# Aleksander Cieślak: The splitting ideal

17/06/24 20:27

Tuesday, June 18, 2024 17:15

*Location:*A.4.1 C-19*Aleksander Cieślak*

Speaker:Speaker:

*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*Jadwiga Świerczyńska*

Speaker:Speaker:

*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

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*Andres Uribe-Zapata (TU Wien)*

Speaker:Speaker:

*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.

# Tomasz Żuchowski: The Nikodym property and filters on \(\omega\). Part II

22/04/24 08:20

Tuesday, April 23, 2024 17:15

In this talk we will study the family \(\mathcal{AN}\) of such ideals \(\mathcal{I}\) on \(\omega\) that the space \(N_{\mathcal{I}^*}\) carries a sequence \(\langle\mu_n\colon n\in\omega\rangle\) of finitely supported signed measures satisfying \(\|\mu_n\|\rightarrow\infty\) and \(\mu_n(A)\rightarrow 0\) for every \(A\in Clopen(N_{\mathcal{I}^*})\). If \(\mathcal{I}\in\mathcal{AN}\) and \(N_{\mathcal{I}^*}\) 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.

*Location:*A.4.1 C-19*Tomasz Żuchowski*

Speaker:Speaker:

*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 \(N_{\mathcal{I}^*}\) carries a sequence \(\langle\mu_n\colon n\in\omega\rangle\) of finitely supported signed measures satisfying \(\|\mu_n\|\rightarrow\infty\) and \(\mu_n(A)\rightarrow 0\) for every \(A\in Clopen(N_{\mathcal{I}^*})\). If \(\mathcal{I}\in\mathcal{AN}\) and \(N_{\mathcal{I}^*}\) 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.

# Krzysztof Zakrzewski: Function spaces on Corson-like compacta

13/04/24 07:08

Tuesday, April 16, 2024 17:15

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.

*Location:*A.4.1 C-19*Krzysztof Zakrzewski (MIM UW)*

Speaker:Speaker:

*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 \(R^{\Gamma}\) consisting of sequences with finitely many nonzero coordinates for some set \(\Gamma\).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.