Continuum-wise expansive measures

The principal result of this paper is the introduction of a readily verifiable sufficient condition for the existence of an invariant measure on transient Markov chains. Specifically, subject to mild additional regularity conditions, it is enough to check that for every compact set $A$ in the state space $X$, there exists a compact $C \supset A$ such that \begin{equation*}\tag{0.1} \lim_{y\in F, y\rightarrow\infty} \frac{P(y, A)}{P(y, C)} = 0\quad\text{if} \bar{F} \text{is not compact and} F = \{y: P(y, A) > 0\},\end{equation*} $P(y, A)$ denote the 1-step transition probabilities of the Markov chain. (0.1) is readily implied by simple conditions on the generating functional of the offspring distribution (Theorems 2.1 and 2.2) for discrete time multi-type Markov processes. These conditions together with some additional regularity-hypotheses are verified for (i) A 1-dimensional neutron branching model of Harris, (ii) Discrete-time age dependent branching processes, (iii) Galton-Watson processes with immigration. A generalization (to discrete-time temporally homogeneous Markov processes with $\sigma$-compact metric state space) of a condition of T. Harris (1957) for existence of invariant measures on transient Markov chains is also given. This condition is, unfortunately, difficult to check in specific examples. The sufficient condition (0.1) involves only 1-step transition probabilities, as opposed to the $n$-step transitions incorporated in the Harris condition, which is also necessary. Throughout this paper, all subsets of a topological space considered are assumed Borel measurable. The closure, interior, boundary and complement of a set $A$ are denoted by $\bar{A}, A^0, \partial A$ and $A^c$ respectively, and $I_A$ denotes the indicator function of $A$. The sets of positive integers, nonnegative integers, are denoted by $I$ and $I_0$ respectively. For any $\sigma$-compact metric space $X$, the following notations are used: $M(X) =$ set of regular measures (Borel measures finite on compact sets) on $X$. $B(X) =$ set of bounded measurable functions on $X$ with sup norm. $B'(X) = \{s \in B(X): \|s\| < 1\}$. $C(X) =$ set of bounded continuous functions on $X$. $C_{00}(X) = \{s\in C(X): s$ has compact support$\}$. $C_0(X) = \{s \in C(X):\lim_{x\rightarrow\infty}s(x) = 0\}.$ $C'(X) = C(X) \cap B'(X)$, similarly for $C_0'(X)$ and $C'_{00}(X)$. For any $\mathscr{L} \subset B(X), \mathscr{L}^+ = \{s \in \mathscr{L}: s \geqq 0\}$. If $\mu \in M(X)$ and $\in B(X)$, then $\mu f = \int_xf(x)\mu(dx)$ when defined.

Let m be a Radon measure on a Hausdorff topological space X.Corresponding to three kinds of outer measures, three kinds of m-negligible sets are considered.The main theorem states that in a metacompact space X each locally m-negligible set is m-negligible.For a Radon measure in a HausdorfT topological space X we distinguish three kinds of negligible sets corresponding to three kinds of outer measures: strictly negligible, negligible and locally negligible sets, respectively.It is shown that two classes of negligible sets coincide iff the corresponding outer measures coincide.Further, each locally negligible set is strictly negligible if X is Lindelf.We give two examples that both in locally compact separable spaces and in locally compact metrizable spaces a negligible set need not be strictly negligible.It is an unsolved problem whether a locally negligible set is always negligible, or equivalently, whether the corresponding outer measures are always the same.In this latter form the problem is due to Schwartz [5, p. 17].The main purpose of this note is to prove that in a metacompact space, and hence in a paracompact space, each locally negligible set is negligible.It will turn out that this is even true for compact inner regular Borel measures which are only locally a-finite.At least in this slightly more general situation it can be shown that the assumption " metacompact" cannot be replaced by "locally compact".By the compact inner regularity of Radon measures the necessity to distinguish several kinds of negligible sets arises with the infinity of the measures.That is, for finite, compact inner regular Borel measures the three kinds of negligible sets coincide.In the case of finite Borel measures (not necessarily compact inner regular) there are at most two kinds of negligible sets.The class of topological spaces in which these two kinds coincide for each finite or each finite regular Borel measure was considered by Gardner in [3] (see also [4]).Throughout this paper X is a Hausdorff topological space.(X), \}(X) and Si(X) denote, respectively, the families of all open, closed and compact subsets of X.The Borel a-algebra is denoted by %5(X).A Borel measure in X, i.e. a nonnegative, countably additive set function on 93 (X), is called locally finite if each point in X has a neighbourhood of finite measure.

In this paper, we primarily introduce the concepts of random measure-theoretic N-expansivity and random measure-theoretic cw-expansivity for a Borel probability measure, as well as the notions of random N-expansivity and random cw-expansivity for a random dynamical system. Subsequently, we prove that a random dynamical system is randomly N-expansive (res., randomly cw-expansive) if and only if every Borel probability measure is randomly measure-theoretic N-expansive (res., randomly measure-theoretic cw-expansive) of f and all Borel probability measures share a common random measure-theoretic N-expansive characteristic (res., random measure-theoretic cw-expansive characteristic), namely, there is a measure random variable δ : Ω → ( 0 , + ∞ ) such that δ is a random measure-theoretic N-expansive characteristic (res., random measure-theoretic cw-expansive characteristic) of every Borel probability measure.

Optimization approaches to quadrature: New characterizations of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions

Admissible Representations of Probability Measures

Using the notion of topologically stable points, it is proved that every equicontinuous pointwise topologically stable homeomorphism of a compact metric space is persistent. Also, using the notion of strong topologically stable points of a Borel probability measure, it is shown that every pointwise strong topologically stable Borel probability measure with respect to an equicontinuous homeomorphism of a compact metric space is strong persistent. Further, it is established that any homeomorphism of as well as that of (0,1) does not admit any uniformly expansive point. Finally, these results are used to show that the unit circle does not admit any expansive homeomorphism.

Chapter 24 - Geometric Measure Theory: Selected Concepts, Results and Problems

A subset X of a Polish group G is called Haar null if there exist a Borel set B ⊃ X and Borel probability measure μ on G such that μ(gBh) = 0 for every g; h ∊ G. We prove that there exist a set X ⊂ R that is not Lebesgue null and a Borel probability measure μ such that μ (X + t) = 0 for every t ∊ R. This answers a question from David Fremlin’s problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set B. (The answer was already known assuming the Continuum Hypothesis.)This result motivates the following Baire category analogue. It is consistent with ZFC that there exist an abelian Polish group G and a Cantor set C ⊂ G such that for every non-meagre set X ⊂ G there exists a t ∊ G such that C ∩ (X + t) is relatively non-meagre in C. This essentially generalizes results of Bartoszyński and Burke–Miller.

Expansive measures for flows

In this paper, given a homeomorphism f of a compact metric space X, we show that the set of all asymptotic average shadowable points of f is an open and invariant set and f has the asymptotic average shadowing property if and only if the set of all asymptotic average shadowable points of f is X if and only if any Borel probability measure μ of X has the asymptotic average shadowing property.

Radon, Baire, and Borel measures on compact spaces. I

Borel probability measures living on metric spaces are fundamental mathematical objects. There are several meaningful distance functions that make the collection of the probability measures living on a certain space a metric space. We are interested in the description of the structure of the isometries of such metric spaces. We overview some of the recent results of the topic and we also provide some new ones concerning the Wasserstein distance. More specifically, we consider the space of all Borel probability measures on the unit sphere of a Euclidean space endowed with the Wasserstein metric $W_p$ for arbitrary $p \geq 1,$ and we show that the action of a Wasserstein isometry on the set of the Dirac measures is induced by an isometry of the underlying unit sphere.

The well-known Cramér-Wold theorem states that a Borel probability measure on $\\mathbb{R}^d$ is uniquely determined by the totality of its one-dimensional projections. In this paper we examine various conditions under which a probability measure is determined by a subset of its $(d - 1)$-dimensional orthogonal projections.

We develop measure theory in the context of subsystems of second order arithmetic with restricted induction. We introduce a combinatorial principleWWKL (weak-weak Konig's lemma) and prove that it is strictly weaker thanWKL (weak Konig's lemma). We show thatWWKL is equivalent to a formal version of the statement that Lebesgue measure is countably additive on open sets. We also show thatWWKL is equivalent to a formal version of the statement that any Borel measure on a compact metric space is countably additive on open sets.

We introduce a new potential-theoretic definition of the dimension spectrum of a probability measure for q > 1 and explain its relation to prior definitions. We apply this definition to prove that if and is a Borel probability measure with compact support in , then under almost every linear transformation from to , the q-dimension of the image of is ; in particular, the q-dimension of is preserved provided . We also present results on the preservation of information dimension and pointwise dimension. Finally, for and q > 2 we give examples for which is not preserved by any linear transformation into . All results for typical linear transformations are also proved for typical (in the sense of prevalence) continuously differentiable functions.