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This question asks the following:

Let $\Omega$ be a countably infinite set and let $\mathcal{F}$ be a field that contains all subsets of $\Omega$ which are finite and also contains 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 is not countably additive on $\mathcal{F}$.

(b) Show that $\Omega$ is the limit of an increasing sequence of sets $A_n\in \mathcal{F}$ with $\mu(A_n)=0\ \forall n$, but $\mu(\Omega)=1$.

I did the second part by defining the sequence of sets $A_n=\{a_1,a_2,\cdots,a_n\},\ n\ge 1$, where $\Omega=\{a_1,a_2,\cdots\}$. Clearly $A_n\uparrow \Omega$, and $\mu(A_n)=0,\ \forall n$ since each of $A_n$ is finite, but $\mu(\Omega)=1$ since $\Omega^C=\emptyset$, which is finite.

However, for some reason I am not getting the first part. If I take $N$ sets $A_n\in \mathcal{F},\ 1\le n\le N$, if all of them are finite, the first part is obvious, however, I am having trouble when a finitely many of them are infinite and have finite complements. Let say, $A_1,A_2$ have finite complements, then $\mu(A_1)+\mu(A_2)=2$, but $\mu(A_1\cup A_2)=1$, as $A_1^C\cap A_2^C$, which contradicts the part (a). Am I missing something? Please help.

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    To show finite additivity, your $A_1$ and $A_2$ need to be disjoint, bu they can't be both infinite and disjoint and be in $\mathcal{F}$ as it is just a field and not $\sigma$-filed and by definition is generated by the finite subsets. So you need to subratc the measure of their intersection – 2017-01-19
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    Thanks a lot! Yes, missing the disjointness is the key to my confusion. – 2017-01-19

1 Answers 1

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For finite additivity, it is sufficient to show the following:

For all $A_1, A_2 \in \mathcal{F}$, if $A_1 \cap A_2 = \emptyset$ then $\mu(A_1 \cup A_2) = \mu(A_1) + \mu(A_2)$.

Show that if $A_1, A_2$ both have finite complement then $A_1 \cap A_2 \ne \emptyset$, and so nothing needs to be done in that case.

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    Oh, correct, missed that disjoint property, silly me. Thanks a lot! – 2017-01-19