Finitely purely atomic measures: Coincidence and rigidity properties

This paper is a natural continuation of the paper 2] by the same author. We shall prove that several coincidence and rigidity phenomena which usually do not appear are possible only in case the underlying measure space is trivial (i.e. is a finite union of atoms). Examples: coincidence of twoL ^p spaces, reflexivity ofL ¹, Radon—Nikodym property ofL ^∞, coincidence of Dunford, Pettis or Bochner integrability, coincidence of theL ^p space and of the weakL ^p space.