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Let $\epsilon>0$ and $n \ge 1$. Then $$\bigcap_{k=1}^\infty\{|S_{n+k}-S_n|<\epsilon = \bigcap_{j \geq n}\{|S_j-S_n|<\epsilon\} \subset \bigcap_{j,k\geq n}\{|S_j-S_k|<2\epsilon\} .$$ Hence, taking comp …

The convergence in $L^2$ of the variation ratios does not yield necessarily a pointwise convergence. For example, consider the case where $f(t,x) = \mathbb{1}_{x = 1/t - \lfloor 1/t \rfloor}$ for all …

The notation $L^2$ for Lebesgue measure is confusing. I denote $\lambda$ and $\lambda_2$ the Lebesgue measure on $\mathbb{R}$ and $\mathbb{R}^2$, respectively. The answer is no. Fix a discrete measure …

I replace $\alpha$ by $x$ below. Restricting ourselves to positive numbers is useless, since the property considered does not depend on $a_0(x) = \lfloor x \rfloor$. Therefore, $B$ is the union of all …

Once we have convergence $F_n \to F$ in measure (this is the case in your situation), the natural condition for convergence in $L^1$ is uniform integrability. This is equivalent to convergence $||F_n| …

For $f \in H$, call $Mf$ the mean value of $f$ on $\mathbb{S}_1$. Let $\phi : \mathbb{R} \to \mathbb{R}$ be any non-Borel function. Call $\mathbb{1} \in H$ the constant function equal to $1$ everywher …

Partial answer For every $a \in [0,1]$, $(\mu_n([0,a]))_{n \ge 0}$ is still a non-negative martingale, hence it converges almost surely to some random variable $L_a$ with values in $[0,+\infty]$. One …

In dimension 2, we have the Schramm and Loewner evolutions, very nice processes which are invariant by conformal maps (up to time-changes). https://en.wikipedia.org/wiki/Schramm%E2%80%93Loewner_evolut …

ADDENDUM. Vokram told me that I answered another question, not his question. Therefore, I give another counterexample (not so different from the previous one) disproving the equality. Choose a functio …

The integral with regard do $\mathrm{d}M^*$ is a pathwise Stieltjès integral, so the question is an analysis problem. Let $f : \mathbb{R}_+ \to \mathbb{R}$ be any continuous function, $F$ its current …

The notations are contradictory. Once $p$ is fixed, and then it varies. Do you set $\mu_{n}:=\frac{1}{n}\sum_{i=0}^{n-1}T_{\ast}^{i}\nu$ for ALL $n \ge 1$ and assume that $T_{\ast}^{p} \nu = \nu$ for …