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Uniform boundedness of operators and barrelledness in spaces with Boolean algebras of projections. (English) Zbl 0849.46002
In a recent paper [Proc. Am. Math. Soc. 114, No. 3, 687-694 (1992; Zbl 0747.46026)], the present authors proved that a quasi-barrelled locally convex space equipped with a “suitable” Boolean algebra of projections modelled on an atomless finite positive measure space is necessarily barrelled. As applications of that result, it was shown in the cited paper that, for such measure spaces, the space of Pettis integrable functions with values in any Banach space, as well as the space of Bochner integrable functions with values in any normed space, are always barrelled. The latter fact has been subsequently generalized to a much wider class of spaces, nonmetrizable in general, consisting of strongly measurable functions with values in any normed space.
The purpose of this paper is to extend the general result mentioned above to quasi-barrelled spaces with Boolean algebras of projections modelled on arbitrary positive measure spaces without atoms of finite measure.

##### MSC:
 46A08 Barrelled spaces, bornological spaces 46A32 Spaces of linear operators; topological tensor products; approximation properties 46G10 Vector-valued measures and integration 46E40 Spaces of vector- and operator-valued functions