The Baire Category of Subsequences and Permutations which preserve Limit Points

Let $$\mathcal {I}$$ I be a meager ideal on $$\mathbf {N}$$ N . We show that if x is a sequence with values in a separable metric space then the set of subsequences [resp. permutations] of x which preserve the set of $$\mathcal {I}$$ I -cluster points of x is topologically large if and only if every ordinary limit point of x is also an $$\mathcal {I}$$ I -cluster point of x. The analogue statement fails for all maximal ideals. This extends the main results in [Topology Appl. 263 (2019), 221–229]. As an application, if x is a sequence with values in a first countable compact space which is $$\mathcal {I}$$ I -convergent to $$\ell $$ ℓ , then the set of subsequences [resp. permutations] which are $$\mathcal {I}$$ I -convergent to $$\ell $$ ℓ is topologically large if and only if x is convergent to $$\ell $$ ℓ in the ordinary sense. Analogous results hold for $$\mathcal {I}$$ I -limit points, provided $$\mathcal {I}$$ I is an analytic P-ideal.

Remarks Connected with the Weak Limit of Iterates of Some Random-Valued Functions and Iterative Functional Equations

The paper consists of two parts. At first, assuming that (Ω, A, P) is a probability space and (X, ϱ) is a complete and separable metric space with the σ-algebra 𝒝 of all its Borel subsets we consider the set 𝒭c of all 𝒝 ⊗ 𝒜-measurable and contractive in mean functions f : X × Ω → X with finite integral ∫ Ωϱ (f(x, ω), x) P (dω) for x ∈ X, the weak limit π f of the sequence of iterates of f ∈ 𝒭c, and investigate continuity-like property of the function f ↦ πf, f ∈ 𝒭c, and Lipschitz solutions φ that take values in a separable Banach space of the equation\varphi \left( x \right) = \int_\Omega {\varphi \left( {f\left( {x,\omega } \right)} \right)P\left( {d\omega } \right)} + F\left( x \right).Next, assuming that X is a real separable Hilbert space, Λ: X → X is linear and continuous with ||Λ || < 1, and µ is a probability Borel measure on X with finite first moment we examine continuous at zero solutions φ : X → 𝔺 of the equation\varphi \left( x \right) = \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \mu } \left( x \right)\varphi \left( {\Lambda x} \right)which characterizes the limit distribution π f for some special f ∈ 𝒭c.

ON THE WEAK-HASH METRIC FOR BOUNDEDLY FINITE INTEGER-VALUED MEASURES

It is known that the space of boundedly finite integer-valued measures on a complete separable metric space becomes a complete separable metric space when endowed with the weak-hash metric. It is also known that convergence under this topology can be characterised in a way that is similar to the weak convergence of totally finite measures. However, the original proofs of these two fundamental results assume that a certain term is monotonic, which is not the case as we show by a counterexample. We clarify these original proofs by addressing the parts that rely on this assumption and finding alternative arguments.

A Universal Separable Diversity

The Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any finite set of points is assigned a non-negative value, extending the notion of a metric which only applies to unordered pairs of points.We construct the unique separable complete diversity that it is ultrahomogeneous and universal with respect to separable diversities.

On computability and disintegration

We show that the disintegration operator on a complete separable metric space along a projection map, restricted to measures for which there is a unique continuous disintegration, is strongly Weihrauch equivalent to the limit operator Lim. When a measure does not have a unique continuous disintegration, we may still obtain a disintegration when some basis of continuity sets has the Vitali covering property with respect to the measure; the disintegration, however, may depend on the choice of sets. We show that, when the basis is computable, the resulting disintegration is strongly Weihrauch reducible to Lim, and further exhibit a single distribution realizing this upper bound.

Further results on images of metric spaces

We introduce the notion of an ls-?-Ponomarev-system to give necessary and sufficient conditions for f:(M,M0) ? X to be a strong wc-mapping (wc-mapping, wk-mapping) where M is a locally separable metric space. Then, we systematically get characterizations of weakly continuous strong wc-images (wc-images, wk-images) of locally separable metric spaces by means of certain networks. Also, we give counterexamples to sharpen some results on images of locally separable metric spaces in the literature.

A note on a Residual Subset of Lipschitz Functions on Metric Spaces

Let (X, d) be a quasi-convex, complete and separable metric space with reference probability measure m. We prove that the set of real-valued Lipschitz functions with non-zero pointwise Lipschitz constant m-almost everywhere is residual, and hence dense, in the Banach space of Lipschitz and bounded functions. The result is the metric analogous to a result proved for real-valued Lipschitz maps defined on ℝ2 by Alberti et al.

Distance Sets of Urysohn Metric Spaces

A metric space M = (M; d) is homogeneous if for every isometry f of a finite subspace of M to a subspace of M there exists an isometry of M onto M extending f . The space M is universal if it isometrically embeds every finite metric space F with dist(F) ⊆ dist(M) (with dist(M) being the set of distances between points in M).A metric space U is a Urysohn metric space if it is homogeneous, universal, separable, and complete. (We deduce as a corollary that a Urysohn metric space U isometrically embeds every separable metric space M with dist(M) ⊆ dist(U).)The main results are: (1) A characterization of the sets dist(U) for Urysohn metric spaces U. (2) If R is the distance set of a Urysohn metric space and M and N are two metric spaces, of any cardinality with distances in R, then they amalgamate disjointly to a metric space with distances in R. (3) The completion of every homogeneous, universal, separable metric space M is homogeneous.