Pettis integration via the Stonian transform

F. Dennis Sentilles and Robert Francis Wheeler

Vol. 107 (1983), No. 2, 473–496

DOI: 10.2140/pjm.1983.107.473

Abstract

Let (Ω,Σ,μ) be a finite measure space, and let f be a bounded weakly measurable function from Ω into a Banach space X. Let S be the Stone space of the measure algebra Σ∕μ⁻¹(0). Then f induces a continuous map f : S → (X^∗∗,weak^∗) in a natural way. Criteria for Pettis integrability of f are investigated in the context of this “Stonian transform” f of f. In particular, some insight is achieved as to how f can be Pettis integrable without being weakly equivalent to any strongly measurable function. The fine structure of (X^∗∗,weak^∗) is also examined in this setting.

Mathematical Subject Classification 2000

Primary: 28C05

Secondary: 28B05

Milestones

Received: 24 June 1981

Revised: 28 May 1982

Published: 1 August 1983