I am (self) studying probability theory and measure using the book from Ash, R. et al. [1]. I am trying to solve one of the basic problems (Section 1.2, problem 3) but with no avail…here is the link to the problem:
Let $\Omega$ be a countable infinite set, and let $\mathcal{F}$ be the field consisting of all finite subsets of $\Omega$ and their complements. If $A$ is finite, set $\mu(A) = 0$, and if $A^{c}$ is finite, set $\mu(A) = 1$.
(a) show that $\mu$ is finitely additive but not countable additive.
(b) show that $\Omega$ is the limit of an increasing sequence of sets $A_{n} \in \mathcal{F}$, with $\mu(A_{n}) = 0$ for all $n$, but $\mu(\Omega)=1$
I would like to get some help as where to begin with…
Best Answer
For finite additivity $\mu (A \cup B) =\mu (A) +\mu (B)$ just consider the cases where both sets are finite sets, bot have finite complements and one is finite , the other having finite complement. Now let $x_1,x_2,...$ be the distinct points of the space and note that $\mu (\{x_1\} \cup \{x_2\} \cup ....)=1$ whereas $\sum \mu (\{x_i\})=0$. ( It is understood that '$A$ has finite complement' includes the case when the complement is empty).