New answers tagged measure-theory
6
votes
Accepted
Do invariant open sets generate the $\sigma$-algebra of invariant sets?
This is false already for any irrational rotation of the circle, meaning with $G = \mathbb{Z}$ and $X = S^1$:
Using the fact that every orbit is dense, it is easy to see that the only invariant open ...
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2
votes
Is "the purely probabilistic version of Freiling's axiom of symmetry" disprovable in ZFC?
In any model of ZFC where there is no such set for $c=1$ (e.g., Friedman's model mentioned in Gro-Tsen's answer), there is no $S \subset [0,1]^2$ such that all vertical slices $S_x$ are null and all ...
- 3,747
4
votes
Accepted
Is this theorem true in the case of a general measure space?
$\newcommand{\ep}{\varepsilon}\newcommand{\R}{\mathbb R}\newcommand{\N}{\mathbb N}$The answer is yes, the $f_n$'s are uniformly integrable wrt to $\mu$.
Indeed, let us follow the proof of Theorem 4.5....
- 110k
2
votes
Accepted
Metropolis-Hastings kernel in measure theory
You asked another question that seems somewhat likely to be closed for not being research-level mathematics. I will include my posted answer to that here because that one may get closed.
One does not ...
- 11.4k
3
votes
Accepted
Probabilty measures that are both discrete and continuous
A discrete probability distribution is one made up entirely of point masses; i.e. there is some set $S_1\subseteq S$ for which, for every $s\in S_1,$ the probability $P\{s\}$ assigned to the set $\{s\}...
- 11.4k
5
votes
Probabilty measures that are both discrete and continuous
Your notion of “has a density with respect to another measure” is essentially the notion of “absolute continuity”. Not all measures are absolutely continuous with respect to another, though one can ...
- 1,729
3
votes
Accepted
Is $\sup_{f\in \mathcal{F}}\left|\int _Xfg \, d\mu\right|<\infty$ true for all $g\in L^\infty _\mathbb{C}(\mu )$?
$\newcommand\vpi\varphi\newcommand\F{\mathcal F}$Suppose that $|g|\le C$ $\mu$-almost everywhere for some real $C>0$. Let $f_1:=\Re f\,1(\Re f>0)$, $f_2:=\Re f\,1(-\Re f>0)$, $f_3:=\Im f\,1(\...
- 110k
8
votes
Accepted
Is "the purely probabilistic version of Freiling's axiom of symmetry" disprovable in ZFC?
For $c=1$ the answer is negative (that is, the negation of the statement you give is consistent with ZFC). Indeed, Harvey Friedman's paper “A Consistent Fubini-Tonelli Theorem for Nonmeasurable ...
- 28.4k
6
votes
Is "the purely probabilistic version of Freiling's axiom of symmetry" disprovable in ZFC?
This is not a full answer, but let me just point out that the statement is relatively consistent both with CH and also with $\neg$CH.
CH implies the statement, since we can take $S$ to be the (...
- 219k
24
votes
Accepted
Writing a function on $\mathbb{R}$ as a sum of two injections
The answer is yes. Every function on the reals is the sum of two injective functions, and this can be done in a highly effective manner, constructing the two functions $g,h$ from $f$ without any need ...
- 219k
0
votes
Lebesgue points of a function is not affected by multiplication of the integrand with a smooth function?
Yes, that's a null set.
Pick a sequence of continuous functions $f_n: S^1 \times \mathbb R \to \mathbb{R}$ such that $\|f_n-f\|_1\leq 4^{-n}.$ Let $g_n(t)=\int_{S^1}|f(x,t)-f_n(x,t)|dx.$ Let $Z=\...
- 400
13
votes
Writing a function on $\mathbb{R}$ as a sum of two injections
It works at least for (locally) absolutely continuous functions.
Such a function is the integral of a locally $L^1$ function.
This weak derivative can be written as a sum of a positive and negative ...
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5
votes
Accepted
A nonlinear mapping on $L^2(S^1)$ that commutes with all translation operators is necessarily measurable?
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 $1$ ...
- 3,146
0
votes
Accepted
A complex question related to a certain convergence of Lévy measures
At first, we consider an example.
Let
\begin{gather*}
f(x)=\frac{I_{\{x>0\}}(x)}{2x^2(1\vee x^2)}
=\frac{I_{\{(0,1)\}}(x)}{2x^2} + \frac{I_{\{[1,\infty)\}}(x)}{2x^4},\\
\nu(\mathrm{d}x)=...
- 724
0
votes
Proof of the Dunford-Pettis theorem in the context of probability spaces
You might have a look at the first volume of Probabilités et Potentiel (translated as Probability and Potentials) by C. Dellacherie and P.-A. Meyer; Dunford-Pettis is discussed in II.25.
- 1,714
2
votes
When do the weak-star and compact convergence (compact-open) topology coincide on the dual of a Banach space?
Weak$^*$-convergence of a net means pointwise convergence on singletons, and therefore (uniform) convergence finite sets. Uniform convergence on compact subsets means exactly that. Since finite sets ...
- 41
0
votes
Steinhaus theorem and Hausdorff dimension
In the paper by Feng and Wu, they introduced a notion of thickness, that generalized the Newhouse thickness in the line. They showed that if a set has positive Feng-Wu thickness, then the sumset after ...
2
votes
Accepted
Computation of tangent functional
$\newcommand{\om}{\omega}\newcommand{\Om}{\Omega} \newcommand{\de}{\delta}\renewcommand{\th}{\theta}$For each real $t>0$ and some $\om_t\in\Om$,
\begin{equation*}
\|x+ty\|
=|(x+ty)(\om_t)|=|...
- 110k
2
votes
Computation of tangent functional
The idea is that, for $|x(\omega)|\sim 1$ and $t$ small, $$|x(\omega)+ty(\omega)|\sim|x(\omega)^2+tx(\omega)y(\omega)|\sim 1+tx(\omega)y(\omega)$$
Indeed, if $|x(\omega)|<1-\varepsilon$, then for ...
- 354
2
votes
Accepted
Gateaux differentiability of the norm in Banach spaces
We have the following definitions:
\begin{equation*}
V^*:=\{f\in S^*\colon\exists x\in S\ f(x)=1\},
\end{equation*}
where $S$ and $S^*$ are the unit spheres in the underlying Banach space $E$ and ...
- 110k
2
votes
Accepted
Potentially elementary question on affine functions on Banach spaces
A function $f$ from a linear (or, more generally, affine) space $A$ is affine (see e.g. this post) if for all $a$ and $b$ in $A$ and all real $t$ one has $f((1-t)a+tb)=(1-t)f(a)+tf(b)$. If $f$ is ...
- 110k
6
votes
What is an "open Baire set"?
The function $\varphi$ in H.-J. is also assumed to be lower semi-continous, which is why $\{a<\varphi\}$ is open. This set is a Baire set because $\varphi$ is a Baire function.
- 1,714
3
votes
Gently changing measure
To question 1, there is such a pair. This is a minor reworking of Ashutosh's example you linked.
Start with $L,$ add an $\omega_1$-sequence of random reals $X=\langle r_{\alpha}: \alpha<\omega_1 \...
- 3,747
3
votes
Does convolution with $(1+|x|)^{-n}$ define an operator $L^p(\mathbb R^n) \to L^p(\mathbb R^n)$
No. E.g., let $n=1$ and
$$f(u)=\frac{1(u>0)}{(1+u)^{1/2}\ln(2+u)}$$
for real $u$. Then, letting $I_f(x)$ denote the integral in question and letting $x\to\infty$, we have
$$I_f(x)\ge\int_0^x\frac{...
- 110k
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