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Probabilities on Non-Standard Models.

  1. Oct 26, 2013 #1

    WWGD

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    Hi, I think I read here; maybe not, that , within a non-standard model of the Reals, it is possible to have probabilities , say over an interval, so that each point has non-zero probability.

    AFAIK, the transfer principle ( a.k.a elementary equivalence of models) does not disallow having a convergent uncountable sum ( tho a sum over an uncountable index has to be defined carefully). Anyone know about this and/or have a ref? Thanks,

    WWGD: What Would Gauss Do?
     
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  3. Oct 26, 2013 #2

    Erland

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    It depends on what you mean. For a standard probability measure on an nonstandard set, even on a nonstandard space (for example, a Loeb measure), it is clearly impossible that every point has nonzero measure.

    But for a nonstandard measure, it is possible in the following sense: One can find a nonstandard extension *R of R (the set of real numbers) such that there is a hyperfinite (i.e. *finite) subset S of *R which contains R (i.e. S contains all standard reals). This holds if the extension is an enlargement.
    This hyperfinite set S has a nonstandard cardinality H, which is a hyperfinite number. One can then define a nonstandard probability measure m on *R by stipulating that m({x})=1/H for all x ε S, which includes all x ε R.
     
  4. Nov 3, 2013 #3

    WWGD

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    Thanks, Erland, nice explanation.
     
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