On mappings preserving measurability

Let μ = (μn) be a universal fuzzy measure and let M(μ)M(μ) be the set of all μ  -measurable sets, i.e. sets A⊂NA⊂N 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 π:N→Nπ:N→N such that A∈M(μ)A∈M(μ) implies π(A)∈M(μ)π(A)∈M(μ). Under some assumptions on μ we prove μ∗(π(A)) = λμ∗(A  ) for all A∈M(μ)A∈M(μ), where λ=μ∗(π(N))λ=μ∗(π(N)).