## Probability Foundations – Problem (37/365)

A problem similar to the previous post. Let $\Omega$ be a countable set and $\mathcal{A}$ a collection of all its subsets. Put $\mu(A) = 0$ if $A$ is finite and $\mu(A) = \infty$ if $A$ is inifinite. Show that the set function $\mu$ is finitely additive but not countable additive.

To see that it is finitely additive, let $A,B \in \mathcal{A}$ be disjoint.

$\displaystyle \mu(A \cup B) = \mu(A) + \mu(B) = 0 + 0 = 0 \text{ if both finite} \\ \mu(A \cup B) = \mu(A) + \mu(B) = \infty \text{ if either one infinite}$

To show that it is not countably additive, consider the case where $\Omega = \mathbb{N}$ is the set of natural numbers. Then

$\displaystyle \mu(\{ 1, 2, 3, \dots \}) = \infty \\ \text{But,} \sum_{i=1}^\infty \mu(i) = 0$

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