Almost Everywhere Behavior of Functions According to Partition Measures

This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals. The following summarizes the main results proved under suitable partition hypotheses. If is a cardinal, , , and , then satisfies the almost everywhere short length continuity property: There is a club and a so that for all , if and , then . If is a cardinal, is countable, holds and , then satisfies the strong almost everywhere short length continuity property: There is a club and finitely many ordinals so that for all , if for all , , then . If satisfies , and , then satisfies the almost everywhere monotonicity property: There is a club so that for all , if for all , , then . Suppose dependent choice ( ), and the almost everywhere short length club uniformization principle for hold. Then every function satisfies a finite continuity property with respect to closure points: Let be the club of so that . There is a club and finitely many functions so that for all , for all , if and for all , , then . Suppose satisfies for all . For all , does not inject into , the class of -length sequences of ordinals, and therefore, . As a consequence, under the axiom of determinacy , these two cardinality results hold when is one of the following weak or strong partition cardinals of determinacy: , , (for all ) and (assuming in addition ).