June 2024
Aleksander Cieślak: The splitting ideal
17/06/24 20:27
Tuesday, June 18, 2024 17:15
Location: A.4.1 C-19
Speaker: Aleksander Cieślak
Title: The splitting ideal
Abstract: We will investigate the cardinal invariants and the Katetov position of certain ideal on \(\omega\). As a result we will obtain a new upper boundary of the covering number of the density zero ideal.
Location: A.4.1 C-19
Speaker: Aleksander Cieślak
Title: The splitting ideal
Abstract: We will investigate the cardinal invariants and the Katetov position of certain ideal on \(\omega\). As a result we will obtain a new upper boundary of the covering number of the density zero ideal.
Jadwiga Świerczyńska: On Q- and selective measures
12/06/24 12:16
Tuesday, June 11, 2024 17:15
Location: A.4.1 C-19
Speaker: Jadwiga Świerczyńska
Title: On Q- and selective measures
Abstract: We will present some generalizations of well-known definitions of types of ultrafilters to the realm of finitely additive measures on \(\omega\). We will show a few results similar to the ones for ultrafilters: measure is selective if and only if it is a P-measure and a Q-measure, and that selective measures (Q-measures, respectively) are minimal in the Rudin-Keisler (Rudin-Blass) ordering. We will also show an example of a selective non-atomic measure. The second part will be focused on the integration: we will briefly describe Lebesgue integral with respect to finitely additive measures on \(\omega\) and prove that it is a generalization of an ultralimit. Finally, we will present an idea of further generalizations of these definitions for functionals on \(\ell^{\infty}\).
Location: A.4.1 C-19
Speaker: Jadwiga Świerczyńska
Title: On Q- and selective measures
Abstract: We will present some generalizations of well-known definitions of types of ultrafilters to the realm of finitely additive measures on \(\omega\). We will show a few results similar to the ones for ultrafilters: measure is selective if and only if it is a P-measure and a Q-measure, and that selective measures (Q-measures, respectively) are minimal in the Rudin-Keisler (Rudin-Blass) ordering. We will also show an example of a selective non-atomic measure. The second part will be focused on the integration: we will briefly describe Lebesgue integral with respect to finitely additive measures on \(\omega\) and prove that it is a generalization of an ultralimit. Finally, we will present an idea of further generalizations of these definitions for functionals on \(\ell^{\infty}\).
Andres Uribe-Zapata: Finitely additive measures on Boolean algebras: freeness and integration
04/06/24 12:09
Tuesday, June 4, 2024 17:15
Location: A.4.1 C-19
Speaker: Andres Uribe-Zapata (TU Wien)
Title: Finitely additive measures on Boolean algebras: freeness and integration
Abstract: In this talk, we present an integration theory with respect to finitely additive measures on a field of sets \(\mathcal{B} \subseteq \mathcal(X)\) for some non-empty set \(X\). For this, we start by reviewing some fundamental properties of finitely additive measures on Boolean algebras. Later, we present a definition of the integral in this context and some basic properties of the integral and the integrability. We also study integration over subsets of \(X\) to introduce the Jordan algebra and compare the integration on this new algebra with the integration on \(\mathcal{B}\). Finally, we say that a finitely additive measure on \(\mathcal{B}\) is free if \(\mathcal{B}\) contains any finite subset of \(X\) and its measure is zero. We close the talk by providing some characterizations of free finitely additive measures.
This is a joint work with Miguel A. Cardona and Diego A. Mejía.
References:
[CMU] Miguel A. Cardona, Diego A. Mejía and Andrés F. Uribe-Zapata. Finitely additive measures on Boolean algebras. In Preparation.
[UZ23] Andrés Uribe-Zapata. Iterated forcing with finitely additive measures: applications of probability to forcing theory. Master’s thesis, Universidad Nacional de Colombia, sede Medellín, 2023. https://shorturl.at/sHY59.
Location: A.4.1 C-19
Speaker: Andres Uribe-Zapata (TU Wien)
Title: Finitely additive measures on Boolean algebras: freeness and integration
Abstract: In this talk, we present an integration theory with respect to finitely additive measures on a field of sets \(\mathcal{B} \subseteq \mathcal(X)\) for some non-empty set \(X\). For this, we start by reviewing some fundamental properties of finitely additive measures on Boolean algebras. Later, we present a definition of the integral in this context and some basic properties of the integral and the integrability. We also study integration over subsets of \(X\) to introduce the Jordan algebra and compare the integration on this new algebra with the integration on \(\mathcal{B}\). Finally, we say that a finitely additive measure on \(\mathcal{B}\) is free if \(\mathcal{B}\) contains any finite subset of \(X\) and its measure is zero. We close the talk by providing some characterizations of free finitely additive measures.
This is a joint work with Miguel A. Cardona and Diego A. Mejía.
References:
[CMU] Miguel A. Cardona, Diego A. Mejía and Andrés F. Uribe-Zapata. Finitely additive measures on Boolean algebras. In Preparation.
[UZ23] Andrés Uribe-Zapata. Iterated forcing with finitely additive measures: applications of probability to forcing theory. Master’s thesis, Universidad Nacional de Colombia, sede Medellín, 2023. https://shorturl.at/sHY59.