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.
[Math] Finitely additive probability measure thats not countably subadditive
measure-theoryprobabilityprobability theory
Best Answer
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.