On some questions of Drewnowski and Łuczak concerning submeasures on N

For a given submeasure ϕ on N a sequence (An)n∈N of subsets of N is called a ϕ-sequence if ϕ(⋃n∈NFn)=0 for every choice of finite sets Fn⊂An (n∈N). We show an example of a submeasure ϕ which is not the lim sup of lower semicontinuous submeasures, but for any ϕ-sequence (An)n. Moreover, we show that it is enough to consider only decreasing sequences (An)n in the above. We also construct a submeasure on N which is not the core of a σ-submeasure, but has the property that for every sequence (An)n of subsets of N if then there is a subsequence (nk)k and finite sets Enk⊂Ank such that (Ank∖Enk)k is a ϕ-sequence. These answer questions of Drewnowski and Łuczak (2008) from [2].