# Questions tagged [measure-theory]

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

2,102
questions

**-1**

votes

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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(...

**0**

votes

**0**answers

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**

votes

**2**answers

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}...

**0**

votes

**1**answer

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 $TV(f) \le M$}\}
$?

**1**

vote

**0**answers

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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: $t+E=\{t+e|e\in E\}$

**3**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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 ...

**0**

votes

**2**answers

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**

votes

**1**answer

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.
...

**0**

votes

**1**answer

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 $ ...

**2**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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^...

**-1**

votes

**1**answer

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?
$$\int_{\mathbb R} u^3 dx \lesssim \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$$

**2**

votes

**0**answers

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**

vote

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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**

votes

**4**answers

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**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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\...

**2**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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**

vote

**1**answer

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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vote

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

vote

**0**answers

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 ...

**0**

votes

**0**answers

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**

votes

**1**answer

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

**1**answer

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....