Handbook of Set Theory

pp 491-555


Invariants of Measure and Category

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The purpose of this chapter is to discuss various results concerning the relationship between measure and category. The focus is on set-theoretic properties of the associated ideals, particularly, their cardinal characteristics. The key notion is the Tukey reducibility which compares partial orders with respect to their cofinality type. We define small sets of reals associated with cardinal invariants and discuss their properties. We present a number of ZFC results and forcing-like constructions of various small sets of reals assuming continuum hypothesis. We also present a proof of the result of Shelah that the smallest cardinality of a family of Lebesgue measure zero sets covering the real line may have countable cofinality.