© 1996, Topology Atlas

1. Tau-smooth measures.

By a space, we mean a completely regular Hausdorff space, and by a measure, a countably additive nonnegative finite measure. Recall that the Borel sets are the sigma-algebra generated by the open sets, and the Baire sets, the sigma algebra generated by the zero sets of continuous real valued functions. Measures can be rather ill behaved, so one often focuses on more restricted classes, such as radon measures (measures that are inner regular with respect to the compact sets), or tau-smooth measures. Essentially, any measure that is not "pathological" is tau-smooth. The class of tau-smooth measures contains properly the class radon measures, and has has better stability properties. For instance, an arbitrary product of tau-smooth probability measures is always tau-smooth, while this is not the case for radon measures (consider an uncountable product of copies of the open interval (0,1) with Lebesgue measure. Any measure extension to the Borel sets of the product must assign zero measure to the compact sets).

The idea of the defintion of tau-smooth (or tau-additive) measures, consists of strengthening the following property: "The measure of a countable union of measurable sets can be approximated by using finite unions", so that it will also be valid for arbitrary unions of certain collections of sets. But, which collections should one consider? Of course, an arbitrary union of measurable sets need not be measurable, and even if it is measurable, the requirement maybe so strong as to exclude interesting cases. For instance, consider Lebesgue measure on [0,1], and the collection of finite subsets. Their union is measurable and has measure one, but it cannot be approximated by unions of finite subcollections. In the case of Borel measures, the right notion asks that the measure of an arbitrary union of open sets be approximated by finite unions. In the case of Baire measures, one considers only collections of cozero sets whose union is again a cozero set, and requires the same approximation property. The canonical example of a Borel measure that is not tau-smooth is the Dieudonne measure on the first uncountable cardinal: a Borel set has measure one if it contains a closed unbounded set, and measure zero otherwise. Then the union of all countable initial open segments is the whole space, so it has measure one. But the union of any finite subcollection has measure zero. For a Baire example, restrict the Dieudonne measure to the countable and cocountable sets.

2. Measure compact spaces.

A space is measure compact if every Baire measure defined on it is tau-smooth. While the definition is given in terms of measures, this is a topological property, so a natural question is to give topological characterizations of such spaces. Trivially, every Lindelof space is measure compact; for a while the term almost-Lindelof was used in place of measure compact. It had been thought that measure compactness was equivalent to realcompactness, and some proofs to this effect appeared in the literature. Such "proofs" ceased from being published when W. Moran gave an example of a realcompat not measure compact space (J. London Math. Soc., 43 (1968), 633-639). There is, however, a characterization of realcompactnes using tau-smoothness: A space is realcompact iff every two valued Baire probability measure is tau-smooth. Regarding characterizations of measure compact spaces, G. Plebanek has reciently provided one (Trans. A.M.S., Vol. 332, Number 1,(1992),181-191) in terms of strongly positive families.

3. The Borel extension problem.

Given a Baire measure, when can it be extended to a Borel measure? If the space is normal and every closed set is a G-delta, then there is no problem: The Borel and the Baire sets coincide. But in general they do not, and this is an area of current research. For tau-smooth measures the answer is both known and pleasent: Every tau-smooth Baire measure has a unique tau-smooth Borel extension (though there may be other non tau-smooth Borel extensions).

4. Survey articles and open problems.

In the area of Baire measures, we mention the paper by R. F. Wheeler, "A survey of Baire measures and strict topologies" (Expo. Math., 2 (1983) 97--190). Regarding Borel measures, the interested reader may want to consult the articles "The regularity of Borel measures" by R. J. Gardner (Springer Lecture Notes in Math., Vol. 945, pp. 42-100) and "Borel measures", By R. J. Gardner and W. F. Pfeffer (Handbook of Set-Theoretic Topology, K. Kunen and J.E. Vaughan, Eds., (North-Holland, Amsterdam-New York-Oxford, 1984), 961-1043). The first two papers mentioned above contain several problems, some of which have already been solved, while others remain open. In D. H. Fremlin's Problem list one can also find problems that sit in the intersection of Analysis, Set Theory and Topology.