Additive but not $\sigma$-additive function

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?

Solution 1:

Hint Consider $\Omega = \mathbb{N}$, the power set $F=\mathcal{P}(\mathbb{N})$ and the mapping $\mu: F\to [0,\infty]$, $$\mu(A) := \begin{cases} 0, & \text{$A$ is a finite set} \\ \infty, & \text{otherwise}. \end{cases}$$

Solution 2:

Take $\Omega$ to be the set of all natural numbers, $F$ to be the family of all subsets of $\Omega$ and let $\mu(A) = 0$ if $A$ is a finite set and $\mu(A) = \infty$ if $A$ is infinite, I leave it to you to check that it's additive but not $\sigma$-additive.