Related papers: On the range of a vector measure

We prove that if an analytic subset $A$ of a linear metric space $X$ is not contained in a $\sigma Z_\omega$-subset of $X$ then for every Polish convex set $K$ with dense affine hull in $X$ the sum $A+K$ is non-meager in $X$ and the sets…

We prove that for every $n\in \mathbb{N}$ there exists a metric space $(X,d_X)$, an $n$-point subset $S\subseteq X$, a Banach space $(Z,\|\cdot\|_Z)$ and a $1$-Lipschitz function $f:S\to Z$ such that the Lipschitz constant of every function…

In this paper the weak topology on a normed space is studied from the viewpoint of infinite-dimensional topology. Besides the weak topology on a normed space $X$ (coinciding with the topology of uniform convergence on finite subsets of the…

We prove that the sequence of cones of metric measure spaces converges if the sequence of base spaces converges in Gromov's box, concentration, and weak topologies. As an application, we show that the generalized Cauchy distribution with…

In this paper, the notion of semi-compact perturbation of a closed linear subspace is introduced. Then for a of pair of closed linear subspace of a Banach space such that one is a semi-compact perturbation of the other, it is proved that…

It is shown that given a metric space $X$ and a $\sigma$-finite positive regular Borel measure $\mu$ on $X$, there exists a bounded continuous real-valued function on $X$ that is one-to-one on the complement of a set of $\mu$ measure zero.

We provide a framework for proofs of structural theorems about sets with positive Banach logarithmic density. For example, we prove that if $A\subseteq \mathbb{N}$ has positive Banach logarithmic density, then $A$ contains an approximate…

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of…

Let $\pi:X\to Y$ be a factor map, where $(X,\sigma_X)$ and $(Y,\sigma_Y)$ are subshifts over finite alphabets. Assume that $X$ satisfies weak specification. Let $\ba=(a_1,a_2)\in \R^2$ with $a_1>0$ and $a_2\geq 0$. Let $f$ be a continuous…

We prove density results for neural-network classes on compact sets \(K\subset X\), where \(X\) is a topological vector space whose continuous dual \(X^*\) separates points. Let \(\Psi:\mathbb R\to\mathbb R\) be a continuous squashing…

We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence of diameter 2 properties. We prove that the strong diameter 2 property prevents the bidual from being strictly convex and being smooth, and we…

Let vphi:C rightarrow C be a bilipschitz map. We prove that if E\subset\C is compact, and gamma(E), alpha(E) stand for its analytic and continuous analytic capacity respectively, then C^{-1}\gamma(E)\leq \gamma(\vphi(E)) \leq C\gamma(E) and…

Let $X$ and $Y$ be separable Banach spaces. Suppose $Y$ either has a shrinking basis or $Y$ is isomorphic to $C(2^\mathbb{N})$ and $A$ is a subset of weakly compact operators from $X$ to $Y$ which is analytic in the strong operator…

We introduce the {\em $\mu$-topological stability}. This is a type of stability depending on the measure $\mu$ different from the set-valued approach \cite{lm}. We prove that the map $f$ is $m_p$-topologically stable if and only if $p$ is a…

A Banach space (or its norm) is said to have the diameter $2$ property (D$2$P in short) if every nonempty relatively weakly open subset of its closed unit ball has diameter $2$. We construct an equivalent norm on $L_1[0,1]$ which is weakly…

We prove that if $K$ is a compact space and the space $P(K\times K)$ of regular probability measures on $K\times K$ has countable tightness in its $weak^*$ topology, then $L_1(\mu)$ is separable for every $\mu\in P(K)$. It has been known…

We study Riemannian manifolds $(M^n,g)$ with mean-convex boundary whose Ricci curvature is nonnegative in a spectral sense. Our first main result is a sharp spectral extension of a rigidity theorem by Kasue: we prove that under the…

Given a family $\mathcal{C}$ of closed bounded convex sets in a Banach space $X$, we say that $X$ has the $\mathcal{C}$-MIP if every $C \in \mathcal{C}$ is the intersection of the closed balls containing it. In this paper, we introduce a…

A uniformly continuously integrable sequence of real-valued measurable functions, defined on some probability space, is relatively compact in the $\sigma(L^1,L^\infty)$ topology. In this paper, we link such a result to weak convergence…

Given any asymptotically flat 3-manifold $(M,g)$ with smooth, non-empty, compact boundary $\Sigma$, the conformal conjecture states that for every $\delta>0$, there exists a metric $g' = u^4 g$, with $u$ a harmonic function, such that the…