Example of measure for some algebra

Question

Can you give an example of a finitely additive measure given on $\mathcal A$, not having a countably additive extension to a $\sigma$-algebra generated by $\mathcal A$ for some algebra $\mathcal A$ over $\mathbb N$?

Answer 1

On $\mathbb N$ with the sigma algebra of all subsets let $\mu (A)=0$ if $A$ is a finite set and $\infty$ if it is an infinite set. Then $\mu (\mathbb N)=\infty$ and $\sum _n \mu(\{n\})=0$ so $\mu$ is not countably additive. It is trivial to check that $\mu$ is finitely additive.