A probability measure defined on a sample space $\Omega$ has the following properties:
- For each $E \subset \Omega$, $0 \le P(E) \le 1$
- $P(\Omega) = 1$
- If $E_1$ and $E_2$ are disjoint subsets $P(E_1 \cup E_2) = P(E_1) + P(E_2)$
The above definition defines a measure that is finitely additive (by induction) but not necessarily countably additive.
What is a probability measure that would be finitely additive but not countably additive (for a countable sample space $\Omega$)?
The example that I have seen most commonly on forums (this and elsewhere) is to set $P(E) = 0$ if $E$ is finite and $P(E) = 1$ if $E$ is co-finite. But that is not a probability measure as defined above since it is not defined on every subset of $\Omega$.
So an example of such a probability measure, or what is the reasoning that a finitely additive probability measure is not always countably additive?
Best Answer
Let $\mathcal{U}$ be a free ultrafilter on $\mathbb{N}$. Let $P(A)=1$ if $A\in\mathcal{U}$ and $P(A)=0$ if $A\notin\mathcal{U}$. I think it is impossible to give an explicit example of a finitely additive measure on a $\sigma$-algebra that is not countably additive, but our resident set theorists might be able to tell you more about that.