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No, those sets are not open. Indeed, take any non-isolated point $x$ in $X$ and a sequence $(x_n)_{n\in\omega}$ that converges to $x$. For every $n$, consider the sign measure $\mu_n=\delta_{x}-\delta …

Yes, there exists such a function: Consider the real line as a linear space over the field $\mathbb Q$ and find a linearly independent Cantor set $C\subseteq \mathbb R$ (using the Kuratowski-Mycielski …

The answer to this problem is negative: For the compact Polish group $X=S_3^\omega$ we have $Sn(X)\le\mathfrak r$ where $$\mathfrak r=\min\{|\mathcal R|:\mathcal R\subseteq [\omega]^\omega\;\wedge\;\f …

$\newcommand\Sn{\mathit{Sn}}$A subset $A$ of a group $X$ is called algebraic if $A=\{x\in X: a_0xa_1x\dotsm xa_n=1\}$ for some elements $a_0,a_1,\dotsc,a_n\in X$. Let $\mathcal A_X$ be the family of a …

This is a very interesting question whose answer depends on dimension properties of the spaces $X,Y$. First we introduce a suitable terminology. A function $f:X\to Y$ between topological spaces is cal …

The set $MS=\{(\mu,K)\in P_X\times K_X:\mathrm{supp}(\mu)=K\}$ is of type $G_\delta$ in $P_X\times K_X$ and hence the weak+Hausdorff topology on $P_X$ is Polish. Indeed, fix any countable base $\{U_n\ …

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 …

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_\# \ …

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 …

The Borel $\sigma$-algebras generated by these two topologies seem to be equal. The idea of the proof is as follows. Let $\mathcal M_+$ be the subspace of $\mathcal M$ consisting of measures. It is kn …

I am curious whether somebody ever tried to generalize the classical theory of Lebesgue integral to functions and measures with values in Hausdorff topological rings. The generalization of a measure i …

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 …

There exists a counterexample under the Continuum Hypothesis, which implies that the unit interval $[0,1]$ admits a well-order $\preceq$ such that for every $x\in[0,1]$ the initial interval ${\downarr …

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 …

In the Frechet space $X:=\mathbb R^\omega$ consider the dense linear subspace $$L_0:=\{(x_n)_{n\in\omega}\in\mathbb R^\omega:|\{n\in\omega:x_n\ne0\}|<\omega\}.$$ Fix a countable base $\{V_n\}_{n\in\o …