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Topic: Non standard probability theory
Replies: 12   Last Post: Dec 19, 2013 8:03 PM

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Posts: 1,845
Registered: 11/29/07
Re: Non standard probability theory
Posted: Dec 11, 2012 3:32 PM
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On Nov 11, 6:36 pm, kir <> wrote:
> On Sunday, November 11, 2012 4:18:15 PM UTC-5, Ray Vickson wrote:
> > On Sunday, November 11, 2012 7:38:09 AM UTC-8, kir wrote:
> > > What kind of work is being done inprobabilitytheory beyond the standard normally used? I know that negative probabilities have been discussed. Is there anything else?
> > I'm not sure this is what you want, but there has been work inProbabilitytheory using "non-standardanalysis", so one can talk about actual infinitesimals, etc. Perhaps the most accessible introduction to this is the old but still good little book by Edward Nelson ("Radically ElementaryProbability", Princeton U. Press). You can download a free pdf version from Edward Nelson's home page: just go to his  publications list and click on the book title.
> > RGV
> Primarily dealing withnonstandardanalysis.

Robin Pemantle's "Probability and Hyperreals"
Bernstein and Wattenberg (1969). In an early paper, B & W construct a
measure that assigns a "probability" in *[0,1) to every subset of real
[0,1), even those that are not Lebesgue-measurable. (E.g., Vitali
sets get infinitesimal probability.)

The reference is to
Bernstein, Allen R. and Wattenberg, Frank. (1969) "Nonstandard
measure theory." In W.A.J. Luxemburg, ed., Applications of Model
Theory to Algebra, Analysis, and Probability (NY: Holt, Rinehart and
Winston), pp. 171-186.

Further details on the Bernstein and Wattenberg "measure" can be found

Brian Skyrms "Zeno's Paradox of Measure" In R.S. Cohen, L. Laudan,
eds., Physics, Philosophy and Psychoanalysis: Essays in Honor of Adolf
Grünbaum, pp. 223-254. Boston Studies in the Philosophy of Science,
76. Dordrecht, Holland; Boston; Lancaster: Reidel, 1983.

<quote from page 242>
They show that one can construct a measure defined for all subsets of
the unit interval, which takes its values in a non-standard of the
reals, which is finitely additive, translation invariant up to an
infinitesimal which is infinitesimally close to Lebesgue measure on
the Lebesgue measurable sets, and which is regular (i.e. only yhe null
set gets measure zero). The Vitali sets of the last section, and the
sets containing one point will then both have infinitesimal measure.

See also Nigel Cutland's "Non-Standard Analysis and Its Applications"
wherein H Jerome Keisler authors a section "Infinitesimals
in Probability Theory"

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