Finitely additive probability measure thats not countably subadditive 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. 
 A: 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.
A: 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.
