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One reason is that probabilists often consider more than one measure on the same space, and then a negligible set for one measure (added in a completion) might be not negligible for the other. The sit …

The answer is negative: It is possible that there is no good choice of $i,j$. Let $T$ be a uniform spanning tree in the infinite ladder ${\bf Z} \times \{0,1\}$. To be precise, this is a weak limit o …

Yes, the local time (at zero) maps the zero set of Brownian motion to an interval. See e.g. Lemma 6.9 page 159 in [1] for continuity. [1] Brownian motion, by Peter Mörters and Yuval Peres. Cambridge U …

$N(p)$ can grow arbitrarily quickly. Given a sequence $a_m \downarrow 0$ with $a_0=1$ and $a_m<a_{m-1}/2$ for all $m$, define $f(x)=x^{-1/m}$ for all $x \in (a_m,a_{m-1}]$ and $m \ge 1$. Then $N(p)< …

With respect to the norm topology, the space of Radon measures on a domain $\Omega$ is not separable. Indeed, for any two distinct points $x,y$ in $\Omega$, the Dirac measures $\delta_x$ and $\delta_ …

There is$^*$ a counterexample in the atomic case, see below, so we will assume that $(\Omega, \mathrm{P})$ is a non-atomic standard Lebesgue probability space (so it is Isomorphic to the unit interval …

It is true and clear if the metric space $X$ has a finite diameter, but false in general: Take $\beta_i=2^{-i}$ and $\mu_i$ the point mass at $3^i$. Details: In the case $D=$diam$(X)<\infty$, write $s …

We assume that $f$ is not identically zero, whence $f*g(x)>0$ for all $x \in \mathbb{R}^d$. The answer is positive, and this is also true when the density f is replaced by an arbitrary positive fini …

False for p infinite. True for finite. See e.g. the book by Krengel, Ergodic theorems. Other sources (that also go further) are [1, Sec. I.2.1] or [2, Theorem 8.8]. [1] T. Eisner, Stability of opera …

The answer is negative. Take $E={\mathbb R}$ and $\Omega={\mathbb R}$, with $\cal A$ the Lebesgue (or Borel) $\sigma$-algebra. Let $V$ be a subset of ${\mathbb R}$ which is not $\cal A$-measurabl …

For given $\alpha$ the corresponding set of nontangential boundary points is a $G_{\delta}$ set, since the cone must contain points of $S$ in annuli arbitrarily close to the unit circle. Here $G_n$ c …

Let $f_n$ be independent random variables where $f_n$ takes values $0, 2n$ with $\mu(f_n=2n)=1/n$ for each $n$. Take $g_n=0$ for all $n$. Then $F_n(f_n)=0$ for all $n$ so the hypothesis holds, but th …

Here is an answer for the case that $X$ is countable and all its subsets are measurable. Let $Y \subset X$ be nonempty, suppose $\{p_y : y \in Y\}$ are strictly positive numbers with $p= \sum_{y \in …

Let $D$ consist of all numbers $d$ in $[0,1]$ such that in their binary expansion for every $k$ the digit at location $2^k$ vanishes, i.e., $d_{2^k}=0$. Then it is easy to check that $D$ has Lebesgue …

Let $$f_n(x) = \sup_{r \in {\mathbb Q} \cap [\frac{1}{n+1},\frac{1}{n})} \frac{|B(x,r)\setminus E|}{|B(x,r)|}\,,$$ so that $f_n(x) \to 0$ for a.e. $x \in E$. By Egorov's theorem [1], for every $\epsi …