Fair-coin cylinder content is a premeasure

Let $A={\bigcup }_{n\ge 0}{A}_n$ with disjoint $A_n\in \mathcal{C}$ and $A\in \mathcal{C}$. All these sets are clopen, being finite unions of clopen cylinders. The family consisting of $\mathrm{\Omega }\setminus A$ and all A_n is an open cover of Omega. Compactness gives finitely many members covering Omega and hence finitely many A_n covering A. Using their least indices if the same member repeats gives a finite index set F with $A={\bigcup }_{n\in F}{A}_n$. This uses ambient compactness only of Omega itself, not an unstated compact-subset criterion.

For every n outside F, disjointness gives $A_n\subseteq A\setminus {\bigcup }_{j\in F}{A}_j=\mathrm{\varnothing }$. Thus all other terms have zero content, and finite additivity gives $p_0(A)={\sum }_{n\in F}{p}_0(A_n)={\sum }_{n\ge 0}{p}_0(A_n)$. If A is empty every A_n is empty and the same identity is zero equals zero. Along with $p_0(\mathrm{\varnothing })=0$ and $p_0(\mathrm{\Omega })=1$, this is precisely a finite premeasure.