A representation of additive functionals on L^p-spaces, 0 < p < 1

Judith Ann Palagallo

Vol. 66 (1976), No. 1, 221–234

DOI: 10.2140/pjm.1976.66.221

Abstract

Mizel and Sundaresan have given an integral representation for a class of nonlinear functionals, called additive functionals, on the Banach spaces L^p, p ≧ 1. In this paper, analogous results for these additive functionals on the spaces L^p, 0 < p < 1, are presented. The convergence of additive functionals is also investigated whenever three types of convergence are imposed on the members of L^p: almost everywhere convergence, convergence in measure, and convergence in the metric d, where d(x,y) = ∫ |x−y|^p dμ. In all three cases an integral representation for the functional is obtained, and necessary and sufficient conditions are given for the continuity of the functional.

Mathematical Subject Classification 2000

Primary: 46E30

Milestones

Received: 23 September 1975

Revised: 3 June 1976

Published: 1 September 1976