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.)
