Martin Doležal — Page 2

arXiv2015

Haar meager sets revisited

In the present article we investigate Darji's notion of Haar meager sets from several directions. We consider alternative definitions and show that some of them are equivalent to the original one, while others fail to produce interesting notions. We define Haar meager sets in nonabelian Polish groups and show that many results, including the facts that Haar meager sets are meager and form a $σ$-ideal, are valid in the more general setting as well. The article provides various examples distinguishing Haar meager sets from Haar null sets, including decomposition theorems for some subclasses of Polish groups. As a corollary we obtain, for example, that $\mathbb Z^ω$, $\mathbb R^ω$ or any Banach space can be decomposed into a Haar meager set and a Haar null set. We also establish the stability of non-Haar meagerness under Cartesian product.

math.GN↗

arXiv2014

Classification of the spaces $C_p^*(X)$ within the Borel-Wadge hierarchy for a projective space $X$

We study the complexity of the space $C^*_p(X)$ of bounded continuous functions with the topology of pointwise convergence. We are allowed to use descriptive set theoretical methods, since for a separable metrizable space $X$, the measurable space of Borel sets in $C^*_p(X)$ (and also in the space $C_p(X)$ of all continuous functions) is known to be isomorphic to a subspace of a standard Borel space. It was proved by A. Andretta and A. Marcone that if $X$ is a $σ$-compact metrizable space, then the measurable spaces $C_p(X)$ and $C^*_p(X)$ are standard Borel and if $X$ is a metrizable analytic space which is not $σ$-compact then the spaces of continuous functions are Borel-$Π^1_1$-complete. They also determined under the assumption of projective determinacy (PD) the complexity of $C_p(X)$ for any projective space $X$ and asked whether a similar result holds for $C^*_p(X)$. We provide a positive answer, i.e. assuming PD we prove, that if $n \geq 2$ and if $X$ is a separable metrizable space which is in $Σ^1_n$ but not in $Σ^1_{n-1}$ then the measurable space $C^*_p(X)$ is Borel-$Π^1_n$-complete. This completes under the assumption of PD the classification of Borel-Wadge complexity of $C^*_p(X)$ for $X$ projective.

math.FA↗