arXiv subjects

Fernando Barrera

Publications and source records attributed to Fernando Barrera.

2 recordsLinked to original sources

arXiv2026

On the problem of generalized measures: an impossibility result

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.

math.LO↗

arXiv2025

The $\lambda$-PSP at $\lambda$-$\Pi^1_1$ sets

Given a strong limit cardinal $\lambda$ of countable cofinality, we show that if every (boldface) $\lambda\hyp\boldsymbol{\Pi}^1_1$ subset of the generalised Cantor space ${}^{\lambda}2$ has the $\lambda$-$\mathsf{PSP}$, then $0^\dagger$ exists. We show too that if every (lightface) $\lambda\hyp\Pi^1_1$ subset of ${}^\lambda 2$ has the $\lambda\hyp\mathsf{PSP}$, then there is an inner model with a measurable cardinal. The paper, a contribution to the ongoing research on generalised regularity properties in generalised descriptive set theory at singular cardinals of countable cofinality, is aimed at descriptive set theorists, and so it presents its results in as much detail as possible, particularly regarding the inner model-theoretic aspects. In doing so, we intend to provide the community with the tools needed to handle consistency strength arguments at the corresponding levels.

math.LO↗