On some questions of Drewnowski and Luczak concerning submeasures on N
Abstract For a given submeasure ph on N a sequence A n n N of subsets of N is called a ph sequence if ph n N F n 0 for every choice of finite sets F n A n n N We show an example of a submeasure ph which is not the lim sup of lower semicontinuous submeasures, but lim ph A n 0 for any ph sequence A n n Moreover, we show that it is enough to consider only decreasing sequences A n n in the above. We also construct a submeasure on N which is not the core of a s submeasure, but has the property that for every sequence A n n of subsets of N if lim ph A n 0 then there is a subsequence n k k and finite sets E n k A n k such that A n k E n k k is a ph sequence. These answer questions of Drewnowski and Luczak (2008) from [2]