Über den Zusammenhang zwischen Funtionenalgebren von Zahlentheoretischen Funktionen und Mengenalgebren

Abstract. Spaces of number theoretical functions are often determinated by an algebra of sets of \(\mathbb N\). In other examples one starts from an algebra \(\mathcal F\) of bounded functions on \(\mathbb N\). By taking the sets \(f^{-1}((-\infty,a])\), \(f\) real-valued, \(f \in \mathcal F\), and the set-algebra generated by these sets one gets an easy connection between function algebras and set algebras. But even in the case, that \(\mathcal F\) is selfadjoint, \(\mathcal F\) separates the points, \(\mathcal F\) contains the constants, \(\mathcal F\) is complete in the sup-norm and each \(f \in \mathcal F\) possesses a mean-value, there can occur, that for some \(f \in \mathcal F\), \(f\) real-valued, \(a \in \mathbb R\), the set \(f^{-1}((-\infty,a])\) does not possess an asymptotic density.
Now, if the function-algebra is taken as above, each \(f \in \mathcal F\), \(f\) real-valued, possesses a limit distribution \(\rho_f\). For the set-algebra \(W\) generated by the sets \(f^{-1}((-\infty,a])\), \(f \in \mathcal F\), \(f\) real-valued, \(a\) point of continuity of \(\rho_f\), it can be proved, that each \(w \in W\) possesses an asymptotic density.