How can I prove that an ultrafilter induces a finitely additive measure? If $\mathcal{U}$ is an ultrafilter on a set $X$, it can be defined a function $\mu_{\mathcal{U}}\colon\mathcal{P}(X)\to\{0,1\}$ such that, for all $A\subseteq X$ it holds $\mu_{\mathcal{U}}(A)=1$ iff $A\in\mathcal{U}$. 
Are $\mu_{\mathcal{U}}(\emptyset)=0$ and finite additivity enough to prove that $\mu_{\mathcal{U}}$ is well-defined? Of course, given a subset $E$ of $X$, we can split $E$ into disjoint sets $E_1,\dots,E_n$.  If $\mu_{\mathcal{U}}(E)=1$, then exactly one $E_i$ belongs to $\mathcal{U}$, so $\mu_{\mathcal{U}}(E_1)+\dots+\mu_{\mathcal{U}}(E_n)=1$. If $\mu_{\mathcal{U}}(E)=0$, then none of the $E_i$'s belongs to $\mathcal{U}$, so $\mu_{\mathcal{U}}(E_1)+\dots+\mu_{\mathcal{U}}(E_n)=0$. Did I prove that $\mu_{\mathcal{U}}$ is actually a function, hence a finitely additive measure?
 A: You just define $\mu(A) = 0$ if $A \notin \mathcal{U}$ and $\mu(A) = 1$ if $A \in \mathcal{U}$. This is well-defined (a set cannot be both in $\mathcal{U}$ and not in $\mathcal{U}$, and so any set has measure $0$ or $1$).
As an ultrafilter does not contain $\emptyset$, $\mu(\emptyset) = 0$, as required.
And if we have finitely many disjoint subsets $E_1,\ldots,E_n$ of $X$, we need to show that for $E = \cup_{i=1}^n E_i$ we have $\mu(E) = \sum_i \mu(E_i)$.
If all $E_i$ are not in $\mathcal{U}$, then $X \setminus E_i \in \mathcal{U}$ for all $i$, which means that $X \setminus E = \cap_{i=1}^n (X \setminus E_i) \in \mathcal{U}$, as filters are closed under finite intersections, and hence $E \notin \mathcal{U}$ as well. So we have equality ($0$'s on both sides) in our equality we have to prove.
On the other hand, if some $E_{i_0} \in \mathcal{U}$, then the other $E_i$ are not in $\mathcal{U}$ by disjointness, and so have measure $0$, and $E$ contains $E_{i_0}$ so also is in $\mathcal{U}$ and has measure $1$. So we have equality as well: $\mu(E) = 1$ and the sum is $1$ as well: one $1$ for $E_{i_0}$ and the others are $0$. 
This shows that $\mu$ is indeed a finitely additive measure.
