Related papers: Discrete probability spaces revisited

In this paper we describe a theory of a cumulative distribution function on a space with an order from a probability measure defined in this space. This distribution function plays a similar role to that played in the classical case.…

We prove that every Lindel\"of scattered subspace of a $\Sigma$-product of first-countable spaces is $\sigma$-compact. In particular, we obtain the result stated in the title. This answers some questions of Tkachuk from [Houston J. Math. 48…

We study homeomorphisms of compact metric spaces whose restriction to the nonwandering set has the pseudo-orbit tracing property. We prove that if there are positively expansive measures, then the topological entropy is positive. Some short…

We introduce the $\Sigma_1$-definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, (i) the sequence is $\Sigma_1$-definable…

We prove the sharp bound for the probability that two experts who have access to different information, represented by different $\sigma$-fields, will give radically different estimates of the probability of an event. This is relevant when…

Let (X,d) be a metric space and (\Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of \Omega. Loosely speaking, these consist of points in \Omega…

In this paper we give a generalization of the discrete complex-valued random variable defined and investigated in \cite{ssa} and \cite{m8}. We prove the statements concerning the expressions for the excepted value and the variance of this…

In this work, we define the notion of unimodular random measured metric spaces as a common generalization of various other notions. This includes the discrete cases like unimodular graphs and stationary point processes, as well as the…

Given any closed Riemannian manifold $M$, we construct a reversible diffusion process on the space ${\mathcal P}(M)$ of probability measures on $M$ that is (i) reversible w.r.t.~the entropic measure ${\mathbb P}^\beta$ on ${\mathcal P}(M)$,…

In this paper we establish that the functor of idempotent probability measures acting in the category of compacta and their continuous mappings is perfect metrizable.

Topology may be interpreted as the study of verifiability, where opens correspond to semi-decidable properties. In this paper we make a distinction between verifiable properties themselves and processes which carry out the verification…

Let $\Omega$ be an open set in a metric measure space $X$. Our main result gives an equivalence between the validity of a weighted Hardy-Sobolev inequality in $\Omega$ and quasiadditivity of a weighted capacity with respect to Whitney…

For a probability measure space $(X,\mathscr{A},\mu)$, we define a pseudometric $\delta$ on the ring $\mathcal{M}(X,\mathscr{A})$ of real-valued measurable functions on $X$ as $\delta(f,g)=\mu(X\setminus Z(f-g))$ and denote the topological…

We consider the differential entropy of probability measures absolutely continuous with respect to a given $\sigma$-finite reference measure on an arbitrary measurable space. We state the asymptotic equipartition property in this general…

We study the values taken by the Riemann zeta-function $\zeta$ on discrete sets. We show that infinite vertical arithmetic progressions are uniquely determined by the values of $\zeta$ taken on this set. Moreover, we prove a joint discrete…

We consider a sequence of identically independently distributed random samples from an absolutely continuous probability measure in one dimension with unbounded density. We establish a new rate of convergence of the $\infty-$Wasserstein…

In this paper, we address the problem of constructing a uniform probability measure on $\mathbb{N}$. Of course, this is not possible within the bounds of the Kolmogorov axioms and we have to violate at least one axiom. We define a…

In this work, we study convergence in probability and almost sure convergence for weighted partial sums of random variables that are related to the class of generalized Oppenheim expansions. It is worth noting that the random variables…

This paper explores the Bergman geometry of bounded domains $\Omega$ in $\mathbb{C}^n$ through the lens of information geometry by introducing a mapping $\Phi: \Omega \rightarrow \mathcal{P}(\Omega)$, where $\mathcal{P}(\Omega)$ denotes a…

This is an introduction to measure theory, integration and function spaces, with all the needed preliminaries included, and with some applications included as well. We first discuss some basic motivations, coming from discrete probability,…