On some questions of Drewnowski and Łuczak concerning submeasures on

For a given submeasure ϕ on N a sequence ( A n ) n ∈ N of subsets of N is called a ϕ-sequence if ϕ ( ⋃ n ∈ N F n ) = 0 for every choice of finite sets F n ⊂ A n ( n ∈ N ). We show an example of a submeasure ϕ which is not the lim sup of lower semicontinuous submeasures, but lim ϕ ( A n ) = 0 for any ϕ-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 σ-submeasure, but has the property that for every sequence ( A n ) n of subsets of N if lim ϕ ( 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 ϕ-sequence. These answer questions of Drewnowski and Łuczak (2008) from [2].