On some questions of Drewnowski and Łuczak concerning submeasures on \(\mathbb N\) (Q990848)

The questions in the title of the paper deal with properties of submeasures introduced in [\textit{L. Drewnowski} and \textit{T. Łuczak}, J. Math. Anal. Appl. 347, No. 2, 442--449 (2008; Zbl 1153.28001)] in the study of cores of submeasures. A \(\phi\)-sequence for a submeasure~\(\phi\) is a sequence \(\langle A_n\rangle_n\) of sets such that \(\phi(\bigcup_nF_n)=0\) for every sequence \(\langle F_n\rangle_n\), where each \(F_n\) is a finite subset of~\(A_n\). The properties alluded to are (A): \(\lim_n\phi(A_n)=0\) for every \(\pi\)-sequence; (B): if \(\phi(Z)=0\) for every \(Z\) such that \(Z\setminus A_n\) is finite for all~\(n\) then \(\lim_n\phi(A_n)=0\); and (C): every sequence \(\langle A_n\rangle_n\) such that \(\lim_n\phi(A_n)=0\) has a subsequence that is a \(\phi\)-sequence upon removal of a finite subset from each term. Properties (A) and (B) are equivalent if singletons have submeasure zero. From a maximally almost disjoint family one can construct a \(0\)-\(1\)-valued submeasure with property~(A) that is not a \(\limsup\) is lsc submeasures. A selective ultrafilter yields a (sub)measure with property~(C) that is not the core of a countably subadditive submeasure.