Conglomerated filters, statistical measures, and representations by ultrafilters

TL;DR

This paper introduces the concept of conglomerated filters to show that certain filters cannot be derived from a single statistical measure or represented as intersections of ultrafilters, advancing the understanding of ultrafilter structures.

Contribution

It presents a new combinatorial approach using conglomerated filters to analyze the limitations of Erdős-Ulam and summable filters in terms of statistical measure representations.

Findings

Erdős-Ulam and summable filters cannot be generated by a single statistical measure.

Such filters cannot be expressed as intersections of countable ultrafilters.

The paper explores minimal families of ultrafilters and discusses open questions.

Abstract

Using a new concept of conglomerated filter we demonstrate in a purely combinatorial way that none of Erd\"{o}s-Ulam filters or summable filters can be generated by a single statistical measure and consequently they cannot be represented as intersections of countable families of ulrafilters. Minimal families of ultrafilters and their intersections are studied and several open questions are discussed.