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The purpose of this paper is to study non-L0-linear perturbations of random isometries in random normed modules. Let (Ω,F,P) be a probability space, K the scalar field R of real numbers or C of complex numbers, L0(F,K) the equivalence classes of K-valued ℱ-measurable random variables on Ω, (E1,∥⋅∥1) and (E2,∥⋅∥2) random normed modules over K with base (Ω,F,P). In this paper, we first establish the Mazur-Ulam theorem in random normed modules. Making use of this theorem and the relations between random normed modules and classical normed spaces, we show that if f:E1→E2 is a surjective random ε-isometry with f(0)=0 and has the local property, where ε∈L0(F,R) and ε≥0, then there is a surjective L0-linear random isometry U:E1→E2 such that ∥f(x)−U(x)∥2≤4ε, for all x∈E1. We do not obtain a sharp estimate as the classical result, since random normed modules have a complicated stratification structure, which is the essential difference between random normed modules and classical normed spaces.MSC:46A22, 46A25, 46H25.

On non- L 0 -linear perturbations of random isometries in random normed modules