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Results tagged with measure-theory
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user 169474
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
1
vote
Accepted
Weak convergence of random measures generated by non-negative martingales?
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 …
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0
votes
Existence of the limit of periodic measures
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 …
- 3,808
5
votes
Accepted
A nonlinear mapping on $L^2(S^1)$ that commutes with all translation operators is necessaril...
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 …
- 3,808
4
votes
Accepted
Integral of $M^\text{*} - M$ with respect to $M^\text{*}$ is zero for $M^\text{*}$ the runni...
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 …
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2
votes
Accepted
If $f : [0,1] \to H$ has $t$-derivative with respect to the norm of $H$, and $H=L^2[0,1]$ it...
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 …
- 3,808
3
votes
Conditional expectation: commuting integration and supremum
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 …
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1
vote
Sufficient conditions for L1 convergence of exponentials
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| …
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3
votes
Accepted
Equivalence of unions in probability theory
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 …
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1
vote
Accepted
Measurability of Brjuno numbers
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 …
- 3,808
1
vote
Is there something like a "self-avoiding Markov chain" on a continuous space?
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 …
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2
votes
Accepted
On partial absolute continuity
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 …
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