On mappings preserving measurability

Let μ = (μn) be a universal fuzzy measure and let image be the set of all μ-measurable sets, i.e. sets image for which the limit μ∗(A) = limn→∞μn(A ∩ {1, 2, … , n}) exists. We are studying properties of measurability preserving injective mappings, i.e. injective mappings image such that image implies image. Under some assumptions on μ we prove μ∗(π(A)) = λμ∗(A) for all image, where image.