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Claim: For every $f \in L^1[0,1]$, the set $A_f$ is convex. Proof: Let $\mu_1=\mu$ be Lebesgue measure on $[0,1]$ and consider the signed measures $\mu_2,\mu_3$ on $[0,1]$ defined using the real and …

In fact the correct conclusion is $ \, \dim(A \times A) \ge 2p$, while the reverse inequality only holds under additional assumptions (like equality of the Hausdorff and packing dimensions of $A$). …

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 …

A measurable function $f$ taking values in $[-M,M]$ can be approximated above and below by simple functions: $f_n \le f \le f_n+1/n$ where $f_n(x)=\lfloor nf(x) \rfloor/n$. Note that $f_n$ takes on le …

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 …

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 …

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 …

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 …

The Champernowne constant $C_2$ ( see https://en.wikipedia.org/wiki/Champernowne_constant ) has the stronger property of normality (see https://en.wikipedia.org/wiki/Normal_number#Properties) for pr …

The function you propose is related to the L'evy concentration function, studied by Kolmogorov, Rogozin, Esseen and others. See the special volume [1] https://link.springer.com/chapter/10.1007/97 …

Here is an example for question 2. Given $\varepsilon>0$, consider $X=[-1,0] \cup (\varepsilon,1+\varepsilon]$ with the usual metric. Then $A=[-1,0]$ is closed and open in $X$, and $ A^\varepsilon =A …

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 …

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 …

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. …