Give an example of a measure space $(\Omega, \mathit{F})$ and a function $\mu$ on $\mathit{F}$ that is additive but not $\sigma$-additive, i.e. $\mu(\cup A_i)= \sum\mu(A_i)$ for a finite collection of disjoint $A_i$ but not for some infinite collections.
I know a measure function defined on $\sigma$-algebra is $\sigma$-additive, but I struggle finding a function that would not be additive for infinite collections. Can someone give me an example and show me why?
Best Answer
Hint Consider $\Omega = \mathbb{N}$, the power set $F=\mathcal{P}(\mathbb{N})$ and the mapping $\mu: F\to [0,\infty]$, $$\mu(A) := \begin{cases} 0, & \text{$A$ is a finite set} \\ \infty, & \text{otherwise}. \end{cases}$$