maria Roginskaya
      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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