Vol. 107, No. 2, 1983

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Pettis integration via the Stonian transform

F. Dennis Sentilles and Robert Francis Wheeler

Vol. 107 (1983), No. 2, 473–496

Let ,Σ) 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∕μ1(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
Received: 24 June 1981
Revised: 28 May 1982
Published: 1 August 1983
F. Dennis Sentilles
Robert Francis Wheeler