Example for finitely additive but not countably additive probability measure 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?

In the following note the author shows that a finitely additive diffused measure on $\mathcal{P}(\omega)$ can be used to define a non Ramsey family. Combining this with a result of Mathias, it follows that it is consistent with $ZFC$ that there is no (ordinal) definable finitely additive diffused total measure on $\mathcal{P}(\omega)$.
