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See Corollary 2.8 in this paper: If $X$ is perfect, compact and metrizable, then there is a non-atomic regular Borel measure of full support on $X$.

Attempt number 2. Consider the case $f\ge 0$. For $\alpha\in[0,1]$ let $A_{\alpha}=\{x\in X, f(x)\ge \alpha\}$, which is measurable. For $n\in \mathbb{N}$ define $f_n=\frac{1}{n}\sum_{k=1}^{n}\chi_{ …

I'll rename your $\Omega$ into $X$ to simplify typing. Let $D=\{x\in X,~\forall n\in\mathbb{N}:~ f(\tau^n(x))=f(x)\}=\bigcap\limits_{n\in\mathbb{N}_0} \{x\in X: f(\tau^{n+1}(x))=f(\tau^n(x))\}$ $=\bi …

Let $G$ be a domain in $\mathbb{C}^{d}$ and let $H\left(G\right)$ be the space of all holomorphic functions on $G$. My question is how to characterize all such (Radon) measures $\mu$ on $G$, that $\i …

I'll assume that $X$ is Hausdorff. A Hausdorff topological vector space is metrizable iff it is first countable. Hence, $X$ is metrizable and has a dense subset $\{x_n\}_{n=1}^{\infty}$. Define $f_n( …

Assume that $B$ is a complete boolean algebra endowed with a Hausdorff topology, with respect to which all operations on $B$ are continuous, $0$ has a base of full sets (recall that $A\subset B$ is fu …

This is the proof for the case when $E$ is bounded and Jordan measurable, but not necessarily closed. Also the proof seems to work for any $C^1$ map between manifolds. We know that $\partial E$ is a …