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I would suggest the following axioms. The length in a metric space $X$ is a function $\ell:c(X)\to[0,+\infty]$ defined on the family $c(X)$ of all connected compact subsets of $X$ that satisfies the …

The affirmative answer to this question is given by the following theorem, proved by the technique of continuous selections. Theorem. For any compact metrizable space $X$ there exists a continuous …

A Borel subset $B$ of the unit interval $\mathbb I=(0,1)$ is defined to be a density neighborhood of a set $A\subseteq\mathbb I$ if for every $a\in A$ we have $$\lim_{\varepsilon\to0}\frac{\lambda(B\c …

Professor Wladyslaw Wilczynski kindly informed me that the answer to this problem is negative. Take any Lebesgue null dense $G_\delta$-set $A$ in the real line $\mathbb R$. Choose a countable dense su …

For a compact space $Y$ the answer is affirmative, but in general case of Polish space $Y$ it is negative. Results yielding nice selections of relations $R$ are known in Descriptive Set Theory as Un …

At least under MA+$\neg$CH the answer is negative. It is known that under MA+$\neg$CH the real line contains an uncountable set A such that every subset of A is Borel in A (moreover, it is of type $F_ …

Let $X$ be a finite ultrametric space and $P(X)$ be the space of probability measures on $X$ endowed with the Wasserstein-Kantorovich-Rubinstein metric (briefly WKR-metric) defined by the formula $$\r …

In general the answer to this problem is negative: if the measure $\mu$ has connected support and the measure $\nu$ has disconnected support, then for any continuous map $f:P\to P$ the measure $f_\# \ …

This is a very good (and also well studied) question, especially for homeomorphisms of measures. For example, the Haar measures on the zero-dimensional compact groups $\mathbb Z_2^\omega$ and $\mathbb …

Problem. Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap …