On the problem of generalized measures: an impossibility result
Claudio Agostini, Fernando Barrera, Vincenzo Dimonte
TL;DR
The paper tackles the problem of a satisfactory notion of measure in generalized descriptive set theory by introducing $\lambda^+$-measures valued in positively totally ordered monoids with an infinitary sum. It proves an impossibility theorem: under mild cardinal assumptions, there is no nontrivial continuous $\lambda^+$-measure on subspaces of the generalized Baire space $^{\kappa}\lambda$ (and more generally on $\lambda^+$-Borel or $T_0$ spaces of weight $\lambda$). The work also analyzes trivial Dirac-like measures, the role of the sum’s continuity, and conditions under which $\lambda$-measures can exist in relation to measurable cardinals. Overall, the results delineate the landscape where generalized measures can or cannot behave analogously to classical measure theory, highlighting the separable/Polish-like constraints that arise in the uncountable setting.
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 $λ^+$-measure could be defined in generalized descriptive set theory has remained open. We introduce a broad class of $λ^+$-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 $λ^+$-measure of this kind exists on ${}^κλ$, nor on any $λ^+$-Borel space or $T_0$ topological space of weight at most $λ$. We also show the optimality of these cardinal assumptions.