Search Results

Brownian motion, i.e. Wiener measure, is a good source of ideas and examples here. For instance if $W_t$ is 1-dimensional standard Brownian motion at time $t$ and $$P(\forall x\,F(x)=x^2)=1$$ and $Y=W …

The process $X$ you mention is uniformly continuous on the rationals* in the compact interval $[0,n]$, with probability 1. So you define Brownian motion $B$ to be the unique continuous extension of: $ …

No, let $\mathcal S_0=\{\{0,1\},\emptyset\}$ and let $\mathcal S_1$ be the powerset of $\{0,1\}$. Let $\mathcal F_n=\mathcal S_0^{n-1}\times\mathcal S_1 \times\mathcal S_0^{\infty}$. Let us write $X …

So apparently the Krylov-Bogoliubov theorem says that every continuous function $f:X\to X$ on a compact metrizable space $X$ has an invariant probability measure $\mu$. Of course, if $X$ is just a si …

$U$ is a $G_\delta $ set, hence Borel at level 2 of the Borel hierarchy. It has Lebesgue measure 1 and is also comeager, so it is large both in the sense of measure and of Baire category.

I suppose you can just randomly perturb the independent distribution. That is, let $$\mu(x_i,y_j)=\mu_x(x_i)\mu_y(y_j)+\epsilon_{i,j}$$ where the $\epsilon_{i,j}$ form a random matrix (sufficiently ra …

Let's say we start a 1-dimensional Brownian motion somewhere in the unit interval $[0,1]$, and look at where it hits the middle-third Cantor set. Certain points have higher probability than others. I …

According to Mattila, Geometry of sets and measures in Euclidean spaces, p. 168, the Fourier dimension $\text{dim}_F(A)$ of $A\subseteq \mathbb R^n$ is the unique number in $[0,n]$ such that for any $ …

For $p=\infty$, there is a best possible lower bound of $$\inf_{|s| = n} \alpha_p(s)=\infty$$ since you can change the maximum of a (continuous, say) function without changing its integrals much. Tha …

They're the same, $\mathcal G_t=\mathcal F_t$. Indeed, suppose $A\in\mathcal G_t$. So in particular $A\in\bigcap_{n=1}^\infty(\mathcal F_{t+1/n}\vee\mathcal N)$. Note that for any $\sigma$-algebra $ …

It's not a $\pi $-system. For a counter example let $ A=\{0, 1\} $, $ B=\{2, 3\}$, $ u=v=1$. Show there is no $ C $ and $ w $ with $$ A_u\cap B_v= C_w $$ by first finding out what $ w $ would have to …

Just want to observe that if $X$ is $\mathcal N(0,1)$ and $Y$ has any distribution with a strictly increasing* cdf $F_Y$ such that $F_Y^{-1} \circ F_X$ is computable (in whatever model of computabilit …

Yes, and this is a big part of why Dynkin systems are of interest. $\pi$-systems, on the other hand, have a simpler definition, but $\mu(A\cap B)$ is not determined by $\mu(A)$ and $\mu(B)$.

No, it is not true. It suffices to show that the complement is not necessarily Borel. Let $B=X\setminus A$, which is a general Borel set. We have $$ \{x:H_0x\not\subseteq A\} = \{x:H_0x\cap B\ne\varn …

To answer the second part (Can $b$ be chosen so that it depends only on $a$ and not the function $f$): No, it cannot. First note that there exist infinitely many upper density one subsets $S_n\subse …