Local rearrangement invariant spaces and distribution of Rademacher series

Abstract

We prove that a local version of Khintchine inequality holds for arbitrary rearrangement invariant (r.i.) spaces on an non-empty open set $$E\subset [0,1]$$ . For this, we give a definition of local r.i. space which is compatible with the notion of systems equivalent in distribution and prove that the Rademacher system $$(r_{k+N})_{k=1}^\infty $$ on an non-empty open set E is equivalent in distribution to $$(r_k)_{k=1}^\infty $$ on [0, 1], with N depending on E. The result can be generalized to a wider class of sets.