Additivity of Measure Implies Dominating Reals

Abstract

We show that additivity of measure $A\left ( m \right )$, the union of less than continuum many measure zero sets has measure zero) implies that every family $F \subseteq {\omega ^\omega }$ of cardinality less than continuum is eventually dominated (this is the property $D$). This yields as a corollary from known results that $A\left ( m \right ) + B\left ( c \right ) \to A\left ( c \right )$. $A\left ( c \right )$ is the property that the union of less than continuum many first category sets has first category and $B\left ( c \right )$ is the property that the real line is not the union of less than continuum many first category sets. Also, a new property of measure and category is introduced, the covering property, $C\left ( m \right )$, which says that for any family of measure zero (first category) sets of cardinality less than the continuum there is some measure zero (first category) set not covered by any member of the family. By dualizing the proof that $A\left ( m \right ) \to D$ we show that $wD \to C\left ( m \right )$. The weak dominating property, $wD$, says that no small family contained in ${\omega ^\omega }$ dominates every element of ${\omega ^\omega }$.