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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