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How is it that a finitely additive probability measure on a field may not be countably subadditive? I know that the field must be countably additive and thus finite additivity does not suffice, but I'm struggling with the reasoning.

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    What do you mean by "the field must be countably additive"?2012-09-27

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Revised: Let $\mathscr{F}=\{A\subseteq\Bbb N:A\text{ is finite or }\Bbb N\setminus A\text{ is finite}\}$, and define $\mu(A)=0$ if $A$ is finite and $\mu(A)=1$ if $\Bbb N\setminus A$ is finite.

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    Script(F) isn't a sigma algebra. A(n) = {2n} is a member of Script(F) for all integers n, but the (denumerable) union of all the A(n) isn't. The stated problem was solved by Banach using the Axiom of Choice.2018-05-30
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Examples of measures which cannot be extended to be countably summable are: 1) a uniform distribution on the rational numbers in the interval $[0,1]$. This is easy to see: There are only countably many rationals, so any finite set of rationals will have measure zero (or the total probability will be infinite), yet all of them together have measure 1. This is inconsistent with countable additivity.

2) a uniform distribution on the integers, by the same argument, sinvce any finite interval of inyegers must have probability zero.

More interesting: On some set of increasing sequences of natural numbers, define the density to be the limit (if it does not exist, take the limsup) of $ \#\{n_k \colon n_k\le N\} / N $ when $N$ goes to infinity. It can be shown that density defined in this way is onlyn a finitely additive measure, cannot be extended to countably additive.

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    @kjetilbhalvorsen Measurability is not a big problem if you are willing to extend a finitely additive measure using the Hahn-Banach theorem. But it prevents you from explicitely constructing a canonical finitely additive measure on a $\sigma$-algebra that is not countably additive.2012-09-27