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Haar-null union of dense subsets

Let $\{X_i\}_{i \in \mathbb{R}-\{0\}}$ be a set of subsets of a separable infinite-dimensional Fréchet space $X$ and $I$ be uncountable. Moreover, suppose that (Dense $G_{\delta}$) $X_i$ is a dense ...
MrsHaar's user avatar
  • 63
9 votes
4 answers
4k views

Is the space of Radon measures a Polish space or at least separable?

Background: I work on a SPDE problem where in order to apply Prokhorov's theorem I need that some measure space is Polish space. And additionaly it would be good if that space is Banach space. Earlier ...
Mark's user avatar
  • 657
4 votes
1 answer
189 views

Intrinsic volumes of non-polyconvex, non-compact sets

I am reposting this question I asked and bountied on Math SE, which has been upvoted but not answered or commented on. The intrinsic volumes (AKA Minkowski Functionals or, with different ...
Joe Previdi's user avatar
0 votes
1 answer
133 views

Product of sets with the Radon-Nikodym Property (RNP)

I have read that it is somewhat well-known that if two Banach spaces $X$ and $Y$ have the Radon-Nikodym Property (RNP), then their product $X\times Y$ also has the RNP. Does the above result ...
BigbearZzz's user avatar
  • 1,245
12 votes
1 answer
1k views

List of all known Riesz representation theorems

Due to the history and development of measure and integration theory and different mathematical schools, there is a huge variety and inconsistency of definitions for concepts like tightness of a ...
yada's user avatar
  • 1,773
1 vote
1 answer
83 views

Reference request for (weak*) metrizability of a bounded space of signed Radon measures on a compact set

I know the following is true and I know how to prove it (cf. exercise 50 on page 171 in Folland, Theorem 7.18 in Folland), but per my adviser's instructions, it would be better to find a source to ...
Ben Ciotti's user avatar
3 votes
2 answers
410 views

Is a bounded sequence of $H^1(\Omega)$ tight?

Assume $\Omega$ is a bounded subset of $\Bbb R^d$ and $ (u_n)_n$ is a bounded sequence of the Sobolev space $H^1(\Omega)$. Question: Can we say that $ (u_n)_n$ is tight in $L^2(\Omega)$ namely: ...
Guy Fsone's user avatar
  • 1,101
3 votes
0 answers
227 views

Is there a noncommutative version of von Neumann's ergodic theorem? [closed]

The two most celebrated ergodic theorems are Birkhoff's ergodic theorem and von Neumann's ergodic theorem. E. C. Lance in his remarkable work (Ergodic Theorems for Convex Sets and Operator Algebras) ...
Neil hawking's user avatar
7 votes
1 answer
1k views

Reference request: norm topology vs. probabilist's weak topology on measures

Let $(X,d)$ be a metric space and $\mathcal{M}(X)$ be the space of regular (e.g. Radon) measures on $X$. There are two standard topologies on $\mathcal{M}(X)$: The (probabilist's) weak topology and ...
JohnA's user avatar
  • 710
6 votes
1 answer
580 views

Why is it difficult to solve the Monge problem directly?

I'm trying to understand something about the Monge problem. The Monge problem is: Let $c(x,y): \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ and $$\mathcal{T}(\mu_1,\mu_2) = \{ T: \...
yoshi's user avatar
  • 427
2 votes
1 answer
258 views

Control the oscillation of a function by its total variation

Is it possible to control the oscillation of a BV vector field $u:\mathbb R^N \to \mathbb R^N$ at a point $x_0$ by the total variation of $u$?
user avatar
6 votes
1 answer
575 views

Sub-Gaussian decay of convolution of $L^1$ function with Gaussian kernel

I think it might be helpful to put the new statement at the beginning and put the original post at the end. This new statement is more mathematically elegant. Let $f\geq0$ be in $L^1(\mathbb{R}^d)$ ...
neverevernever's user avatar
7 votes
1 answer
261 views

Comparison of several topologies for probability measures

Let $X$ be a compact metric space and denote $\mathcal M(X)$ the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\operatorname{supp} \mu$ for the support of $\mu$. As is well ...
Kass's user avatar
  • 243
2 votes
1 answer
307 views

Box counting dimension of a set and Lipschitz functions

If $f$ is Lipschitz, then the following holds for the Hausdorff dimension: $$\dim_H f(A) \le \dim_H A.$$ Is the same true for the box counting dimension?
Riku's user avatar
  • 839
3 votes
2 answers
322 views

Hausdorff dimension of the graph of the sum of two continuous functions

How can one prove the following result on the Hausdorff dimension of the graph of the sum of two continuous functions: Let $f,g:[0,1] \to \mathbb R$ be two continuous functions. Suppose that $$\...
user avatar
3 votes
1 answer
162 views

Reference Request: $L^p(x)$/(Musielak–Orlicz space) analogue of classical $L^p$ result

Fix a non-empty open domain $\Omega\subseteq \mathbb{R}^d$ with compact closure, and a finite Borel measure $\mu$ on its closure $\overline{\Omega}$. In Halmos' book it is shown that: Classical ...
ABIM's user avatar
  • 5,405
7 votes
0 answers
177 views

Does this ideal in $B(L_1)$ have a (bounded) right approximate identity?

I will take a roundabout way to defining this ideal, because (a) this route is how my collaborators and I came to it (b) this alternative definition, rather than the standard one, may suggest a ...
Yemon Choi's user avatar
  • 25.8k
4 votes
0 answers
115 views

Box counting dimension of the graph of a BV function

Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function. Is the box counting dimension of the graph of $u$ equal to $N$? How can we prove it? The analogous question for the ...
Riku's user avatar
  • 839
1 vote
2 answers
275 views

Again, proving that specific preorder on the set of measurable functions is symmetric

This question is followup to the previous similar question. There I was trying to find good sufficient condition for abstract preorder to be symmetric, but now, as I have found good formalization of ...
Doktor Diagoras's user avatar
3 votes
2 answers
926 views

Weak convergence of conditional probabilities

Suppose $\mu_n\implies\mu$, i.e. $\mu_n$ converges weakly to $\mu$ where $\mu_n$, $\mu$ are probability measures on some metric space $(X,d)$. Given a Borel set $B$, define $\mu^B$ to be the ...
JohnA's user avatar
  • 710
2 votes
0 answers
71 views

Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is an $s$-dimensional Hausdorff measure restricted to the Koch curve?

Motivated by my previous question Alberti rank-one theorem and irregular jump discontinuities, I'd like to ask the following: Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is an ...
Riku's user avatar
  • 839
2 votes
0 answers
73 views

Alberti rank-one theorem and irregular jump discontinuities

Is it fair to say that Alberti rank one theorem means that a BV functions $u \in BV(\mathbb{R}^2)$ has $Du = D^{cantor}u$ if and only if it has a jump discontinuity across a curve that is not smooth (...
Riku's user avatar
  • 839
0 votes
1 answer
75 views

Fractal set $E$ such that the indicator function $\mathbf{1}_E$ is BV

Is there a "fractal" set $E \subset \mathbb R^2$ such that the indicator function $\mathbf{1}_E$ is in $BV(\mathbb R^2)$?
Riku's user avatar
  • 839
2 votes
0 answers
73 views

Projection of BV function

Let $u \in [BV(\mathbb R^N)]^N$. We have $$D^{jump} u(x) = a(x) \otimes b(x)|D^{jump}u|,$$ where $a,b \in \mathbb S^{N-1}$. What is the projection of $D^{jump}u$ in the direction $a$? And how can ...
Riku's user avatar
  • 839
3 votes
0 answers
73 views

"Almost" absolute continuity of derivative of BV function if ${\rm Tr}\,D_Sf = 0$

Let $f: \mathbb R^N \to \mathbb R^N$ be a $BV$ function. Suppose that $\mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure: $\operatorname{div} f \ll \mathcal L^N$. This ...
Riku's user avatar
  • 839
1 vote
0 answers
100 views

Weak estimate for difference quotient of BV function

In an answer to the question Weak Lebesgue spaces and an estimate for BV functions it was remarked that if $u\in BV(\mathbb R^N)$ then there exists a Lebesgue negligible set $F \subset \mathbb R^N$ ...
Riku's user avatar
  • 839
5 votes
0 answers
198 views

Heuristic and graphic representation of BV functions and their singularities

This question is about some heuristics and graphs of BV functions. In 1-dimensional setting, two key examples of $BV$ functions $u: \mathbb R \to \mathbb R$ are the Heaviside function, whose ...
Riku's user avatar
  • 839
0 votes
0 answers
63 views

Coarea-like formula for BV functions (not their derivative)

Let $\Omega \subset \mathbb R^N$ and $f \in BV(\Omega)$. The coarea formula states that $$Df = \int_{\mathbb R} D \chi_{\{f >h\}} \, dh.$$ Unfortunately, the formula $$f = \int_{\mathbb R} \...
Riku's user avatar
  • 839
1 vote
1 answer
178 views

Growth assumption and example of finite (arbitrarily small) time blow up for ODE

Consider the following ODE initial value problem \begin{align*} &\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\ &\Phi(0,x) = x, & x \in \...
Riku's user avatar
  • 839
4 votes
1 answer
365 views

Lusin Lipschitz approximation in BV and Sobolev space

Theorem 5.34 in Functions of bounded variation by L. Ambrosio, N. Fusco and D. Pallara states that Let $u \in [BV(\mathbb{R}^N)]^m$. Then there exists a constant $\kappa>0$ such that for every $...
Riku's user avatar
  • 839
5 votes
1 answer
220 views

Alberti rank one theorem and a blow-up argument

In this paper, it is written that Alberti’s rank says that the singular part $D^s u$ with respect to $\mathcal L^d$ of the distributional derivative $Du$ of a function $u \in BV_{loc}(\mathbb R^d; \...
user avatar
1 vote
0 answers
88 views

If $u$ is $BV$ then $\operatorname{curl} Du = 0$ in the sense of distributions

Let $u\in BV(\mathbb{R}^N; \mathbb{R}^M)$. How does one prove that $$\operatorname{curl} Du = 0$$ holds in the sense of distributions?
Riku's user avatar
  • 839
1 vote
0 answers
47 views

Consistency of the definition of total variation for functions of one or several variables

Where can I find a proof that the definition of total variation for functions of several variables is consistent with the definition of total variation for functions of one variable?
Riku's user avatar
  • 839
2 votes
0 answers
199 views

Convergence of the difference quotient of a BV function

Consider a BV function $u \in BV(\mathbb{R}^N; \mathbb{R}^N)$. What can be said about the difference quotient $$ \frac{u(x+\epsilon y)-u(x)}{\epsilon} $$ regarding its convergence as $\epsilon \to 0$...
Riku's user avatar
  • 839
1 vote
0 answers
107 views

Level sets of a BV function and its derivative

Given $u \in BV(\Omega; \mathbb{R}^M)$, where $\Omega \subset \mathbb{R}^N$, what is the relationship between its level sets and its distributional derivative $Db$? More specifically, does Alberti ...
Riku's user avatar
  • 839
2 votes
0 answers
279 views

Relationship between $p$-capacity and Riesz $s$-capacity of a set

What is the relationship between the definitions of $s$-capacity (page 13 here) and $p$-capacity (here) of a set? Are they equivalent? If not, what inequalities hold? What is the difference (in terms ...
Riku's user avatar
  • 839
1 vote
1 answer
92 views

Limit of doubly indexed functions

Let $(\Omega,\mu)$ be a $\sigma$-finite measure space and $f_{n,j}$ be a doubly indexed sequence of positive functions in $L^p(\Omega),$ $1<p<\infty.$ Suppose $f_{n,j}$ converges pointwise a.e. ...
A beginner mathmatician's user avatar
6 votes
1 answer
1k views

Prove that the flow of a divergence-free vector field is measure preserving

On page 3 of this preprint, after recalling the definition of flow generated by a vector field, the authors remark that "a necessary condition for a flow $\varphi_t(\cdot)$ generated by $a(t, \cdot)$ ...
Riku's user avatar
  • 839
1 vote
0 answers
92 views

Alberti rank-one theorem and reduction of the study of BV function to the two-dimensional case

By Alberti rank-one theorem, could it be possible to reduce the study of a function $u \in BV(\mathbb{R}^N, \mathbb{R}^N)$ to the study of a function $\tilde{u} \in BV(\mathbb{R}^2, \mathbb{R}^2)$? At ...
user avatar
1 vote
1 answer
154 views

BV function with absolutely continuous divergence

Let $f:\Omega \subset \mathbb{R}^N \to \mathbb{R}^N$ be a vector field such that $f \in BV(\Omega)$. Suppose that $\mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure and ...
Riku's user avatar
  • 839
2 votes
1 answer
328 views

Hausdorff dimension of the graph of a BV function (in 1 dimensional setting)

Let $u: \Omega\subset \mathbb{R} \to \mathbb{R}$ be a function of bounded variation. Question 1. How can we prove that the Hausdorff dimension of the essential graph of $u$ equal to $1$? Question ...
Riku's user avatar
  • 839
2 votes
1 answer
118 views

Control the derivative of a BV function by its symmetric part

Can the derivative of a BV function $f:\mathbb{R}^n\to\mathbb{R}^n$ be controlled by the symmetric part of the derivative $\frac{1}{2}(Df+(Df)^T)$?
Riku's user avatar
  • 839
5 votes
2 answers
321 views

If the Hausforff dimension of the graph of a function $u$ is $N$ and $\tilde u = u$ a.e. then $\dim_H \mathrm{graph} \, \tilde u = N$ too

Let $\Omega$ be an open (non empty) set and $u:\Omega \subset \mathbb{R}^N \to \mathbb{R}^M$ be a function such that the Hausdorff dimension of its graph is $N$. Let $\tilde u = u$ a.e. Is it true ...
Riku's user avatar
  • 839
6 votes
1 answer
2k views

Sobolev functions on $\mathbb{R}^N$ cannot be discontinuous on a $(N-1)$-dimensional submanifold

How can one prove (or where can I find a proof) that if $u \in W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$, then $u$ cannot have a $(N-1)$-manifold of discontinuity points?
Riku's user avatar
  • 839
5 votes
1 answer
499 views

Hausdorff dimension of the graph of a BV function

Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function. Is the Hausdorff dimension of the graph of $u$ equal to $N$? How can we prove it? Update. In an answer to this post, it ...
Riku's user avatar
  • 839
2 votes
1 answer
1k views

Proving that family of sets has non-empty intersection

Let's say I have an object which can be viewed as family of sets $\mathfrak{S} \subseteq 2^S$, and I want to prove that its intersection is non-empty. What is known already: $S$ is set of measurable ...
Doktor Diagoras's user avatar
1 vote
1 answer
247 views

Equivalent notion of approximate differentiability

Is it true that the definition of approximate differentiability presented here of a function $f: \mathbb{R}^N \to \mathbb{R}$ is equivalent to the following one? $$\lim_{r \to 0} \rlap{-}\!\!\int_{...
Riku's user avatar
  • 839
3 votes
0 answers
75 views

Functional characterization of local correlation matrices?

Definition: A matrix $C\in\mathbb R^{m\times n}$ is local correlation matrix iff there exists real random variables $x_1,\dots,x_m,y_1,\dots,y_n$ defined on a common probability space which takes ...
VS.'s user avatar
  • 1,826
3 votes
0 answers
200 views

Largest weak(-like) topology with respect to which continuous functions are dense in the space of Borel functions

Let $X$ denote the space of bounded Borel functions $f\colon [0,1] \to \mathbb{R}$. Let $M$ denote the space of finite Borel measures on $[0,1]$. What is the largest family $F \subset M$ such that for ...
Skeeve's user avatar
  • 1,277
7 votes
1 answer
856 views

Compactness of set of indicator functions

Let $\chi_A(x)$ denote an indicator function on $A\subset [0,1]$. Consider the set $$K=\{\chi_A(x): \text{ A is Lebesgue measurable in }[0,1]\}.$$ Is this set compact in $L^\infty(0,1)$ with respect ...
Saj_Eda's user avatar
  • 395

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