I know if $u$ is set function on algebra $A$ then $u$ is a measure iff $u$ is finitely additive and countably subadditive. But will the statement hold if $A$ is any collection of subsets of a set $X$?
I feel certainly not since proving the above fact for algebra we are using properties like if two sets are in $A$ then their difference is also in $A$ and clearly this won't hold for any collection.
So I basically want a counter example for this: If $u$ is set function on $C$ ( any collection of subsets of $X$) which is a measure but $u$ is not monotone and countably subadditive. Any hints?
Best Answer
Consider $\{\emptyset , \{1\}, \{1,2\}\}$ in $\mathbb R$. Let $\mu (\emptyset)=0, \mu \{1\}=1,\mu (\{1,2\})=0$. Then $\mu $ is countably additive but not monotone.