Rosenthal families, pavings and generic cardinal invariants

Following D. Sobota we call a family $\mathcal F$ of infinite subsets of $\mathbb N$ a Rosenthal family if it can replace the family of all infinite subsets of $\mathbb N$ in classical Rosenthal's Lemma concerning sequences of measures on pairwise disjoint sets. We resolve two problems on Rosenthal families: every ultrafilter is a Rosenthal family and the minimal size of a Rosenthal family is exactly equal to the reaping cardinal $\mathfrak r$. This is achieved through analyzing nowhere reaping families of subsets of $\mathbb N$ and through applying a paving lemma which is a consequence of a paving lemma concerning linear operators on $\ell_1^n$ due to Bourgain. We use connections of the above results with free set results for functions on $\mathbb N$ and with linear operators on $c_0$ to determine the values of several other derived cardinal invariants.Comment: Modified titl

Convergence of measures in forcing extensions

We prove that if $\mathcal{A}$ is a $\sigma$-complete Boolean algebra in a model $V$ of set theory and $\mathbb{P}\in V$ is a proper forcing with the Laver property preserving the ground model reals non-meager, then every pointwise convergent sequence of measures on $\mathcal{A}$ in a $\mathbb{P}$-generic extension $V[G]$ is weakly convergent, i.e. $\mathcal{A}$ has the Vitali--Hahn--Saks property in $V[G]$. This yields a consistent example of a whole class of infinite Boolean algebras with this property and of cardinality strictly smaller than the dominating number $\mathfrak{d}$. We also obtain a new consistent situation in which there exists an Efimov space.Comment: 22 page

Small cardinals and small Efimov spaces

We introduce and analyze a new cardinal characteristic of the continuum, the \emph{splitting number of the reals}, denoted $\mathfrak{s}(\mathbb R)$. This number is connected to Efimov's problem, which asks whether every infinite compact Hausdorff space must contain either a non-trivial convergent sequence, or else a copy of $\beta \mathbb N$

The Josefson--Nissenzweig theorem, Grothendieck property, and finitely supported measures on compact spaces

The celebrated Josefson-Nissenzweig theorem implies that for a Banach space $C(K)$ of continuous real-valued functions on an infinite compact space $K$ there exists a sequence of Radon measures $\langle\mu_n\colon\ n\in\omega\rangle$on$K$which is weakly* convergent to the zero measure on$K$and such that$\big\|\mu_n\big\|=1$for every$n\in\omega$. We call such a sequence of measures \textit{a Josefson-Nissenzweig sequence}. In this paper we study the situation when the space$K$admits a Josefson-Nissenzweig sequence of measures such that its every element has finite support. We prove among the others that$K$admits such a Josefson-Nissenzweig sequence if and only if$C(K)$does not have the Grothendieck property restricted to functionals from the space$\ell_1(K)$. We also investigate miscellaneous analytic and topological properties of finitely supported Josefson-Nissenzweig sequences on general Tychonoff spaces. We prove that various properties of compact spaces guarantee the existence of finitely supported Josefson-Nissenzweig sequences. One such property is, e.g., that a compact space can be represented as the limit of an inverse system of compact spaces based on simple extensions. An immediate consequence of this result is that many classical consistent examples of Efimov spaces, i.e. spaces being counterexamples to the famous Efimov problem, admit such sequences of measures. Similarly, we show that if$K$and$L$are infinite compact spaces, then their product$K\times L$always admits a finitely supported Josefson--Nissenzweig sequence. As a corollary we obtain a constructive proof that the space$C_p(K\times L)$contains a complemented copy of the space$c_0$endowed with the pointwise topology--this generalizes results of Cembranos and Freniche. Finally, we provide a direct proof of the Josefson-Nissenzweig theorem for the case of Banach spaces$C(K)$.Comment: 57 pages, comments are welcom