Probabilities on Non-Standard Models.

[WWGD]

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?

 

[Erland]

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.

 

[WWGD]

Thanks, Erland, nice explanation.