Topological measure theory for completely regular spaces and their projective covers

Robert Francis Wheeler

Vol. 82 (1979), No. 2, 565–584

DOI: 10.2140/pjm.1979.82.565

Abstract

This paper investigates the relationships among tight, τ-additive, and σ-additive Baire measures on a completely regular Hausdorff space X and its projective cover E(X). The most interesting questions arise in the σ-additive case, and lead to the following definitions: the space X has the weak (resp. strong) lifting property if for each σ-additive measure on X, some (resp., every) pre-image measure on E(X) is σ-additive. It is shown that every weak cb space has the strong lifting property, while the Dieudonné plank fails even the weak lifting property. Also, if X is weak cb, then X is measure-compact if and only if E(X) is measure-compact.

Some applications to extensions of measures on lattices and to strict topologies on spaces of continuous functions are given. A relationship between the lifting properties mentioned above and conventional use of the term “lifting” in measure theory is indicated.

Mathematical Subject Classification 2000

Primary: 28C15

Milestones

Received: 14 July 1978

Revised: 19 January 1979

Published: 1 June 1979