Question
Let $\Omega$ be a countable set and $\mathscr{F}$ the collection of all its subsets. Put $\mu(A)=0$ if $A$ is finite and $\mu(A)=\infty$ if $A$ is infinite. Show that the set function $\mu$ is finitely additive but not countably additive.
Step 1
We have a countable set \(\Omega\) and a collection of subsets \(\mathscr{F}\) which includes all subsets of \(\Omega\). The set function \(\mu\) is defined as follows: - \(\mu(A) = 0\) if \(A\) is finite, - \(\mu(A) = \infty\) if \(A\) is infinite. Show more…
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