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Given a probability space (Ω, \( \mathcal{F} \), P), where Ω denotes a non-empty set, \( \mathcal{F} \) a σ-algebra and P a probability measure on Ω: A random variable (r.v.) is a function X : Ω → ℝ such that {ω ∈ Ω : X (ω) ∈ A} ∈ F for any Borel set A in ℝ. A stochastic process is a family (X t ) t≥0 of random variables defined on (Ω, \( \mathcal{F} \), P). The stochastic process (X t ) t≥0 is said to be a discrete-time stochastic process if t takes values in ℕ and a continuous-time stochastic process if t takes values in ℝ+. A filtration on Ω is a family \( \left( {\mathcal{F}_t } \right)_{t \geqslant 0} \) of σ-algebras on Ω such that \( \mathcal{F}_u \subseteq \mathcal{F}_v \) for any u ≤ v.

Let X = {X(t) ∈ ℝd,t ∈ ℝN} be a centered space-time anisotropic Gaussian random field whose components are independent and satisfy some mild conditions. We study the packing dimension of range X(E) under the anisotropic (time variable) metric space (ℝN,ρ) and (space variable) metric space (ℝd, τ), where E ⊂ ℝN is a Borel set. Our results generalize the corresponding results of Estrade, Wu and Xiao (Commun. Stoch. Anal., 5, 41–64 (2011)) for time-anisotropic Gaussian random fields to space-time anisotropic Gaussian fields.

This paper addresses an H2 optimal control problem for a class of discrete-time stochastic systems with Markov jump parameter and multiplicative noises. The involved Markov jump parameter is a uniform ergodic Markov chain taking values in a Borel-measurable set. In the presence of exogenous white noise disturbance, Gramian characterization is derived for the H2 norm, which quantifies the stationary variance of output response for the considered systems. Moreover, under the condition that full information of the system state is accessible to measurement, an H2 dynamic optimal control problem is shown to be solved by a zero-order stabilizing feedback controller, which can be represented in terms of the stabilizing solution to a set of coupled stochastic algebraic Riccati equations. Finally, an iterative algorithm is provided to get the approximate solution of the obtained Riccati equations, and a numerical example illustrates the effectiveness of the proposed algorithm.

Hjorth (Variations on Scott, 1998; The fine structure and Borel complexity of orbits, 2010) introduced a Scott analysis for general Polish group actions, and asked whether his notion of rank satisfies a boundedness principle similar to the one of Scott rank—namely, if the orbit equivalence relation is Borel, then Hjorth ranks are bounded. We answer Hjorth’s question positively. As a corollary we prove the following conjecture of Hjorth—for every limit ordinal $$\alpha $$ , the set of elements whose orbit is of complexity less than $$\alpha $$ is a Borel set.

We study games of length ω2 with moves in ℕ and Borel payoff. These are, e.g., games in which two players alternate turns playing digits to produce a real number in [0, 1] infinitely many times, after which the winner is decided in terms of the sequence belonging to a Borel set in the product space [0,1]ℕ. The main theorem is that Borel games of length ω2 are determined if, and only if, for every countable ordinal α, there is a fine-structural, countably iterable model of Zermelo set theory with α-many iterated powersets above a limit of Woodin cardinals.

Motivated by mathematical tissue growth modelling, we consider the problem of approximating the dynamics of multicolor P\'olya urn processes that start with large numbers of balls of different colors and run for a long time. Using strong approximation theorems for empirical and quantile processes, we establish Gaussian process approximations for the P\'olya urn processes. The approximating processes are sums of a multivariate Brownian motion process and an independent linear drift with a random Gaussian coefficient. Which of the two terms dominates depends on the ratio of the number of time steps $n$ to the initial number of balls $N$ in the urn. We also establish an upper bound of the form $c(n^{-1/2}+N^{-1/2})$ for the maximum deviation over the class of convex Borel sets of the step $n$ urn composition distribution from the approximating normal law., Comment: 16 pages

We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than $1$, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime. In particular, we prove that if $A$ has Hausdorff dimension $>1$, then the set of distances spanned by points of $A$ has Hausdorff dimension at least $40/57 > 0.7$ and there are many $y\in A$ such that the pinned distance set $\{ |x-y|:x\in A\}$ has Hausdorff dimension at least $29/42$ and lower box-counting dimension at least $40/57$. We use the approach and many results from the earlier work of Keleti and Shmerkin, but incorporate estimates from the recent work of Guth, Iosevich, Ou and Wang as additional input., 21 pages, no figures

A subsetXof a Polish groupGisHaar nullif there exists a Borel probability measure μ and a Borel setBcontainingXsuch that μ(gBh) = 0 for everyg,h∈G. A setXisHaar meagerif there exists a compact metric spaceK, a continuous functionf:K→Gand a Borel setBcontainingXsuch thatf−1(gBh) is meager inKfor everyg,h∈G. We calculate (inZFC) the four cardinal invariants (add, cov, non, cof) of these two σ-ideals for the simplest non-locally compact Polish group, namely in the case$G = \mathbb {Z}^\omega$. In fact, most results work for separable Banach spaces as well, and many results work for Polish groups admitting a two-sided invariant metric. This answers a question of the first named author and Vidnyánszky.

For any $x\in\mathbb{R}^d$, $d\geq 2$, denote $\pi^x: \mathbb{R}^d\backslash\{x\}\rightarrow S^{d-1}$ as the radial projection $$\pi^x(y)=\frac{y-x}{|y-x|}. $$ Given a Borel set $E\subset{\Bbb R}^d$, $\dim_{\mathcal{H}} E\leq d-1$, in this paper we investigate for how many $x\in \mathbb{R}^d$ the radial projection $\pi^x$ preserves the Hausdorff dimension of $E$, namely whether $\dim_{\mathcal{H}}\pi^x(E)=\dim_{\mathcal{H}} E$. We develop a general framework to link $\pi^x(E)$, $x\in F$ and $\pi^y(F)$, $y\in E$, for any Borel set $F\subset\mathbb{R}^d$. In particular, whether $\dim_{\mathcal{H}}\pi^x(E)=\dim_{\mathcal{H}}E$ for some $x\in F$ can be reduced to whether $F$ is visible from some $y\in E$ (i.e. $\mathcal{H}^{d-1}(\pi^y(F))>0$). This allows us to apply Orponen's estimate on visibility to obtain $$\dim_{\mathcal{H}}\left\{x\in\mathbb{R}^d: \dim_{\mathcal{H}}\pi^x(E), Comment: Conjecture 1.2 is added

A set $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ is universal for countable subsets of ${\mathbb {R}}$ if and only if for all $x \in {\mathbb {R}}$ , the section $U_x = \{y \in {\mathbb {R}} : U(x,y)\}$ is countable and for all countable sets $A \subseteq {\mathbb {R}}$ , there is an $x \in {\mathbb {R}}$ so that $U_x = A$ . Define the equivalence relation $E_U$ on ${\mathbb {R}}$ by $x_0 \ E_U \ x_1$ if and only if $U_{x_0} = U_{x_1}$ , which is the equivalence of codes for countable sets of reals according to U. The Friedman–Stanley jump, $=^+$ , of the equality relation takes the form $E_{U^*}$ where $U^*$ is the most natural Borel set that is universal for countable sets. The main result is that $=^+$ and $E_U$ for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets, $E_U$ is Borel bireducible to $=^+$ . If one assumes a particular instance of $\mathbf {\Sigma }_3^1$ -generic absoluteness, then for all $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ that are $\mathbf {\Sigma }_1^1$ (continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of $=^+$ into $E_U$ .

In this paper, we follow and extend a group-theoretic method introduced by Greenleaf–Iosevich–Liu–Palsson (GILP) to study finite points configurations spanned by Borel sets in $\mathbb{R}^n,n\geq 2,n\in\mathbb{N}.$ We remove a technical continuity condition in a GILP’s theorem in [Revista Mat. Iberoamer31 (2015), 799–810]. This allows us to extend the Wolff–Erdogan dimension bound for distance sets to finite points configurations with k points for $k\in\{2,\dots,n+1\}$ forming a $(k-1)$ -simplex.

Considered are perfect information games with a Borel measurable payoff function that is parameterized by points of a Polish space. The existence domain of such a parameterized game is the set of parameters for which the game admits a subgame perfect equilibrium. We show that the existence domain of a parameterized stopping game is a Borel set. In general, however, the existence domain of a parameterized game need not be Borel, or even an analytic or co-analytic set. We show that the family of existence domains coincides with the family of game projections of Borel sets. Consequently, we obtain an upper bound on the set-theoretic complexity of the existence domains, and show that the bound is tight.

Mauldin [15] proved that there is an analytic set, which cannot be represented by $B\cup X$ for some Borel set B and a subset X of a $\boldsymbol{\Sigma }^0_2$ -null set, answering a question by Johnson [10]. We reprove Mauldin’s answer by a recursion-theoretical method. We also give a characterization of the Borel generated $\sigma $ -ideals having approximation property under the assumption that every real is constructible, answering Mauldin’s question raised in [15].

We study injective versions of the characterization of sets potentially in a Wadge class of Borel sets, for the first Borel and Lavrentieff classes. We also study the case of oriented graphs in terms of continuous homomorphisms, injective or not.

We prove Fuglede’s conjecture in $$\mathbb {Q}_p$$ which states that a Borel set of positive and finite Haar measure in $$\mathbb {Q}_{p}$$ is a spectral set if and only if it tiles $$\mathbb {Q}_p$$ by translations.

On the Cantor cube { 0 , 1 } N with the standard product topology we construct a finite Choquet capacity with respect to the family of all compact sets such that every compact set of positive capacity contains continuum many pairwise disjoint compact subsets of positive capacity.

We prove that it is true in Sacks, Cohen, and Solovay generic extensions that any ordinal definable Borel set of reals necessarily contains an ordinal definable element. This result has previously been known only for countable sets.

Any free Borel flow is shown to admit a cross section with only two possible distances between adjacent points. Non smooth flows are proved to be Lebesgue orbit equivalent if and only if they admit the same number of invariant ergodic probability measures., Comment: Minor improvements in exposition

In the paper we introduce and study a new family of “small” sets which is tightly connected with two well known σ-ideals: of Haar-null sets and of Haar-meager sets. We define a subset A of a topological group X to be null-finite if there exists a convergent sequence (xn)n∈ω in X such that for every x ∈ X the set {n ∈ ω : x + xn ∈ A} is finite. We prove that each null-finite Borel set in a complete metric Abelian group is Haar-null and Haar-meager. The Borel restriction in the above result is essential as each non-discrete metric Abelian group is the union of two null-finite sets. Applying null-finite sets to the theory of functional equations and inequalities, we prove that a mid-point convex function f : G → ℝ defined on an open convex subset G of a metric linear space X is continuous if it is upper bounded on a subset B which is not null-finite and whose closure is contained in G. This gives an alternative short proof of a known generalization of the Bernstein–Doetsch theorem (saying that a mid-point convex function f: G → ℝ defined on an open convex subset G of a metric linear space X is continuous if it is upper bounded on a non-empty open subset B of G). Since Borel Haar-finite sets are Haar-meager and Haar-null, we conclude that a mid-point convex function f: G → ℝ defined on an open convex subset G of a complete linear metric space X is continuous if it is upper bounded on a Borel subset B ⊂ G which is not Haar-null or not Haar-meager in X. The last result resolves an old problem in the theory of functional equations and inequalities posed by Baron and Ger in 1983.

The literature provides dichotomies involving homomorphisms (like the G 0 dichotomy) or reductions (like the characterization of sets potentially in a Wadge class of Borel sets, which holds on a subset of a product). However, part of the motivation behind the latter result was to get reductions on the whole product, like in the classical notion of Borel reducibility considered in the study of analytic equivalence relations. This is not possible in general. We show that, under some acyclicity (and also topological) assumptions, this is widely possible. In particular, we prove that, for any non-self dual Borel class Γ, there is a concrete finite ⊑ c-antichain basis for the class of Borel relations, whose closure has acyclic symmetrization, and which are not potentially in Γ. Along similar lines, we provide a sufficient condition for ⊑ c-reducing G 0. We also prove a similar result giving a minimum set instead of an antichain if we allow rectangular reductions.

Necessary and sufficient condition governing the boundedness of the multilinear fractional integral operator $$T_{\gamma , \mu }$$ defined with respect to a measure $$\mu $$ on a $$\sigma $$-algebra of Borel sets of quasi-metric space X from the product $$L^{p_1}(X, \mu )\times \cdots \times L^{p_m}(X, \mu )$$ to $$L^q(X, \mu )$$ is established. The related weak type inequality is also obtained. The derived results are used to get appropriate boundedness of $$T_{\gamma , \mu }$$ in Morrey spaces defined with respect to a measure $$\mu $$.

Given a countable Borel equivalence relation, I introduce an invariant measuring how difficult it is to find Borel sets separating its equivalence classes. I evaluate these invariants in several standard generic extensions.

We provide a method for constructing (possibly non-trivial) measures on non-locally compact Polish subspaces of infinite-dimensional separable Banach spaces which, under suitable assumptions, are minimizers of causal variational principles in the non-locally compact setting. Moreover, for non-trivial minimizers the corresponding Euler-Lagrange equations are derived. The method is to exhaust the underlying Banach space by finite-dimensional subspaces and to prove existence of minimizers of the causal variational principle restricted to these finite-dimensional subsets of the Polish space under suitable assumptions on the Lagrangian. This gives rise to a corresponding sequence of minimizers. Restricting the resulting sequence to countably many compact subsets of the Polish space, by considering the resulting diagonal sequence we are able to construct a regular measure on the Borel algebra over the whole topological space. For continuous Lagrangians of bounded range it can be shown that, under suitable assumptions, the obtained measure is a (possibly non-trivial) minimizer under variations of compact support. Under additional assumptions, we prove that the constructed measure is a minimizer under variations of finite volume and solves the corresponding Euler-Lagrange equations. Afterwards, we extend our results to continuous Lagrangians vanishing in entropy. Finally, assuming that the obtained measure is locally finite, topological properties of spacetime are worked out and a connection to dimension theory is established., Comment: 46 pages

We give an example, in the infinite dimensional separable Hilbert space, of a purely unrectifiable Borel set with finite nonzero one dimensional Hausdorff measure, whose projection is nonnegligible in a set of directions which is not Aronszajn null.

We prove the first bifurcation result of time quasi-periodic traveling waves solutions for space periodic water waves with vorticity. In particular we prove existence of small amplitude time quasi-periodic solutions of the gravity-capillary water waves equations with constant vorticity, for a bidimensional fluid over a flat bottom delimited by a space-periodic free interface. These quasi-periodic solutions exist for all the values of depth, gravity and vorticity, and restricting the surface tension to a Borel set of asymptotically full Lebesgue measure., 87 pages

We establish a connection between random set theory and Boolean valued analysis by showing that random Borel sets, random Borel functions, and Markov kernels are respectively represented by Borel sets, Borel functions, and Borel probability measures in a Boolean valued model. This enables a Boolean valued transfer principle to obtain random set analogues of available theorems. As an application, we establish a Boolean valued transfer principle for large deviations theory, which allows for the systematic interpretation of results in large deviations theory as versions for Markov kernels. By means of this method, we prove versions of Varadhan and Bryc theorems, and a conditional version of Cram&eacute, r theorem.

We provide a Rademacher theorem for intrinsically Lipschitz functions $\phi:U\subseteq \mathbb W\to \mathbb L$, where $U$ is a Borel set, $\mathbb W$ and $\mathbb L$ are complementary subgroups of a Carnot group, where we require that $\mathbb L$ is a normal subgroup. Our hypotheses are satisfied for example when $\mathbb W$ is a horizontal subgroup. Moreover, we provide an area formula for this class of intrinsically Lipschitz functions., Comment: New version. We dropped one hypothesis in Theorem 1.1. We added Remark 2.1

We prove that the existence of a Borel lower density operator (a Borel lifting) with respect to the $$\sigma $$σ-ideal of countable sets, for an uncountable Polish space, is equivalent to CH. One of the implications is known (due to K. Musiał) and the remaining implication is derived from a general abstract result dealing with the negation of GCH. We observe that there is no lower density Borel operator with respect to the $$\sigma $$σ-ideal of countable sets, whose range is of bounded level in the Borel hierarchy.