arXiv subjects

Andrés F. Uribe-Zapata

Publications and source records attributed to Andrés F. Uribe-Zapata.

7 recordsLinked to original sources

arXiv2025

Finitely additive measures on Boolean algebras

In this article, we conduct a detailed study of \emph{finitely additive measures} (fams) in the context of Boolean algebras, focusing on three specific topics: freeness and approximation, existence and extension criteria, and integration theory. In the first topic, we present a classification of \emph{free} finitely additive measures, that is, those for which the measure of finite sets is zero, in terms of approximation to uniform probability measures. This inspires a weaker version of this notion, which we call the \emph{uniform approximation property}, characterized in terms of freeness and another well-determined type of fams we call \emph{uniformly supported}. In the second topic, we study new criteria for the existence and extension of finitely additive measures. In particular, we provide an extension of the so-called \emph{compatibility theorem} -- which characterizes when two finitely additive measures can be extended -- yielding a precise and compact characterization of when three finitely additive measures can be simultaneously extended, under the assumption that one of them is an ultrafilter. Finally, we study a Riemann-type integration theory on fields of sets with respect to finitely additive measures, allowing us to extend and generalize some classical concepts and results from real analysis, such as Riemann integration over rectangles in $\mathbb{R}^{n}$ and the Jordan measure. We also generalize the extension criteria for fams allowing desired values of integrals of a given set of functions. At the end, we explore the connection between integration in fields of sets and the Lebesgue integration in the Stone space of the corresponding field, where we establish a characterization of integrability in the sense of the Lebesgue-Vitali theorem, which follows as a consequence of our results.

math.LO↗

arXiv2025

The measure algebra adding $θ$-many random reals is $θ$-FAM-linked

The notion of $θ$-FAM-linkedness, introduced in the second author's master thesis, is a formalization of the notion of strong FAM limits for intervals, whose initial form and applications have appeared in the work of Saharon Shelah, Jakob Kellner, and Anda Tănasie, for controlling cardinals characteristics of the continuum in ccc forcing extensions. This generalization was successful in such a thesis to establish a general theory of iterated forcing using finitely additive measures. In this paper, using probability theory tools developed in the same thesis, we refine Saharon Shelah's proof of the fact that random forcing is $σ$-FAM-linked and prove that any complete Boolean algebra with a strictly positive probability measure satisfying the $θ$-density property is $θ$-FAM-linked. As a consequence, we get a new collection of examples of $θ$-FAM-linked forcing notions: the measure algebra adding $θ$-many random reals.

math.LO↗

arXiv2025

Absoluteness of the Riemann integral

This article explores the concept of absoluteness in the context of mathematical analysis, focusing specifically on the Riemann integral on $\mathbb{R}^{n}$. In mathematical logic, "absoluteness" refers to the invariance of the truth value of certain statements in different mathematical universes. Leveraging this idea, we investigate the conditions under which the Riemann integral on $\mathbb{R}^{n}$ remains absolute between transitive models of ZFC, the standard axiomatic system in which current mathematics is usually formalized. To this end, we develop a framework for integration on Boolean algebras with respect to finitely additive measures and show that the classical Riemann integral is a particular case of this generalized approach. Our main result establishes that the Riemann integral over rectangles in $\mathbb{R}^{n}$ is absolute in the following sense: if $M \subseteq N$ are transitive models of ZFC, $a, b \in \mathbb{R}^{n} \cap M$, and $f \colon [a, b] \to \mathbb{R}$ is a bounded function in $M$, then $f$ is Riemann integrable in $M$ if, and only if, in $N$ there exists some Riemann integrable function $g \colon [a, b] \to \mathbb{R}$ extending $f$. In this case, the values of the integrals computed in each model are the same. Furthermore, the function $g$ is unique except for a measure zero set.

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arXiv2025

Probability Trees

In this article, we introduce a formal definition of the concept of probability tree and conduct a detailed and comprehensive study of its fundamental structural properties. In particular, we define what we term an inductive probability measure and prove that such trees can be identified with these measures. Furthermore, we prove that probability trees are completely determined by probability measures on the Borel $σ$-algebra of the tree's body. We then explore applications of probability trees in several areas of mathematics, including probability theory, measure theory, and set theory. In the first, we show that the cumulative distribution of finitely many dependent and non-identically distributed Bernouli tests is bounded by the cumulative distribution of some binomial distribution. In the second, we establish a close relationship between probability trees and the real line, showing that Borel, measurable sets, and their measures can be preserved, as well as other combinatorial properties. Finally, in set theory, we establish that the null ideal associated with suitable probability trees is Tukey equivalent to the null ideal on $[0, 1]$. This leads to a new elementary proof of the fact that the null ideal of a free $σ$-finite Borel measure on a Polish space is Tukey equivalent with the null ideal of $\mathbb{R}$, which supports that the associated cardinal characteristics remain invariant across the spaces in which they are defined.

math.PR↗

arXiv2024

A general theory of iterated forcing using finitely additive measures

Based on the work of Shelah, Kellner, and Tănasie (Fund. Math., 166(1-2):109-136, 2000 and Comment. Math. Univ. Carolin., 60(1):61-95, 2019), and the recent developments in the third author's master's thesis, we develop a general theory of iterated forcing using finitely additive measures. For this purpose, we introduce two new notions: on the one hand, we define a new linkedness property, called $μ$-$\mathrm{FAM}$-linked and, on the other hand, we generalize the notion of intersection number to forcing notions, which justifies the limit steps of our iteration theory. Our theory also generalizes iterations with ultrafilters, which have played an important role in the proof of the consistency of Cichoń's maximum. We further show that any iteration constructed with our theory preserves strong unbounded families and what we call anti-Bendixson families, which play a central role in preserving witnesses of $\mathrm{cov}(\mathcal{N})$ of singular size (even of countable cofinality). We also show that our iteration method does not increase $\mathrm{non}(\mathcal{E})$, the smallest size of a set of reals that cannot be covered by an $F_σ$ measure zero set. Finally, we apply our theory to prove a new separation of the left-hand side of Cichoń's diagram where $\mathrm{cov}(\mathcal{N})$ is possibly singular, even with countable cofinality.

math.LO↗

arXiv2024

The intersection number for forcing notions

Based on works of Saharon Shelah, Jakob Kellner, and Anda Tănasie for controlling the cardinal characteristics of the continuum in ccc forcing extensions, in the author's master's thesis was introduced a new combinatorial notion: the intersection number for forcing notions, which was used in such thesis to build a general theory of iterated forcing using finitely additive measures. In this paper, we present the definition of such a notion and prove some of its fundamental properties in detail. Additionally, we introduce a new linkedness property called $μ$-intersection-linked, prove some of its basic properties, and provide some interesting examples.

math.LO↗

arXiv2023

Which came first, set theory or logic?

The construction of first-order logic and set theory gives rise to apparent circularities of mutual dependence, making it unclear which can act as a self-contained starting point in the foundation of mathematics. In this paper, we carry out a metatheoretic and finitistic development of a first-order logical system in which we define formal set theory (ZFC). We contend that the techniques employed in constructing this system offer a philosophically sound basis for introducing ZFC and proceeding with the conventional formal development of mathematics. Lastly, by examining the chronology of the aforementioned construction, we attempt to answer the titular question.

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