Some properties of measure and category

TLDR

All of the properties are true if the continuum hypothesis is assumed. The main technique used is iterated forcing. The authors study elementary cardinal properties of measure and category on ℝ, showing that sets of size below the continuum have measure zero, linking several properties to the structure of ℤ^ℤ under eventual dominance, and proving that many, but not all, combinations of these properties are consistent with ZFC.

Abstract

Several elementary cardinal properties of measure and category on the real line are studied. For example, one property is that every set of real numbers of cardinality less than the continuum has measure zero. All of the properties are true if the continuum hypothesis is assumed. Several of the properties are shown to be connected with the properties of the set of functions from integers to integers partially ordered by eventual dominance. Several, but not all, combinations of these properties are shown to be consistent with the usual axioms of set theory. The main technique used is iterated forcing.