Vol. 49, No. 1, 1973

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Convergence of Baire measures

Ronald Brian Kirk

Vol. 49 (1973), No. 1, 135–148

Assume that there are no measurable cardinals. Then E. Granirer has proved that if a net {mi} of finite Baire measures on a completely regular Hausdorff space converges weakly to a finite Baire measure m, then {mi} converges to m uniformly on each uniformly bounded, equicontinuous subset of Cb, the space of bounded continuous functions. In this paper a relatively simple proof of Granirer’s theorem is given based on a recent result of the author. The same method is used to prove the following analogue of Granirer’s theorem. Let {mi} be a net of Baire measures on X each having compact support in the realcompactification of the underlying space X, and assume that Xfdmi Xfdm for every continuous function f on X where m is a Baire measure having compact support in the realcompactification of X. Then {mi} converges to m uniformly on each pointwise bounded, equicontinuous subset of C, the space of continuous functions on X. (The situation in the presence of measurable cardinals is also treated.)

Mathematical Subject Classification 2000
Primary: 28A32
Secondary: 60B10
Received: 7 July 1972
Published: 1 November 1973
Ronald Brian Kirk