On mappings preserving measurability
Let ?=(?n) be a universal fuzzy measure and let M(?) M ( ? ) be the set of all ? -measurable sets, i.e. sets A?N A ? N for which the limit ??(A)=limn???n(A?{1,2,,n }) exists. We are studying properties of measurability preserving injective mappings, i.e. injective mappings ?:N?N ? : N ? N such that A?M(?) A ? M ( ? ) implies ?(A)?M(?) ? ( A ) ? M ( ? ) . Under some assumptions on ? we prove ??(?(A))=???(A ) for all A?M(?) A ? M ( ? ) , where ?=??(?(N)) ? = ? ? ( ? ( N ) ) .
universal fuzzy measure asymptotic fuzzy measure asymptotic density