Questions tagged [measure-theory]

Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.

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31 views

Can the Banach-Tarski paradox or Tarski's circle-squaring problem be done with hinges?

It is known for both the Banach-Tarski paradox and Tarski's circle-squaring problem that you can finitely partition the starting configuration, then continuously move these pieces (without ...
0 votes
0 answers
40 views

Density of pairs of relatively prime integers in $\mathbb{N}^2$

Let $\mathbb{N}$ denote the set of positive integers. For $n\in\mathbb{N}$ let $[n] = \{1,\ldots,n\}$. For $A\subseteq \mathbb{N}^2$ let the (two-dimensional) lower density $d_2(A)$ be defined by $$...
1 vote
0 answers
74 views

What is the limit of a helix as the frequency tends to infinity?

Consider the helix parametrized by $r(t) = (\cos(\omega t), \sin(\omega t), t)$, for a given $\omega > 0$, and $t \in \mathbb{R}$. How can we interpret the limit as $\omega \to \infty?$ My initial ...
3 votes
0 answers
68 views

Is (the generalised) Sard's theorem optimal?

As mentioned in this question (https://math.stackexchange.com/questions/416607/show-that-fc-has-hausdorff-dimension-at-most-zero/446049#446049), in 1965 Sard proved the following result (paraphrased ...
2 votes
1 answer
76 views

$\Psi$ in finite Wiener–Itô Chaos implies existence of continuous representative on neighborhood of Cameron–Martin space?

Let $(\Theta, H, \mu)$ be an abstract Wiener space, i.e. let $(\Theta, \lVert \cdot \rVert_{\Theta})$ be a separable Banach space, let $(H, \langle \cdot, \cdot \rangle_{H})$ be a separable Hilbert ...
2 votes
0 answers
37 views

Equivalent definition of the Kantorovich-Fisher-Rao distance

I am reading this paper "A JKO splitting scheme for Kantorovich-Fisher-Rao gradient flows" (https://arxiv.org/abs/1602.04457) and in the proof of Proposition 2.2, basically, if the measure ...
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6 votes
3 answers
265 views

If the measure theoretic boundary is closed must it coincide with the topological boundary?

$\DeclareMathOperator\Int{Int}\DeclareMathOperator\Ext{Ext}$Suppose $E\subset\mathbb{R}^n$ is a set of finite perimeter and suppose that the measure theoretic boundary $\partial^*E=\mathbb{R}^n\...
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4 votes
1 answer
100 views

Does approximately Fréchet differentiable imply approximately Gateaux differentiable?

In what follows, $\mu^n$ denotes $n$-dimensional Lebesgue measure, and $B_r(x)$ is the open ball with radius $r$ centered at $x$. In elementary calculus, if we have a function $f : \mathbb{R}^n \...
2 votes
0 answers
74 views

Ergodic diffeomorphisms of the circle

From the paper Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889. the following result is known: Let $(E,\Sigma, \mu)$ be a ...
2 votes
1 answer
161 views

Large cardinals and measurability in $L(A)$

Under suitable large-cardinal assumptions, in the inner model $L(\mathbb R)$ one can have $\omega_1$ and $\omega_2$ measurable (this follows from determinacy). I was wondering if it is possible to ...
1 vote
0 answers
29 views

Convex ordering of measures that are obtained by different push-forwards of a same measure

Suppose that we have a probability measure $\rho$ which is supported on $\mathbb{R}^d$ and absolutely continuous w.r.t. the Lebesgue measure. Take two vector fields $F, G : \mathbb{R}^d \rightarrow \...
1 vote
1 answer
79 views

Cameron-Martin space of product space

Suppose you have Banach spaces $\mathcal B_\alpha$ where $\alpha$ is in some index set $I$. Let $\mu_\alpha$ be Gaussian measures on $\mathcal B_\alpha$ with Cameron-Martin spaces $\mathcal H_{\mu_\...
2 votes
1 answer
185 views

The Borel sigma-algebra of a product of two topological spaces

The following problem arose while trying to justify some "known results" in abstract harmonic analysis on noncommutative groups, for which I couldn't find explicit statements in the ...
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4 votes
0 answers
128 views

Existence of a limit of alpha-difference quotient for Hölder functions

Let $f:\mathbb{R}\to \mathbb{R}^d,d\geq 1,$ be an Hölder function with exponent $\alpha\in (0,1)$, meaning that \begin{equation} \sup_{x, y \in \mathbb R, \,x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}<...
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5 votes
1 answer
397 views

Volume of a shape whose boundary consists of portions of spheres symmetrically placed about the origin in $d\gg 1$ dimensions

We are given a convex shape $S$ in the $d$-dimensional Euclidean space, whose boundary is formed by portions of $2d$ different spheres, one portion per sphere. The radius of each sphere is the same, $...
0 votes
1 answer
77 views

Transforming two smooth densities to the same density

I am looking for an example of the following: Find a bijective, differentiable function $f$ and continuous probability density functions $q_1\ne q_2$ such that $f_*q_1=p=f_*q_2$, where $f_*$ is the ...
0 votes
0 answers
102 views

A measure on the group of homeomorphisms of $\mathbb T^2$

Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall ...
1 vote
0 answers
151 views

Building random homeomorphisms of the torus $\mathbb T^2$

In https://arxiv.org/abs/0912.3423, a family of random homeomorphisms of the circle is constructed. Main Question: Can the construction be generalized to higher space dimensions, e.g. to $\mathbb T^2$?...
1 vote
0 answers
80 views

Building random homeomorphisms of the circle

Given a positive Borel measure without atoms $\tau$ on the circle $\mathbb T =\mathbb R /\mathbb Z =[0,1)$ , in https://arxiv.org/abs/0912.3423 a homeomorphism $h:[0,1)\to [0,1)$ is defined as \...
6 votes
2 answers
337 views

If every point is a Lebesgue point of $f$, is $f$ continuous a.e.?

Let $f: \mathbb R^n \to \mathbb R$ be a locally integrable function. Question: Suppose every point $x \in \mathbb R^n$ is a Lebesgue point of $f$. Does it follow that $f$ is continuous almost ...
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3 votes
1 answer
76 views

Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of Baire for every $F$?

Let $F\subset \Bbb{R}$ intersect every closed uncountable subsets of $\Bbb{R}$. Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of ...
8 votes
2 answers
891 views

Kolmogorov 0-1 law counter examples for almost independent variables

According to Kolmogorov 0-1 law, if $ \left(X_{i}\right)_{i=1}^{\infty} $ are independent, then the tail sigma algebra is trivial. I want to construct such variables which are "almost independent&...
8 votes
1 answer
277 views

Why impossible events have some drawbacks or pathologies in probability theory?

It is said by Halmos, P.R.; in "Lectures on ergodic theory" "Many of the difficulties of measure theory and all the pathology of the subject arise from the existence of sets of measure ...
0 votes
1 answer
82 views

Does $\sum_{|S| \text{ even}} f(S) \le \sum_{|T| \text{ even}} f(T)$ hold for all nondecreasing submodular functions f?

Let $f : 2^n \to \mathbb{R}$ be a nondecreasing submodular function, where $2^n$ is the powers of $\{1, \dots, n\}$. Here nondecreasing means that $f(S) \le f(T)$ for all $S \subseteq T$. And ...
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2 votes
0 answers
79 views

Malliavin-Shavgulidze (type) measures on the group of measure-preserving invertible maps on $\mathbb T$?

The Malliavin-Shavgulidze measures on $\operatorname{Diff}^{1}(I)$ (with $I$ an interval of $\mathbb R$) are defined as the image $W_{\sigma} \circ f^{-1}$ of the Wiener measure $W_{\sigma}$ with ...
2 votes
0 answers
121 views

Weakly mixing diffeomorphism

From Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889. the following result is known: Let $(E,\Sigma, \mu)$ be a measure ...
7 votes
1 answer
212 views

Does this "local time" type limit exist a.e. for $C^2$ functions?

For $f: \mathbb R^n \to \mathbb R$ a locally integrable function, $\varepsilon \in (0, \infty)$, and $x \in \mathbb R^n$, define $I(f, \varepsilon, x)$ to be the averaged integral of $f$ over $B_{\...
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0 votes
1 answer
50 views

Bound the conditional expectation of a random matrix under weak dependence

Let $X$ be an $d\times d$ random matrix satisfying $\mathbb{E}[X]=0$ and $\|X\|_2\leq 1$ almost everywhere. Let $\mathcal{F}$ be the $\sigma$-field generated by $X$. Now suppose we have another $\...
2 votes
1 answer
140 views

Bakry-Emery criterion

The most common use of the Bakry-Emery criterion is for the measure $\mu(x)=e^{-u(x)} /Z$ where $u \in \mathcal{C}^2$. I would like to ask for an application to a smaller class. Consider $u(x)=|x|^2 + ...
6 votes
1 answer
178 views

Steinhaus number of a group

$\newcommand\Sn{\mathit{Sn}}$A subset $A$ of a group $X$ is called algebraic if $A=\{x\in X: a_0xa_1x\dotsm xa_n=1\}$ for some elements $a_0,a_1,\dotsc,a_n\in X$. Let $\mathcal A_X$ be the family of ...
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1 vote
0 answers
25 views

If $(\kappa_t)_{t\ge0}$ is a Markov semigroup with invariant measure $μ$, under which assumption is $t\mapsto\kappa_tf$ measurable for $f\in L^p(μ)$?

Let $(E,\mathcal E)$ be a measurable space; $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$; $\mu$ be a finite measure on $(E,\mathcal E)$ which is subinvariant with respect to $(\...
2 votes
0 answers
190 views

Prove or disprove that $u=0$ a.e. on $\Bbb R^d$

Let $\Omega\subset\Bbb R^d$ be an open set. Let $k:\Bbb R^d\to [0,\infty)$ be measurable such that $0\in \operatorname{supp}k$. This implies that $\Omega\subset \Omega_k=\Omega+\operatorname{supp}k$. ...
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23 votes
3 answers
844 views

Are there arbitrarily large families of lines in $\Bbb R^3$ with average angle $\ge \pi/3$?

Question: Can I have an arbitrarily large finite family of lines $\ell_1,\dotsc,\ell_n\subset\Bbb R^3$ so that the average angle between two (distinct) lines is $\ge \pi/3$? We can assume that all ...
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8 votes
5 answers
457 views

Distributions of distance between two random points in Hilbert space

Let $\mu$ be a probability distribution on a separable infinite-dimensional Hilbert space. Let $D$ be the distance between two independent random samples from $\mu$. So $D$ has some probability ...
2 votes
1 answer
223 views

Matching the integral of a function on smaller open sets

Let $f: [0, 1] \to \mathbb R$ be Lebesgue integrable with $\int_0^1 f \, d \mu = C.$ Question: For every $K$ with $0 < K \leq 1$, does there exist an open subset $U$ of $[0, 1]$ of Lebesgue measure ...
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1 vote
0 answers
42 views

Is a Boolean algebra with an order continuous topology a measure algebra?

Assume that $B$ is a complete boolean algebra endowed with a Hausdorff topology, with respect to which all operations on $B$ are continuous, $0$ has a base of full sets (recall that $A\subset B$ is ...
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0 votes
1 answer
91 views

Condition for set of the type $\{(a,b)|a \in A, \ b = f(a)\}$ to have empty interior if $A$ has empty interior

Let us consider $$\mathcal X = \{(a,b)|a \in A, \ b = f(a)\}, $$ where $A \subset L^1(\mathbb R)$ has empty interior and $f:L^1 \to L^1$ is a bijective map. Does $\mathcal X$ also have empty interior? ...
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4 votes
1 answer
224 views

Bounds on discrepancy metric of product measures

Consider two measurable spaces $X_1 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_1)$ and $X_2 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_2)$ and the product spaces $$X_1^{q} = (\times_{i=1}^q\...
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2 votes
1 answer
129 views

Problem regarding set of positive Lebesgue measure in $\mathbb{R}^2$

Let $T_{p,q}$ be line joining $(0,0)$ and $(p,q).$ Now let us define the set $$L= \bigcup_{p\in[0,1]\cap \mathbb{Q}}T_{p,1} \bigcup_{q\in[0,1]\cap \mathbb{Q}}T_{1,q} $$ and consider $P=[0,1]\times[0,...
1 vote
1 answer
108 views

Coarea formula for measure of epsilon neighbourhood

I am trying to use the coarea formula to get estimates on the measure of an epsilon-neighbourhood of a set. Specificly, given a compact 'nice' set $A\subseteq \mathbb{R^d}$, possibly with more than ...
1 vote
2 answers
145 views

Connection between invariant measure and positive recurrence for continuum state space markov chain

Let $\{ X_n(\omega,x)\}_{n \ge 0}$ be a Markov chain with and underlying probability space $(\Omega,\Sigma,\mathbb{P})$ and state space $X= \mathbb{S}^1$. Suppose this markov chain admits unique ...
6 votes
1 answer
222 views

Are all quasi-regular points on Polish spaces generic points?

Let $X$ be a Polish space and $T\colon X\to X$ be a continuous map. We say that a point $x\in X$ is quasi-regular if for every bounded continous function $\varphi\colon X\to\mathbb{R}$ the sequence $...
1 vote
0 answers
133 views

Study of the class of functions satisfying null-IVP

$\mathcal{N}_u$ : Class of all uncountable Lebesgue-null set i.e all uncountable sets having Lebesgue outer measure $0$. Let $f:\Bbb{R}\to \Bbb{R}$ be a function with the following property : $\...
0 votes
0 answers
46 views

Measure of noncompactness and multivalued functions

The Hausdorff measure of noncompactness of a nonempty and bounded subset $Q$ of $X$, denoted by $\alpha(Q)$, is the infimum of all numbers $\varepsilon>0$ such that $Q$ can be covered by a finite ...
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2 votes
1 answer
239 views

Qualitative difference between "continuous" and "discontinuous" states on $M(G)$

Let $G$ be a locally compact Abelian group (we can think that $G={\mathbb R}$). Let $C_0(G)$ be the space of continuous functions $u:G\to{\mathbb C}$ vanishing at infinity with the usual $\sup$-norm, ...
1 vote
0 answers
44 views

Basin of attraction comparative statics* using local energy functions?

Let $\dot{\boldsymbol{x}}=\boldsymbol{F}(\boldsymbol{x};p)$ be an autonomous dynamical system defined on $[a,b]^n$ ($-\infty<a<B<\infty)$; $p\in\mathbb{R}$ is some fixed parameter. Suppose ...
2 votes
2 answers
78 views

Regular Lagrangian flow for explicit ODE with discontinuous right-hand side

Consider the problem $$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \begin{cases} - 1 & \text{ if } X(t,x) >0, \\ 1 & \text{ if } X(t,x) < 0 \end{cases}, &t \in [0,T],\\ X(0,x) ...
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5 votes
1 answer
226 views

Boolean algebra of ambiguous Borel class

Suppose $X$, $Y$ are uncountable compact metric spaces and $\Delta^0_\xi(X)$, $\Delta^0_\xi(Y)$ ($2\le\xi\le\omega_1$) are the Boolean algebras of Borel sets of ambiguous class $\xi$. So for $\xi=2$ ...
1 vote
1 answer
98 views

A different version of Besicovitch Covering Theorem involving balls of half radius

I am trying a find a reference to/proof of the following result: Let $(M, g)$ be a compact Riemannian manifold. Then there is $b$ so that the following holds: for any $r>0$, there is a covering $\...
2 votes
1 answer
97 views

Can we say that there exists a measurable function $f$ such that $ \nu=f_{\#}\mu$?

Define a coupling $\pi\in \Pi(\mu,\nu)$ on the product space $(X\times X,\mathcal{F}\times\mathcal{F})$. let $\pi_x$ be the disintegration of $\pi$ with respect to the $\mu$, i.e. there exists a Borel ...
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