Invariants of Measure and Category

The purpose of this chapter is to discuss various results concerning the relationship between measure and category. We are mostly interested in set-theoretic properties of these ideals, particularly, their cardinal characteristics. This is a very large area, and it was necessary to make some choices. We decided to present several new results and new approaches to old problems. In most cases we do not present the optimal result, but a simpler theorem that still carries most of the weight of that original result. For example, we construct Borel morphisms in the Cichoń diagram while continuous ones can be constructed. We believe however that the reader should have no problems upgrading the material presented here to the current state of the art. The standard reference for this subject is [8], and this chapter updates it as most of the material presented here was proved after [8] was published. Measure and category have been studied for about a century. The beautiful book [31] contains a lot of classical results, mostly from analysis and topology, that involve these notions. The role played by Lebesgue measure and the Baire category in these results is more or less identical. There are, of course, theorems indicating lack of complete symmetry but they do not seem very significant. For example, Kuratowski’s theorem (cf. Theorem 3.7) asserts that for every Borel function f : 2 −→ 2 there exists a meager set F ⊆ 2 such that f↾(2 \ F ) is continuous. The dual proposition stating that for every Borel function f : 2 −→ 2 there exists a measure one set G ⊆ 2 such that f↾G is continuous is false. We only have a theorem of Luzin which guarantees that a such G’s can have measure arbitrarily close to one. The last 15 years have brought a wealth of results indicating that hypotheses relating to measure are often stronger than the analogous ones relating to category. This chapter contains several examples of this phenomenon. Before we delve into this subject let us give a little historical background. The first result of this kind is due to Shelah [43]. He showed that • If all projective sets are measurable then there exists an inner model with an inaccessible cardinal. • It is consistent with ZFC that all projective sets have the property of Baire. In 1984 Bartoszynski [2] and Raisonnier and Stern in their [35] showed that additivity of measure is not greater than additivity of category, while Miller [29] showed that it can be strictly smaller. In subsequent years several more results of that kind were found. Let us mention one more (cf. [5]) concerning filters on ω (treated as subsets of 2):