\subsection{General background}

Allowing 
probabilities that are finitely but not countably additive permits 
extension to the power set at the cost that many other useful results 
that are true for countable additive probabilities do not hold.
The following theorem, from \citet{kadane-ohagan1995} (relying on results of \citet{rao-rao1983}) gives a necessary and sufficient condition for such an extension of a finitely additive probability.

\begin{thm}\label{thm:three}
Let $\mathcal{C}$ be any collection of subsets of a set $\Omega$ such that $\Omega \in \mathcal C$. Let $\mu$ be a nonnegative real function defined on $\mathcal{C}$ such that $\mu(\Omega) = 1$. Then $\mu$ can be extended to a finitely additive probability on all subsets of $\Omega$ if and only if, for all collections of sets $A_1, \ldots, A_a$ and $B_1,\ldots, B_b$ in $\mathcal{C}$,
\begin{equation}\label{eq:two}
\sum^a_{i=1} I_{A_i} \leq \sum^b_{j=1} I_{B_j}
\end{equation}
implies that
\begin{equation}\label{eq:three}
\sum^a_{i=1} \mu(A_i) \leq \sum^b_{j=1} \mu(B_j),
\end{equation}
where $I_A$ is the indicator function of $A$.
\end{thm}
A second result, also in \citet{kadane-ohagan1995}, gives upper and lower bounds on the probability of a set $D$ (not in general in $\mathcal{C}$):

\begin{thm}\label{thm:four}
Let $\mathcal{C}$ be any collection of subsets of a set $\Omega$ such that $\Omega \in \mathcal{C}$. Let $\mu$ be a nonnegative real function defined on $\mathcal{C}$ such that $\mu(\Omega) = 1$, and let $\mu$ be extendable to a finitely additive probability on all subsets of $\Omega$. Let $\mathcal{M}$ be the set of such extensions. Consider a further set $D \subset \Omega$. Then
$$
\{\mu(D): \mu \in \mathcal{M}\} = [\ell(D,\mathcal M), u(D,\mathcal M)],
$$
where $\ell(D,\mathcal M)$ ($u(D,\mathcal M)$) is the supremum (infimum) of
\begin{equation}\label{eq:four}
h^{-1} \left\{\sum^a_{i=1} \mu(A_i) - \sum^b_{j=1} \mu(B_j)\right\}
\end{equation}
over all $A_1, A_2, \ldots, A_a, B_1, B_2, \ldots, B_b \in \mathcal{C}$ and all $a, b, h = 1,2,3,\ldots,$ such that
\begin{equation}\label{eq:five}
\sum^a_{i=1} I_{A_i} - \sum^b_{i=1} I_{B_j} \leq(\geq)hI_D.
\end{equation}
\end{thm}

\subsection{Finitely additive uniform probabilities on $\mathbb N$}
While there is only one sense of uniformity on a finite set (each element has the same probability), the same is not true on $\mathbb{N}$. Several such senses have been studied in the literature.

\begin{enumerate}
  \item \emph{Limiting relative frequency.} Define 
  $$\mathcal C_F=\{C\subseteq \mathbb N:~\lim_{n\rightarrow\infty}\#(C\cap [n])/n\text{ exists}\}$$ 
  be the collection of subsets of $\mathbb N$ with a limiting relative frequency.  Then it is natural to require $\mu(C)=\lim_{n\rightarrow\infty}\#(C\cap [n])/n$ for $C\in\mathcal C_F$.  \cite{kadane-ohagan1995} proved that such a $\mu$ is extendable. We denote the collection of all such finitely additive measures by $\mathcal M_F$.
\item \emph{Shift invariance.} Another way of defining uniform measure on $\mathbb N$ is to require $\mu$ to be shift invariant. Formally, let $s:\mathbb N\mapsto\mathbb N$ be $s(x)=x+1$. Shift invariance requires $\mu(A)=\mu(s^{-1}(A))$ for all $A\subseteq \mathbb N$. Denote the set of finitely additive shift invariant probabilities by $\mathcal M_S$.
\item \emph{Residue class.} Let $\mathcal C_R$ be the residue class, consisting of sets of the form 
\begin{equation}\label{eq:R}
C=R_{j,k}=\{x:x\equiv j\mod k\}  
\end{equation}
 for some $j\in [k]-1$ and $k\in \mathbb N$.  Uniformity naturally requires that $\mu(R_{j,k})=k^{-1}$ for all $k\in\mathbb N$ and $j\in [k]-1$.  \cite{kadane-ohagan1995} proved that such a $\mu$ is extendable.  We denote the collection of all such extended finitely additive measures by $\mathcal M_R$.
\end{enumerate}
The results in \cite{kadane-ohagan1995} and \citet{schirokauer-kadane2007} jointly imply that that
$$
\mathcal M_F \subset \mathcal M_S \subset\mathcal M_R 
$$
and that each of these inclusions is strict.

\subsection{Finitely additive uniform probabilities on $\mathbb N^2$} Now we extend the three types of finitely additive uniform probabilities to $\mathbb N^2$, and present our main result for finitely additive uniform distributions. The proof of the main result and some intermediate claims, such as extendability, are deferred to later sections.
\begin{enumerate}
  \item \emph{Limiting relative frequency on $\mathbb N^2$.} Define
  \begin{equation}\label{eq:rel_freq_n2}
    \mathcal C_F^2 = \left\{C\subseteq \mathbb N^2: \lim_{n_1\wedge n_2\rightarrow\infty}\frac{\#(C\cap ([n_1]\times [n_2]))}{n_1n_2}\text{ exists}\right\}\,,
  \end{equation}
  and $\mu(C)$ be the limit in \eqref{eq:rel_freq_n2} for $C\in\mathcal C_F^2$.  \Cref{thm:rel_freq_ext} below ensures that $(\mathcal C_F^2,\mu)$ can be extended to $2^{\mathbb N^2}$.  Denote the collection of all such extensions by $\mathcal M_F^2$.
  \item \emph{Shift invariance on $\mathbb N^2$.}  For $j=1,2$, define $s_j:\mathbb N^2\mapsto\mathbb N^2$ as the shift function that increases the $j$th coordinate by one.  Denote $\mathcal M_S^2$ the set of finitely additive shift invariant probabilities on $\mathbb N^2$ (i.e., those satisfy $\mu(A)=\mu(s_j^{-1}(A))$ for all $A\subseteq \mathbb N^2$ and $j=1,2$).
  \item \emph{Residue class on $\mathbb N^2$.} Let $\mathcal C_R^2=\mathcal C_R\times \mathcal C_R$ be the residue class on $\mathbb N^2$ and define $\mathcal M_R^2$ be the set of finitely additive probabilities on $\mathbb N^2$ extended from $(\mathcal C_R^2,\mu)$ with $\mu(R_{j_1,k_1}\times R_{j_2,k_2})=(k_1k_2)^{-1}$ for all $R_{j_1,k_1},R_{j_2,k_2}\in\mathcal C_R$.
\end{enumerate}
The following lemma extends its counterpart in $\mathbb N$, with an almost identical proof.
\begin{lem}\label{lem:hiararchy_n2}
    $\mathcal M_F^2\subseteq \mathcal M_S^2 \subseteq \mathcal M_R^2\,.$
\end{lem}
It is possible to also establish strict inclusions by considering direct products of the examples given in \cite{kadane-ohagan1995,schirokauer-kadane2007}.  Now we state our main result for finitely additive probabilities.
\begin{thm}\label{thm:main_additive}
  Let $G=\{(x,y)\in\mathbb N^2:{\rm gcd}(x,y)=1\}$ be the set of pairs of positive integers that are co-prime. Then
  \begin{align}
  \ell(G,\mathcal M^2_F)= u(G,\mathcal M^2_F)=u(G,\mathcal M^2_S)=u(G,\mathcal M^2_R)=6/\pi^2\,\label{eq:main_1}
  \end{align}
  and
  \begin{align}
    \ell(G,\mathcal M^2_S)=\ell(G,\mathcal M^2_R)=0\,,\label{eq:main_2}
  \end{align}
  where the numbers $u(G,\mathcal M)$, $\ell(G,\mathcal M)$ are defined in \Cref{thm:four}.
\end{thm}
\begin{proof}[Proof of \Cref{thm:main_additive}]
The proof of \Cref{thm:main_additive} essentially contains the organization of results proved in the next three sections.

First, \Cref{lem:hiararchy_n2} implies that
\begin{align}\label{eq:chain}
  \ell(G,\mathcal M_R^2)\le \ell(G,\mathcal M_S^2)\le \ell(G,\mathcal M_F^2)\le
  u(G,\mathcal M_F^2)\le u(G,\mathcal M_S^2)\le u(G,\mathcal M_R^2)\,.
\end{align}

To prove \eqref{eq:main_1}, \Cref{thm:rel_freq_G} implies that $\ell(G,\mathcal M_F^2)=u(G,\mathcal M_F^2)=6/\pi^2$, while 
  \Cref{thm:res_bound} proves that $u(G,\mathcal M_R^2)=6/\pi^2$.  Therefore, \eqref{eq:main_1} follows from \eqref{eq:chain}.
  
Next, \eqref{eq:main_2} is a direct consequence of \eqref{eq:chain} and \Cref{thm:shift}, which proves $\ell(G,\mathcal M_S^2)=0$.
\end{proof}


\Cref{thm:main_additive} implies that if we interpret uniformity by limiting relative frequency, then $G$ has measure $6/\pi^2$ in all finitely additive uniform probabilities on $\mathbb N^2$.  However, if we interpret uniformity by either shift invariance or proportion of residue classes, then the measure of $G$ can be anywhere between $0$ and $6/\pi^2$. Both the lower and upper bounds in these cases are new.
{"config": "arxiv", "file": "1806.00053/finitely.tex"}