The Nikodym property and cardinal characteristics of the continuum

Abstract

We present a general method of constructing Boolean algebras with the Nikodym property and of some given cardinalities. The construction is dependent on the values of some classical cardinal characteristics of the continuum. As a result we obtain a consistent example of an infinite Boolean algebra with the Nikodym property and of cardinality strictly less than the continuum c. It follows that the existence of such an algebra is undecidable by the usual axioms of set theory. Besides, our results shed some new light on the Efimov problem and cofinalities of Boolean algebras.