About positive linear functionals on spaces of arithmetical functions

Abstract. Let \(\mathcal F\) be an algebra of real-valued bounded functions on \(\mathbb N\) which separates the points, which contains the constants and which is complete in the sup-norm. If \(L\) is a positive linear functional on \(\mathcal F\), then, for each \(f\in \mathcal F\), \(L(f)\) can be represented as an integral of \(\overline{f}\) on \(\beta\mathbb N\) where \(\overline{f}\) is the unique extension of \(f\) to the Stone–Čech compactification \(\beta \mathbb N\) of \(\mathbb N\).

Key words and phrases. Probabilistic number theory, spaces of arithmetical functions, positive linear functionals.