Abstract

Let X be a completely regular space. The customary σ-field is the coarsest σ-field on the space of Bairemeasures on X which makes μ→μ(A) measurable for any Baire set A. We compare the customary σ-field with the Baire and Borel σ-field induced by the weak* topology which lies on the dual space C(X)′. In (2.3) it is shown that the customary σ-field is just the Baire σ-field. In part 3 necessary and sufficient conditions are given under which the set of τ-smooth measures is measurable with respect to the Borel σ-field which lies on the positive cone of the space of finitely additive, regular measures C(X)′. Finally, a decomposition theorem for generalized kernels is proved. The τ-smooth part of a generalized kernel is a kernel again if certain conditions are fulfilled.

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