Related papers: Idempotent probability measures, I

Consider a measurable space with a finite vector measure. This measure defines a mapping of the $\sigma$-field into a Euclidean space. According to Lyapunov's convexity theorem, the range of this mapping is compact and, if the measure is…

In recent times, there has been a growing interest in a structuralist understanding of probability, measure and integration theory. The present thesis contributes to this programme in three ways. First, we construct a commutative…

We investigate certain geometric properties of the spaces of idempotent measures. In particular, we prove that the space of idempotent measures on an infinite compact metric space is homeomorphic to the Hilbert cube.

Let $(X,\mu)$ be a standard probability space. An automorphism $T$ of $(X,\mu)$ has the weak Pinsker property if for every $\varepsilon > 0$ it has a splitting into a direct product of a Bernoulli shift and an automorphism of entropy less…

We consider a system of weak* closed sets of finite-dimensional distributions. We show that a corresponding system of random variables can be defined on a probability space with a probability measure determined up to some set of measures,…

We show that the Continuum Hypothesis implies that for every $0<d_1\leq d_2<n$ the measure spaces $(\RR^n,\iM_{\iH^{d_1}},\iH^{d_1})$ and $(\RR^n,\iM_{\iH^{d_2}},\iH^{d_2})$ are isomorphic, where $\iH^d$ is $d$-dimensional Hausdorff measure…

The category of Hilbert modules may be interpreted as a naive quantum field theory over a base space. Open subsets of the base space are recovered as idempotent subunits, which form a meet-semilattice in any firm braided monoidal category.…

The main purpose of this article is to provide a common generalization of the notions of a topological and Kolmogorov-Sinai entropy for arbitrary representations of discrete amenable groups on objects of (abstract) categories. This is…

Given a compact metric space (X,d) equipped with a non-atomic, probability measure m and a real, positive decreasing function p we consider a `natural' class of limsup subsets La(p) of X. The classical limsup sets of `well approximable'…

The notion of $\ast$-measure on a compact Hausdorff space can be defined for arbitrary continuous triangular norm $\ast$. The well-known Hutchinson-Barnsley theory deals with the iterated function systems (IFSs) of probability measures and…

In this paper, we give precise mathematical form to the idea of a structure whose data and axioms are faithfully represented by a graphical calculus; some prominent examples are operads, polycategories, properads, and PROPs. Building on the…

We characterize the (sequentially) weak and strong closure of planar diffeomorphisms in the Sobolev topology and we show that they always coincide. We also provide some sufficient condition for a planar map to be approximable by…

We work with the *-product introduced in [GHK25] and f-generic types to describe the minimal ideals of invariant types and to classify ergodic Keisler measures in amenable NIP theories. Moreover, we analyze the situation around the…

We introduce a category Prob of probability spaces whose objects are all probability spaces and arrows are corresponding to measurable functions satisfying an absolutely continuous requirement. We can consider a Prob-arrow as an evolving…

Let $k \ge 1$ be an integer and $f$ a holomorphic endomorphism of $\mathbb P^k (\mathbb C)$ of algebraic degree $d\geq 2$. We introduce a volume dimension for ergodic $f$-invariant probability measures with strictly positive Lyapunov…

Given any compact connected manifold $M$, we describe $C^2$-open sets of iterated functions systems (IFS's) admitting fully-supported ergodic measures whose Lyapunov exponents along $M$ are all zero. Moreover, these measures are…

Under the assumption of a natural subadditive potential, the so called cylinder function, working on the symbol space we prove the existence of the ergodic invariant probability measure satisfying the equilibrium state. As an application we…

Topological measures and deficient topological measures generalize Borel measures and correspond to certain non-linear functionals. We study integration with respect to deficient topological measures on locally compact spaces. Such an…

We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of "self-similarity" under the operation of re-scaling, the dimension of linear images of the measure behaves in a semi-continuous way. We…

We consider capacity (fuzzy measure, non-additive probability) on a compactum as a monotone cooperative normed game. We introduce topological analogues of well known class of exact games and show that these classes form subfunctors of the…