Finding a measure, a ring theoretical approach

Let \(S\) be a nonempty set and \(F\) consists of all \(Z_{2}\) characteristic functions defined on \(S\). We are supposed to introduce a ring isomorphic to \((P(S),\triangle,\cap)\), whose set is \(F\). Then, assuming a finitely additive function $m$ defined on \(P(S)\), we change \(P(S)\) to a pseudometric space \((P(S),d_{m})\) in which its pseudometric is defined by \(m\). Among other things, we investigate the concepts of convergence and continuity in the induced pseudometric space. Moreover, a theorem on the measure of some kinds of elements in \((P(S),m)\) will be established. At the end, as an application in probability theory, the probability of some events in the space of permutations with uniform probability will be determined. Some illustrative examples are included to show the usefulness and applicability of results.