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Yes, there are such inequalities, which follow from the Kac Lemma in Ergodic Theory. (Although the form of the inequality at the end of the post is too optimistic- did you want the supremum to be tak …

Here is another variation of the fine solution suggested by Mike and made precise by Iosif Pinelis. Let $Z$ be a polish (i.e., separable and complete) metric space with metric $d(\cdot,\cdot).$ Given …

$N(p)$ 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)< …

There are two interpretations of the OP's question. If the W_i are fixed once for all, or one is allowed to pick them afresh in each step. In The first interpretation we obtain a substitution dynamica …

The property does not hold if the measure $\mu$ is atomic, so we assume that the given standard probability space $(X, \mathcal{B}, \mu)$ is nonatomic, whence it is isomorphic to he unit interval …

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 …

The answer, in general, is negative. That is, there exist continuous maps $T:X \to X$ of the type you describe and Borel probability measures $\mu$ such that, for small $\epsilon$, the set $B_\epsilo …

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 …

Some of the relevant considerations can be found in the book [1] and the classic paper [2]. Look e.g. at Lemma 1.4.4 in [2] for $\dim^*$. Billingsley's lemma, that you can find on [1] or in [3], Lem …

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 …

The notion of Haar null sets is the most natural; indeed it was rediscovered in [1], see also [2] and the references therein. The fact that Brownian motion plus a fixed continuous function is nowhere …

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 …

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 …

See e.g. [1] for the basic definitions (Note that functions that agree $\mu$-a.e. are identified.) Given a collection $H$ of extended real-valued measurable functions with supremum $h^* in the a.e. …

The extension you want was proved in the 1990s by Bergelson and Leibman. See [1] and also further developments in [2]. [1] Bergelson, Vitaly, and Alexander Leibman. "Polynomial extensions of van der W …