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    Home/Original/inverse
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    Inverse View

    It is not the case that Countably additive Kolmogorovian probability distributions must violate regularity when the sample space is uncountable

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.Hyperreal-valued probability functions (as developed by Bernstein & Wattenberg 1969) assign infinitesimal positive probability to each point in uncountable spaces.
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    • 2.Regularity is preserved under hyperreal probability when countable additivity is replaced by the weaker condition of hyperfinite additivity.
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    • 3.Therefore the conflict between regularity and uncountable sample spaces is an artifact of restricting probability values to the reals, not a necessary feature of probability theory.
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    Reason for 2 of 2
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    • 1.Kolmogorov's axioms presuppose a sigma-algebra over the sample space, which need not include every singleton of an uncountable set.
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    • 2.Regularity is only violated when undefined singletons are illicitly treated as measurable events requiring positive probability assignment.
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    • 3.The claim conflates the measure-theoretic structure of a probability space with metaphysical claims about possibility, committing a category error identified by Williamson (2007).
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    Reasons Against

    1 perspective
    Reason against
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    • 1.A Kolmogorovian probability function is real-valued
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    • 2.Real-valued probability functions cannot assign positive probability to every point in an uncountable sample space while remaining countably additive and summing to 1
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