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    If each point receives a positive infinitesimal probabili... — Carmelics
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    Challenges→Continuous probability distributions require a restriction to countable additivity rather than full additivity

    If each point receives a positive infinitesimal probability and hyperfinite summation is employed, full additivity is recoverable within a well-defined mathematical framework endorsed by Bernstein and Wattenberg (1969).

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    Key Terms

    Bernstein and Wattenberg (1969)(the scholars whose mathematical framework is being referenced)
    Two mathematicians (Allen Bernstein and William Wattenberg) who published influential work on nonstandard analysis, a branch of math dealing with infinitesimals.
    Hyperfinite summation(the method used to combine infinitesimal probabilities)
    A mathematical technique (from nonstandard analysis) that allows you to add up an impossibly large or infinite number of tiny quantities in a controlled way.
    additivity(Kolmogorov Axiom; breaks down at subatomic/quantum level)
    A property shared between probability measures and the concept of information, which holds at the macroscopic level but loses validity at the subatomic level in favor of a more subtle notion
    infinitesimal(Nonstandard analysis)
    A hyperreal a whose absolute value |a| is less than 1/(n+1) for every natural number n

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    nonstandard analysis(Built on the expressive strength (can formalize calculus) and expressive weakness (cannot rule out infinitesimals) of first-order logic)
    Abraham Robinson's framework that uses the compactness theorem of first-order logic to construct a model R' elementarily equivalent to the real numbers R but containing infinitesimals, thereby giving rigorous foundations to Leibniz-style infinitesimal reasoning
    probability(as used in mathematics and logic)
    A number between 0 and 1 that describes how likely something is to happen; 0.5 means 50% chance, 1 means certain, 0 means impossible.

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    Continuous probability distributions require a restriction to countable additivi...

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