On the problem of generalized measures: an impossibility result

TL;DR

This paper demonstrates the impossibility of defining a satisfactory generalized measure in non-separable structures, extending measure theory to broader spaces and showing such measures cannot exist under certain conditions.

Contribution

It introduces a broad class of $oldsymbol{ extlambda^+}$-measures and proves their non-existence under specific cardinal assumptions in generalized descriptive set theory.

Findings

No continuous $oldsymbol{ extlambda^+}$-measure exists on ${}^oldsymbol{ extkappa}oldsymbol{ extlambda}$

Such measures do not exist on $oldsymbol{ extlambda^+}$-Borel or $T_0$ spaces of weight at most $oldsymbol{ extlambda}$

The cardinal assumptions for non-existence are shown to be optimal

Abstract

This paper investigates the problem of extending measure theory to non-separable structures, from generalized descriptive set theory to a broader class of spaces beyond this framework. While various notions, such as the ideal of measure zero sets, have been generalized, the question of whether a satisfactory notion of $\lambda^+$-measure could be defined in generalized descriptive set theory has remained open. We introduce a broad class of $\lambda^+$-measures as functions taking values in arbitrary positively totally ordered monoids equipped with an infinitary sum. This definition relies on minimal assumptions and captures most natural generalizations of measures to this context. We then prove that, under certain cardinal assumptions, no continuous $\lambda^+$-measure of this kind exists on ${}^\kappa\lambda$, nor on any $\lambda^+$-Borel space or $T_0$ topological space of weight at most $\lambda$. We also show the optimality of these cardinal assumptions.