A note on extensions of asymptotic density (Q2731926)

By a density the authors mean an arbitrary extension of the usual asymptotic density to a (nonnegative) finitely additive measure on \({\mathcal P}(\omega)\), where \(\omega\) stands for the set of natural numbers. They prove that there is a density \(\nu\) with the following property \({\mathbf A}{\mathbf P}\)(null): given \(A_1\subset A_2\subset\cdots\) in \({\mathcal P}(\omega)\), there exists \(B\) in \({\mathcal P}(\omega)\) with \(\nu(A_i\setminus B)= 0\) for all \(i\) and \(\nu(B)= \lim_i \nu(A_i)\). This answers a question of \textit{S. Gangopadhyay} and \textit{B. V. Rao} [Colloq. Math. 80, No. 1, 83-95 (1999; Zbl 0940.28005)], since \(L_1(\nu)\) is in that case complete. The authors also give a short proof of the following related result due to \textit{A. H. Mekler} [Proc. Am. Math. Soc. 92, 439-444 (1984; Zbl 0561.28001)]: If \(P\)-points exist, then there is a density \(\nu\) with the property that, given \(A_1\subset A_2\subset\cdots\) in \({\mathcal P}(\omega)\), there exists \(B\) in \({\mathcal P}(\omega)\) with \(A_i\setminus B\) finite for all \(i\) and \(\nu(B)= \lim_i \nu(A_i)\). Clearly, the latter property implies \({\mathbf A}{\mathbf P}\)(null). The converse does not hold, as is shown in the paper. The desired densities are obtained by using generalized limits of sequences \(|A\cap\{0,1,\dots, n-1\}|/n\), \(n= 1,2,\dots\), where \(A\subset\omega\), with respect to special ultrafilters on \(\omega\). A result of D. Fremlin, which shows that that procedure does not always yield densities with property \({\mathbf A}{\mathbf P}\)(null), is also presented.NEWLINENEWLINENEWLINELet us note that completeness-type properties of \(L_1(\nu)\) for a general finitely additive measure \(\nu\) have been recently discussed in papers by \textit{A. Basile} and \textit{K. P. S. Bhaskara Rao} [J. Math. Anal. Appl. 248, No. 2, 588-624 (2000; Zbl 0976.46017)] and \textit{C. Swartz} [Ric. Mat. 49, 307-315 (2000)].