On nonatomic submeasures on ${\mathbb{N}}$

Abstract.

A submeasure μ defined on the subsets of ${\mathbb{N}}$ is nonatomic if for every ℓ ≥ 1 there exists a partition of ${\mathbb{N}}$ into a finite number of parts on which μ is bounded from above by 1/ℓ. In this paper we answer several natural questions concerning nonatomic submeasures d _F that are determined (like the standard density) by a family F of finite subsets of ${\mathbb{N}}$. We first show that if the number of n-element sets in F grows at most exponentially with n, then d _F is nonatomic; but if this growth condition fails, then d _F need not be nonatomic in general. We next prove that, for a nonatomic submeasure d _F , the minimal number of sets in a 1/ℓ-small partition of ${\mathbb{N}}$ can grow arbitrarily fast with ℓ. We also give a simple example of a nonatomic submeasure that is not equivalent to a submeasure of type d _F .