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.

Tags: Aleksander Cieślak

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 ${\ell }^{\mathrm{\infty }}$.

Tags: Jadwiga Świerczyńska

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.

Tags: Andres Uribe-Zapata

Tuesday, April 23, 2024 17:15

Location: A.4.1 C-19

Speaker: Tomasz Żuchowski

Title: The Nikodym property and filters on $\omega$. Part II

Abstract: For a free filter $F$ on $\omega$, we consider the space $N_F=\omega \cup \{p_F\}$, where every element of $\omega$ is isolated and open neighborhoods of $p_F$ are of the form $A\cup \{p_F\}$ for $A\in F$.
In this talk we will study the family $\mathcal{AN}$ of such ideals $\mathcal{I}$ on $\omega$ that the space $N_{{\mathcal{I}}^{\ast }}$ carries a sequence $⟨{\mu }_n:n\in \omega ⟩$ of finitely supported signed measures satisfying $\Vert {\mu }_n\Vert \to \mathrm{\infty }$ and ${\mu }_n(A)\to 0$ for every $A\in Clopen(N_{{\mathcal{I}}^{\ast }})$. If $\mathcal{I}\in \mathcal{AN}$ and $N_{{\mathcal{I}}^{\ast }}$ is embeddable into the Stone space $St(\mathcal{A})$ of a given Boolean algebra $\mathcal{A}$, then $\mathcal{A}$ does not have the Nikodym property.

Tags: Tomasz Żuchowski

Tuesday, April 16, 2024 17:15

Location: A.4.1 C-19

Speaker: Krzysztof Zakrzewski (MIM UW)

Title: Function spaces on Corson-like compacta

Abstract: Recall that a compact space is Eberlein compact if it is homeomorphic to a subspace of some Banach space equipped with the weak topology. A compact space is $\omega$-Corson compact if it embeds into a $\sigma$-product of real lines, that is a subspace of the product $R^{\mathrm{\Gamma }}$ consisting of sequences with finitely many nonzero coordinates for some set $\mathrm{\Gamma }$.
Every $\omega$-Corson compact space is Eberlein compact. For a Tichonoff space $X$, let $C_p(X)$ denote the space of real continuous functions on $X$ endowed with the pointwise convergence topology.
During the talk we will show that the class $\omega$-Corson compact spaces $K$ is invariant under linear homeomorphism of function spaces $C_p(K)$ and other related results.

Tags: Krzysztof Zakrzewski