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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38 views

System of null set [closed]

Let $X$ be a non-empty set, $f:X \to[0, \infty)$ is a function. Let's define a measure on the power set $\mathcal{P}(X)$: $$\mu(A) = \sum_{x\in A}f(x),$$ when $A$ is a subset of $X$. ($\sum_{x\in A}f(...
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0answers
23 views

Random stationary set with prescribed variance

Let $\Psi$ be a non-vanishing continuous function $\mathbb{R}_+\to\mathbb{R}_+$ such that $\Psi(R)\leq R^{2d}$. Is it always possible to find $X$ a random stationary set of $\mathbb R^d$ (for ...
13
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2answers
196 views

Nontrivial signed measure on Lebesgue measurable sets being trivial on Borel sets

Let $\mathfrak{L}(\mathbb{R})$ be the collection of Lebesgue measurable sets and $\mathfrak{B}(\mathbb{R})$ be the Borel sets. Question: Is there a nontrivial signed measure on $\mathfrak{L}(\mathbb{R}...
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1answer
64 views

Kolmogorov entropy of a subset of $L^1$

How can we estimate the Kolmogorov $\epsilon$-entropy $$H_\epsilon (A,L^1(\mathbb R))$$ where $ A = \{f:\mathbb R \to [0,K] \text{ s.t. $f \in L^1$ and has total variation $TV(f) \le M$}\} $?
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57 views

Example of a strictly proper scoring rule defined on the set of all probability measures on $[0,1]$

This question is closely related to another question I asked recently but is more to the point than that other question. Let $\mathcal P$ be the set of all probability measures on the Borel algebra of ...
13
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1answer
625 views

Was Cantor aware of Lebesgue theory of integration? [closed]

Georg Cantor died in 1919, more than ten years after appearance of the Lebesgue theory of measure and integration at the beginning of the twentieth century. Lebesgue theory has a deep connection with ...
3
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2answers
82 views

Example of a (strictly) proper scoring rule on a general measurable space?

Most of the literature on scoring rules that I know of deals with discrete measurable spaces, but in this paper by Gneiting and Raferty a very general definition of a scoring rule is given. I don't ...
4
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0answers
81 views

Can we show equivalence of two distributions based on their statistics?

Let $p,q$ be two distributions on $\mathbb{R}^d$. Let $f:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}$. Under what conditions does $\mathbb{E}_{x\sim p}f(x,z)=\mathbb{E}_{x\sim q}f(x,z)\ \...
2
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0answers
30 views

Binary law on pairs of finite unions of segments

Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does ...
3
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1answer
85 views

To show a set is a set of positive Lebesgue measure in $ \mathbb{R}$

Let $E\subset \mathbb{R}$ be a set of positive Lebesgue measure. Can we find $l>0$ such that $$\bigcap_{-l\leq t \leq l}t+E$$ is a set of positive Lebesgue measure? Notation: $t+E=\{t+e|e\in E\}$
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1answer
73 views

Measurability of superposition operator with non-separable Banach space

Let $f\colon I \times X \to \mathbb{R}$ be a map where $I \subset \mathbb{R}$ is an interval, $X$ is a Banach space (possibly non-separable) and we have $$t \mapsto f(t,x) \text{ is measurable}$$ $$x \...
3
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1answer
89 views

Spectrum of a self-adjoint operator and spectral measures

Let $T$ be a self-adjoint operator on a Hilbert space $\mathcal{H}$, with spectrum $\sigma(T)$. For any $x,y\in \mathcal{H}$, denote by $\mu_{xy}$ the spectral measure of $T$ with respect to $x$ and $...
3
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0answers
71 views

Recursive expression of Lebesgue measure for balls with removed intersection

This is not the most theoretically challenging question; rather it is more of a reference request for a simple formula (which must be known). Let $\left\{B_{\epsilon_n}(x_n)\right\}_{n=0}^N$ be a set ...
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2answers
101 views

Induced probability measure on a finite orbit under a group action

Suppose we have a discrete group $G$ acting on a compact set $X \subseteq \mathbb{R}^d$ via measure-preserving homeomorphisms, and suppose we have a point $x$ whose orbit $Gx$ is finite (say $|Gx| = n$...
12
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1answer
212 views

Regarding a positive Lebesgue measure set in $\mathbb{R}^2$

Let $P\subset \mathbb{R}^2$ be a positive Lebesgue measure set. Then $P$ does not necessarily contain a subset of the form $A\times B$ where $A,B\subset \mathbb{R}$ are of positive Lebesgue measure. ...
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1answer
54 views

Convergence in weak dual topology $\sigma(L^\infty, L^1)$

Let $f\in L^\infty(\mathbb{R})\cap C(\mathbb{R})$, that is $f$ is continuous and bounded on $\mathbb{R}$. Let $S_r$ denote the shift by $r\in \mathbb{R}$: $S_r f=f(\cdot-r)$. Suppose $S_{r} f $ ...
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0answers
53 views

How to define Lebesgue Integrability of functions assuming values in an arbitrary topological vector space over an arbitrary topological field?

I have already asked this question in this MSE thread, but some people suggested me to ask to the MO community also. Preliminaries An algebra of sets in a set $X$ is an $\mathcal{X}\subseteq\mathcal{P}...
5
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1answer
166 views

Is there a standard way of defining the integral of an extended real function with respect to a finitely additive probability measure?

Let $X$ be a set, and let $\mu$ be a finitely additive probability measure defined on $2^X$. Let $\Phi$ be the set of functions from $X$ to $\mathbb R \cup \{-\infty, \infty\}$. Is there a standard ...
4
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1answer
257 views

Do Borel subsets of the plane with null sections have Borel projections?

This might be a very easy question, and it might be better for mathstackexchange in which case I apologize. I'm stuck on something an anonymous referee wrote to me about a paper of mine and I'm hoping ...
2
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1answer
104 views

Unique solution of a 1-D ODE with a bounded positive right-hand-side

Consider the initial value problem $$\dot x(t) = F(t,x), \quad t \in (0,T)$$ with given initial datum $$x(0) = x_0 \in \mathbb R.$$ More precisely we consider the integral equation $$x(t)=x(0)+\int_0^...
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1answer
95 views

Interpolation inequality $\int_{\mathbb R} u^3 dx \le \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$ [closed]

Let $u \in C^\infty(\mathbb R)$. Is it true that the following interpolation inequality holds? $$\int_{\mathbb R} u^3 dx \lesssim \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$$
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0answers
59 views

Covering R2 by rectifiable curves and null sets

We have a family indexed by $t \in \mathbb R$ of rectifiable Jordan curves $(\gamma_t)$, such that $\bigcup_{t\in \mathbb R}\gamma_t = \mathbb R^2 \setminus \{0\}$. Moreover, the family is monotone, ...
1
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1answer
56 views

Compute limit of $\mathbb P(Y \le X_n)$ using limiting information on the sequence of random variables $X_n$

Let $Y$ be a symmetric random variable, $(X_n)_n$ be a sequence of nonnegative random variables, and set $p_n = \mathbb P(Y \le X_n)$. It is known from Slutsky's theorem that, if $c$ is a constant ...
4
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1answer
124 views

Integration theory for functions and values with values in topological rings

I am curious whether somebody ever tried to generalize the classical theory of Lebesgue integral to functions and measures with values in Hausdorff topological rings. The generalization of a measure ...
2
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0answers
40 views

Entropy of flow and time-1 map

Let $\Phi=(\phi_t)_{t\in \mathbb{R}}$ be a continuous flow on a compact metric space $X$. Let $\mu$ be a $\phi_1$-invariant measure. Then it is not hard to verify tht $\int_{0}^{1} \phi_t\mu dt$ is $\...
5
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1answer
175 views

An example that the sum of two Borel sets which is not a Borel set in n-dimensional Euclidean space

By sum of two sets I mean $A+B := \{x+y:x \in A \quad y \in B\}$, and there is a tip in a book of real analysis by Zhou Minqiang which says: “If $A,B$ are Borel sets in $\mathbb{R}^{n}$, $A+B$ may not ...
4
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0answers
43 views

Divergence as infinitesimal volume change on a Finsler manifold

Let $M$ be a smooth manifold and $Z$ a smooth vector field on it. It generates a family of diffeomorphisms $\phi_t:M\to M$ by demanding that $\phi_0=\operatorname{id}$ and $\partial_t\phi_t(x)=Z(\...
2
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0answers
52 views

Prove integral inequality for divergence-free vector fields

Let $u$ be a divergence-free vector field $u:\mathbb R^n \to \mathbb R^ n$. Does the following inequality hold? $$\Big( \int_{\mathbb R^n} |u|^2 dx\Big)^2 \le C\Big(\int_{\mathbb R^n} |u|^2|x|^2 dx \...
4
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1answer
82 views

Covering of discrete probability measures

Let $\mathcal{P}_{n:+}(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$ where $k_i>0$. Then any measure in $\mathcal{P}_{n:+}(\...
4
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1answer
63 views

Continuous selection parameterizing discrete measures

Let $\mathcal{P}_n(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$. Then any measure in $\mathcal{P}_n(\mathbb{R})$ is in the image of ...
0
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0answers
75 views

Weak topology on spaces of measures and Borel sets

Let $K$ be a compact Hausdorff space (not necessarily metric or even separable). Let $M(K)$ be the space of all Radon measures on $K$ (that is, finite signed regular Borel measures) endowed with the ...
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0answers
38 views

A local base for space of probability measures with Prohorov metric

Let $S$ be a Polish space. Let $P(S)$ denote the space of probability measures on $(S,\mathcal{B})$, where $\mathcal B$ is the Borel-$\sigma$-algebra over $S$. Equip $P(S)$ with the Prohorov metric. I ...
8
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4answers
551 views

Self-contained formalization of random variables?

I have not been able to find any formalization of random variables that supports construction of new random variables dependent on previously constructed ones. In what I have found, a random variable $...
6
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1answer
119 views

Are lattice operations in a Lipschitz space sequentially continuous in the weak* topology?

This is a follow-up on this (answered) question on math.SE, but involves a different topology. I think this time it is more appropriate for MO. I will repeat the background from the question cited ...
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0answers
48 views

Reference request: convergence of cadlag stochastic processes at $t=\infty$

Let $D\equiv D([0,\infty))$ be the space of cadlag functions (right continuous with left limits) on $[0,\infty)$. Consider a sequence of stochastic processes $\big(X^n\equiv (X^n(t))_{t\ge 0}\big)_{n\...
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0answers
119 views

Does a spectral theorem exist for linear operator pencils?

I was wondering if a version of the spectral theorem (the projection valued measure case) holds for linear pencils of the form $$ A-\lambda B $$ where $A,B$ are self-adjoint on some Hilbert space $\...
2
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1answer
58 views

Is the set of almost surely continuous points dense?

Denote by $D(0,T)$ the space of right continuous functions with left limits defined on $[0,T]$. Let $\mathbb P$ be a probability measure on $D(0,T)$. Define $$cont(\mathbb P):=\Big\{t\in [0,T]:~ \...
4
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0answers
68 views

A selection principle in measure theory

A Borel subset $B$ of the unit interval $\mathbb I=(0,1)$ is defined to be a density neighborhood of a set $A\subseteq\mathbb I$ if for every $a\in A$ we have $$\lim_{\varepsilon\to0}\frac{\lambda(B\...
0
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1answer
91 views

Question/References on the Skorokhod M1 topology

Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for ...
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0answers
24 views

$2$-dimensional density of the cone

I'm reading Morgan's Geometric Measure Theory: A Beginner's Guide and he says that the $m$-dimensional density of a set $A$ is given by $$ \begin{equation} \Theta^m (A,a) = \lim_{r \rightarrow 0} \...
2
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1answer
265 views

About Dirac function

In Vladimirov's book "A Collection of Problems on the Equations of Mathematical Physics", p129, 11.16, there is a equality about Dirac function, which is the fundamental solution of three ...
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0answers
23 views

Sufficient conditions for the continuity of an improper integral concerning the finite-time stability of a dynamical system

Consider the initial value problem \begin{equation}\label{fainait ve} \dot{\boldsymbol{x}}(t) = \boldsymbol{f}(\boldsymbol{x}(t)), \;\; \boldsymbol{x}(t) \in \mathbb{R}^n, \;\; t \geq 0, \; \...
1
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1answer
39 views

Scaling behavior of Wasserstein distances

Let $p>1$ and $\mu\neq \nu$ be two probability measures on $\Omega\subset \mathbb{R}^d$ a bounded set. For $\alpha \geq 0$, we let $$C_\alpha(\mu,\nu) = \inf_\sigma \frac{W_p(\mu+\sigma,\nu+\sigma)}...
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0answers
41 views

Perimeter decreases under intersection with half spaces

The main thing i need to prove is the following assertion: Let $E\subset R^N$ be a set of finite perimeter and $H=\{x\in R^N : x\cdot e < t \}$ for $t\in R$ and $e\in S^{N-1}$. Then prove that $$ ...
0
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0answers
50 views

Are these boolean subalgebras always the clopen sets of some topology?

What would one call a family $\mathcal{F}$ of subsets $X$ such that: $$(1):X\in\mathcal{F}$$ $$(2):A,B\in\mathcal{F}\implies A\setminus B\in\mathcal{F}$$ $$(3):P\subseteq\mathcal{F}\text{ is a ...
0
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1answer
95 views

Average over spheres finite

Let $X_1,...,X_N$ be random variables that are iid with the uniform distribution over $\mathbb S^n.$ I am curious how to see that $f(X_1,..,X_N):=\left \lvert \sum_{i=1}^N X_i \right\rvert^{-1}$ has ...
1
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0answers
76 views

Lebesgue dominated convergence for currents

I am learning current theory presently. In Chap 3.3-Definition of Monge-Ampère Operators, J.-P. Demailly, Complex analytic and differential geometry, I am a little confused as follows. Let $X$ be a ...
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0answers
47 views

Sequence of random variables: relation between convergence and joint distributions

Let $X, X_1, X_2, \ldots $ be a sequence of $\mathbb{R}^d$-valued random variables defined on a common probability space $(\Omega, \mathscr{F}, \mathbb{P})$ such that the pairs $$\tag{1}(X_n,X) \quad \...
7
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1answer
165 views

A group where the Weil topology induced by the Haar measure does not coincide with the original topology

Let $(G,\tau)$ be a locally compact Hausdorff topological group that is $\sigma$-finite with respect to the Haar measure $\mu:\mathcal{B}(G)\to[0,\infty]$ ($\mathcal{B}(G)$ is the Borel $\sigma$-...
5
votes
1answer
285 views

Spherical average of $\frac{1}{x}$

Let $X_1,...,X_n$ be points on $\mathbb S^1.$ We then define the expectation value $E(X)=\frac{1}{n}\sum_{i=1}^n X_i.$ Let $\frac{dS(X_1)}{2\pi}$ be the normalized surface measure of $\mathbb S^1,$ i....

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