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Probability maps are additive and normalised maps taking values in the unit interval of a lattice ordered Abelian group. They appear in theory of affine representations and they are also a semantic counterpart of Hajek's probability logic.…

It is well known that the space of invariant probability measures for transitive sub-shifts of finite type is a Poulsen simplex. In this article we prove that in the non-compact setting, for a large family of transitive countable Markov…

The probabilistic method is a technique for proving combinatorial existence results by means of showing that a randomly chosen object has the desired properties with positive probability. A particularly powerful probabilistic tool is the…

While there exists a well-developed asymptotic theory of Fr\'echet means of random variables taking values in a general "finite-dimensional" metric space, there are only a few known results in which the random variables can take values in…

We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all…

We discuss various infinite-dimensional configuration spaces that carry measures quasiinvariant under compactly-supported diffeomorphisms of a manifold M corresponding to a physical space. Such measures allow the construction of unitary…

For a continuous action of a countable discrete group $G$ on a Polish space $X$, a countable Borel partition $P$ of $X$ is called a generator if $G \cdot P := \{ gC : g \in G, C \in P \}$ generates the Borel $\sigma$-algebra of $X$. For $G…

We will show that, consistently, every uncountable set can be continuously mapped onto a non measure zero set, while there exists an uncountable set whose all continuous images into a Polish space are meager.

This article reviews a generous sampling of both classical and more recent results on the interplay between measurable and topological dynamics. In the first part we have surveyed the strong analogies between ergodic theory and topological…

We define and study entanglement of continuous positive definite functions on products of compact groups. We formulate and prove an infinite-dimensional analog of Horodecki Theorem, giving a necessary and sufficient criterion for…

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 $K$…

For a continuous map $T$ of a compact metrizable space $X$ with finite topological entropy, the order of accumulation of entropy of $T$ is a countable ordinal that arises in the context of entropy structure and symbolic extensions. We show…

We show from a categorical point of view that probability measures on certain measurable or topological spaces arise canonically as the extension of probability distributions on countable sets. We do this by constructing probability monads…

Let $\mathbf{\eta}$ be a positive Radon measure on the space of continuous paths from R into a locally compact Polish space $Y$, and assume that $\mathbf{\eta}$ admits an invariant measure. We explicit conditions on $\mathbf{\eta}$ such…

Construct a random set by independently selecting each finite subset of the integers with some probability depending on the set up to translations and taking the union of the selected sets. We show that when the only sets selected with…

We investigate some basic descriptive set theory for countably based completely quasi-metrizable topological spaces, which we refer to as quasi-Polish spaces. These spaces naturally generalize much of the classical descriptive set theory of…

We investigate four partial orderings on the space of quantum measurements (i.e on POVMs or positive operator valued measures), describing four notions of coarse/fine-ness of measurement. These are the partial orderings induced by: (1)…

This paper presents a constructive proof of the existence of a regular non-atomic strictly-positive measure on any second-countable non-atomic locally compact Hausdorff space. This construction involves a sequence of finitely-additive set…

We consider a transitive action of a finitely generated group $G$ and the Schreier graph $\Gamma$ defined by this action for some fixed generating set. For a probability measure $\mu$ on $G$ with a finite first moment we show that if the…

We consider the isometry group of the infinite dimensional separable hyperbolic space with its Polish topology. This topology is given by the pointwise convergence. For non-locally compact Polish groups, some striking phenomena like…