Related papers: Spaces of measurable functions

In this paper, some features of countably $\alpha$-compact topological spaces are presented and proven. The connection between countably $\alpha$% -compact, Tychonoff, and $\alpha$-Hausdorff spaces is explained. The space is countably…

In this manuscript, we claim that the newly introduced $\mathcal{F}$-metric spaces are Hausdorff and also first countable. Moreover, we assert that every separable $\mathcal{F}$-metric space is second countable. Additionally, we acquire…

Let the space $\mathbb{R}^n$ be endowed with a Minkowski structure $M$ (that is $M\colon \mathbb{R}^n \to [0,+\infty)$ is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class $C^2$),…

Let $\mathcal{M} (X)$ denote the space of complete Riemannian metrics with non-positive sectional curvature and with negatively curved ends, on a manifold $X$. We show that $\mathcal{M} (\mathbb{R} \times S ^{1}) $ and $\mathcal{M}…

We study geodesically complete and locally compact Hadamard spaces X whose Tits boundary is a connected irreducible spherical building. We show that X is symmetric iff complete geodesics in X do not branch and a Euclidean building…

Let $(\Omega,\mu)$ be a $\sigma$-finite measure space. Given a Banach space $X$, let the symbol $S(X)$ stand for the unit sphere of $X$. We prove that the space $L^{\infty} (\Omega,\mu)$ of all complex-valued measurable essentially bounded…

We examine measure preserving mappings $f$ acting from a probability space $(\Omega, F,\mu) $ into a probability space $% (\Omega ^{*},F^{*},\mu ^{*}) ,$ where $\mu ^{*}=\mu (f^{-1})$. Conditions on $f$, under which $f$ preserves the…

In a previous paper the authors developed a H^1-BMO theory for unbounded metric measure spaces $(M,\rho,m)$ of infinite measure that are locally doubling and satisfy two geometric properties, called "approximate midpoint" property and…

We prove that for any topological space $X$ of countable tightness, each \sigma-convex subspace $\F$ of the space $SC_p(X)$ of scatteredly continuous real-valued functions on $X$ has network weight $nw(\F)\le nw(X)$. This implies that for a…

We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We prove that a Borel CDH space is completely metrizable and give a complete list of zero-dimensional Borel CDH spaces. We also show that for a…

Let $f\colon X\to Y$ be a perfect surjective map of metrizable spaces. It is shown that if $Y$ is a $C$-space (resp., $\dim Y\leq n$ and $\dim f\leq m$), then the function space $C(X,\uin^{\infty})$ (resp., $C(X,\uin^{2n+1+m})$) equipped…

A metric space $X$ is quasisymmetrically co-Hopfian if every quasisymmetric embedding of $X$ into itself is onto. We construct the first examples of metric spaces homeomorphic to the universal Menger curve and higher dimensional…

We show that there are homotopy equivalences $h:N\to M$ between closed manifolds which are induced by cell-like maps $p:N\to X$ and $q:M\to X$ but which are not homotopic to homeomorphisms. The phenomenon is based on construction of…

Adapting a homotopy reconstruction theorem for general metric compacta, we show that every countable metric or ultrametric compact space can be topologically reconstructed as the inverse limit of a sequence of finite $T_0$ spaces which are…

We prove that every continuous function on a separable infinite-dimensional Hilbert space X can be uniformly approximated by smooth functions with no critical points. This kind of result can be regarded as a sort of very strong approximate…

We prove that: 1. If a Hausdorff M-space is a continuous closed image of a submetrizable space, then it is metrizable. 2. A dense-in-itself open-closed image of a submetrizable space is submetrizable if and only if it is functionally…

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y),\] and set $M(X) = \sup…

We characterize all compact and Hausdorff spaces $X$ which satisfy that for every multiplicative bijection $\phi$ on $C(X, I)$, there exist a homeomorphism $\mu : X \to X$ and a continuous map $p: X \to (0, +\infty)$ such that $$\phi (f)…

We consider the space $M(X)$ of separable measures on the Borel $\sigma$-algebra ${\cal B}(X)$ of a metric space $X$. The space $M(X)$ is furnished with the Kantorovich-Rubinshte\u{i}n metric known also as the ``Hutchinson distance''. We…

Let $(M,g)$ be a Riemannian manifold with Riemannian distance $\mathsf{d}_g$, and $\mathcal{M}(M)$ be the space of all non-negative Borel measures on $M$, endowed with the Hellinger-Kantorovich distance $\mathsf{H\! K}_{\mathsf{d}_g}$…