# Questions tagged [measure-theory]

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

2,332
questions

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### Local dimension of measures

For a Borel prob measure $\mu$ in $\mathbb{R}^n$, define the local dimension of $\mu$ at $x$ by
$$
{\rm dim}_*(\mu, x)=\liminf_{r\to 0} \frac{\log \mu(B(x,r))}{\log r}, {\rm dim}^*(\mu, x)=\limsup_{r\...

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181 views

### An outer measure defined on $\mathbb {R}$

Let $s, \delta \in (0,1)$. Consider the outer measure on $\mathbb{R}$, $\mu^s_{\delta}$, defined by
\begin{align*}
\mu^s_{\delta}(E):=\inf \left\{\sum_{j}\lvert I_{j}\rvert^s: E \subset \bigcup_{j}...

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+200

### Conditions for the SDE be transitive

This question was previously posted on MSE.
Let $f:\mathbb R^3 \to \mathbb R^3$ be a smooth Lipschitz function (bounded if needed), and $W_t$ a $3$-dimentional Brownian motion. Consider the SDE on $\...

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230 views

### A quantity associated to a probability measure space

Let $(S,P)$ be a (finite) probability space. We associate to $(S,P)$ a quantity $n(S,P)$ as follows:
The probability of two randomly chosen events $A,B\subset S$ being independent is denoted by $n(S,P)...

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128 views

### What to call a continuous function with preimage preserving nowhere-density?

Currently I am reading some basic literature on descriptive set theory and boolean algebras. And this comes out a lot, for example in results like:
Let $X$ and $Y$ be topological spaces, and $f:X \to ...

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**1**answer

169 views

### A question about the range of a positive measure

It is a well-known fact that the range of a positive measure $\mu$ on a measure space $(X,\cal M)$ is the interval $[0,\mu(X)]$ provided $\mu$ is atomless (i.e., there is no measurable set $A\in \cal ...

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77 views

### What is the number of finite Dynkin systems?

(This is a spin-off of Determine the minimal elements of a Dynkin system generated by a finite set of finite sets)
Let $\Omega$ be a finite set. A Dynkin system on $\Omega$ is a subset of the power ...

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102 views

### Conditions for the existence of von Neumann-Morgenstern utility on a Polish space

Let $X$ be a Polish space, i.e. a separable complete metric space. Any Borel probability measure on $X$ must be locally finite, outer regular and tight. Let $\mathcal{P}(X)$ be the set of all Borel ...

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41 views

### Baire 1 function equivalence in measure

I am trying to prove (or disprove) the following assertion:
Consider a probability triple $(X,\mathcal{B},\mu)$, $X$ separable Banach space (complete), $\mathcal{B}$ the Borel $\sigma-$algebra and $\...

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68 views

### Convergence of sequences for Baire-1 functions

Let X and Y be separable Banach spaces.
Let $f:X\rightarrow Y$ be a Baire-1 function, which is the pointwise limit of a sequence of continuous functions $f_n:X\rightarrow Y$.
Define $E$ as the set of $...

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21 views

### Measure of set of singular points of a generalized discrete Fourier transform

Assume we are given a function that can be expressed in a Fourier-like expansion
$$
f\colon \mathcal{X} = [0, 2\pi]^d \to \mathbb{C}, \ \boldsymbol{x}\mapsto \sum_{i = 1}^K c_{i} e^{-i \boldsymbol{\...

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53 views

### Is the lattice of bounded Henstock Kurzweil integrable functions countably complete?

The set of HK integrable functions with an integrable upper bound $f$ forms a lattice, and satisfies the MCT and DCT. Does this mean that the lattice is countably complete?
Indexing any countable set, ...

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### Equivalence between notions of dynamical coupling as defined by Villani in his book Optimal Transportation: Old and New

$\DeclareMathOperator\law{law}$In Villani's book he presents the following notions of dynamical couplings:
Let $(X,d)$ be a Polish space. A dynamical transference plan $\Pi$ is a probability measure ...

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146 views

### Set where the speed of convergence is uniform in Lebesgue's density theorem

Let $B \subset \mathbb R^n$ be the unit ball.
Consider a Borel measurable set $E \subset B$ with positive Lebesgue measure $|E|>0$ (say $|E| = |B|/2$).
Then, Lebesgue's density theorem, says that ...

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114 views

### Well-definedness of maximum likelihood estimation

Consider a family $\{\mu_\theta:\theta\in\Theta\}$ of probability measures on a measurable space $X$. Given $x\in X$, the maximum likelihood estimate is the value of $\theta$ which maximizes the ...

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118 views

### Confusion around uniform integrability and Vitali convergence theorem

Motivation
The notion of uniform integrability is important for formulating the Vitali convergence theorem. Unfortunately, different authors define uniform integrability differently, which causes ...

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42 views

### A MNC with maximum property but not singular

Let $E$ be a Banach space, $\mathfrak{M}_E$ indicate the family of all nonempty bounded subset of $E$, $\mathfrak{N}_E$ the family of all relatively compact sets, and $Ker \mu=\{X\in \mathfrak{M}_E$ ...

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70 views

### Supremum with respect to the order of measures on $(X,A)$

Suppose that $(X,\leq )$ is an ordered set, we can define the maximum and the infimum of this set,now let $(X,A)$ be a measurable space and let $M(X,A)$ be the set of all measures on $(X,A)$, we now ...

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88 views

### Does $\mathbb R^n$ equipped with a sum of Dirac delta measures admit nowhere locally constant continuous integrable functions?

For any point $x \in \mathbb R^n$, denote by $\delta_x$ the Dirac Delta measure centered at $x$.
Let $a_n$ be an sequence of positive numbers with $\lim_{n \to \infty} a_n = 0$, and let $d_i$ be a ...

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52 views

### Space of functions and the Coordinate process

I have the following question:
Let a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with a stochastic process $X$ be given and define $(\mathcal{F}_t) = \sigma(X_t;t \geq 0).$
Further, we have ...

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129 views

### The infinity Wasserstein distance $W_\infty$ and the weak topology

Let $X$ be a compact metric space (without isolated points). The $\infty$-Wasserstein distance $W_\infty$ on the space of Borel probability measures on $X$ can be described as $$W_\infty(\mu,\nu) = \...

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55 views

### If $\lambda_i$ is symmetric with $\lambda_i\{0\}=0$, why does $\int_B1-\cos\langle x,x'\rangle\:(λ_1-λ_2)({\rm d}x)=0$ imply $λ_1=λ_2$?

Let $E$ be a separable $\mathbb R$-Banach space and $\lambda_i$ be a finite symmetric measure on $\mathcal B(E)$ with $\lambda_i(\{0\})=0$ and $$\int_B1-\cos\langle x,x'\rangle\:\underbrace{(\lambda_1-...

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73 views

### Is the singular value decomposition a measurable function?

$\DeclareMathOperator\svd{svd}$Consider the SVD of rectangular matrices as operators
$$\svd:\mathbb C^{n\times m}\to \mathbb U_n\times \mathbb D_{n, m}\times\mathbb U_m$$
where $\mathbb U_n$ is the ...

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106 views

### When does the measure integral of the form $\int_{\log(S)} f d \mu$ exist?

When does the measure integral of the form $\int_{\log(S)} f d \mu$ exist ?
Here $\mu$ can be any measure (Lebesgue, Borel, Haar etc), $f$ is a measurable function, $S$ is any measurable set with ...

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204 views

### What is the 'right' definition of zero measure subsets of Banach spaces?

Question. There are several ways of defining a notion of a 'zero measure' subset of a Banach space $X$. Which one is the 'right' or failing that, the preferred notion? [See below for a more precise ...

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202 views

### A strong Borel selection theorem for equivalence relations

In Kechris' book "Classical Descriptive Set Theory" there is the following theorem (12.16):
Let $X$ be a Polish space and $E$ an equivalence relation such that every equivalence class is ...

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86 views

### Averaging and fractional Laplacian

Let $u,\phi:\mathbb R \to \mathbb R$ be smooth functions and $\Omega_\epsilon$ be a bounded domain in $\mathbb R$ with diameter $\epsilon>0$ (consider for exaple the ball $B_{\epsilon/2}(0)$). Is ...

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152 views

### On the weak convergence of probability measures on $\mathbb R$

Let $\mathcal P(\mathbb R)$ be the set of probability measures. Set for $\mu,\nu\in\mathcal P(\mathbb R)$
$$d(\mu,\nu) := \inf\left\{\varepsilon>0:~ F_{\mu}(x-\varepsilon)-\varepsilon \le F_{\nu}(x)...

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164 views

### Existence of a continuous ergodic dynamical system for a given distribution?

It seems to me that given a distribution (which is well-behaved), there should be at least an ergodic dynamical system that its time average would create this distribution. Is this question already ...

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106 views

### Does the space of Lipschitz functions have the Radon-Nikodym property?

Context.
Space of Lipschitz functions. Denote by $Lip_0(D)$ the space of all Lipschitz functions on a metric space $D$ vanishing at some base point $e \in D$. The norm in $Lip_0$ is defined as ...

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### Are open subsets of a $\sigma$-compact LCH space $\mathcal{K}$-analytic?

I'm reading Guedj and Zeriahi's Degenerate Complex Monge-Ampère Equations Chapter 4 which talks about capacities. Specifically Corollary 4.13 claims that when $X$ is a locally compact Hausdorff $\...

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117 views

### Are there any measurable spaces of functions

I am approaching this question from a probability perspective, and am hoping for some kind of framework to help understand all of this. I believe I may have even asked a similar question on here in ...

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141 views

### The mean ergodic theorem for weakly mixing extension

I asked this question in https://math.stackexchange.com/q/4236870/528430, but did not get any help.
I got stuck with the following while going through the proof of Lemma 3.21 from the book 'Ergodic ...

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163 views

### Small ball Gaussian probabilities with moving center

I would like to prove (if possible, otherwise find a counterexample for) the following lemma:
Let $(X,\|\cdot \|_X)$ be a separable Banach space. Additionally, we have a centred Gaussian measure $\mu$ ...

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70 views

### Ergodic action on product spaces

Let $(X_1 \times X_2,d\mu)$ be a measure space with $X_2$ compact. Suppose that we have a continuous (diagonal) action of a topological group $G$ on $X=X_1 \times X_2$. I know that the action of $G$ ...

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661 views

### Is there an increasing function on $[a, b]$ which is differentiable, but not absolutely continuous？

Is there an increasing function on
$[a, b]$ which is differentiable,
but not absolutely continuous？

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**1**answer

249 views

### Integral on level sets

Let $g_\epsilon : K \subset \mathbb{R}^d \rightarrow \mathbb{R}$ (more regularity can be assumed if necessary) be defined on a compact set (with regular boundary) $K \subset \mathbb{R}^d$, and the ...

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179 views

### Understanding Kelley's intersection number (Boolean algebras)

It is known that:
Theorem (Kelley, 1959). There exists a finite, strictly positive, finitely additive measure on a Boolean algebra $A$ if and only if $A^+$ is the union of a countable number of ...

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155 views

### How to compute $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{[-1,1]^n}\exp[2\pi i(\theta_1 v_1.x+\theta_2v_2.x)]d^nx d\theta_1d\theta_2$

Let $\mathbf{v}_1, \mathbf{v}_2$ be two vectors in $\mathbb{R}^n$. I would like to compute the following singular integral:
$$\int_{-\infty}^{ \infty} \int_{-\infty}^{\infty}
\int_{[-1,1]^n}
e(\...

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200 views

### Reference request: large generalized probability measures

I'm interested in references relevant to the following: what is the right generalization, if there is one, of a probability measure that takes on values in an structure of more than continuum size?
I'...

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### Continuity of surface integrals on level sets

Let $\phi:\mathbb{R}^2\to\mathbb{R}$ such that $\phi^{-1}(0)\neq\emptyset$ and $\phi\in C^1(W)$ where $W$ is a compact neighborhood of $\phi^{-1}(0)$, with $\nabla\phi\neq 0$ in $W$. So there is some $...

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447 views

### Converse of mean value theorem almost everywhere?

Let $f: \mathbb R \to \mathbb R$ be a $C^1$ function.
We say a point $c \in \mathbb R$ is a mean value point of $f$ if there exists an open interval $(a,b)$ containing $c$ such that $f’(c) = \frac{f(b)...

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84 views

### Exponential mixing for subshifts

I asked this question on Math.StackExchange some time ago and got no responses.
Let $G=(V,E)$ be a finite graph with adjacency matrix $A$. Let us consider the associated subshift of finite type
$$
\...

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55 views

### Is speaking about a fraction of the Mandelbrot's set meaningful?

Sorry if my question is vague, as I have very little background with fractals and measure theory. My question is inspired by a tweet, where a light shone onto the mandelbrot set, and certain rays were ...

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178 views

### Question concerning an inequality on probabilities of hitting times in a paper

Let $\ell^n: [0,\infty)\to [0,1]$ be right-continuous and increasing functions s.t. $\ell^n(0)=0$. Given $x>0$ and Brownian motion $(B_t)_{t\ge 0}$, can we prove
$$\limsup_{n\to\infty}\mathbb P[\...

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50 views

### (Generalized) Uncentered Maximal Function $\tilde Mf$ in Stein's Harmonic Analysis

It is well known that on $\Bbb R^n$, equipped with the usual Lebesgue measure, the standard Hardy-Littlewood maximal function $Mf(x)$ (with respect to averaging on cubes or balls centered at $x$) is ...

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44 views

### Subspace of RKHS generated by kernel mean embeddings

Suppose $\mathcal{H_k}$ is a reproducing kernel Hilbert space (RKHS) with reproducing kernel $k: \mathcal{X} \times \mathcal{X} \rightarrow \mathbb{R}$. I am looking for results characterising the ...

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31 views

### "Orthogonalization" of bilinear operator

I would like to ask if the following fact correct/known:
Let $D$ is dyadic grid, $ֿ\mu$ and $\nu$ are positive measures, $\{\alpha_I : I \in D\}$ is set of positive numbers indexed by cubes in the ...

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71 views

### Change variables in Gaussian integral over subspace $S$

I have been thinking about a problem and I have an intuition about it but I don't seem to know how to properly address it mathematically, so I'm sharing it with you hoping to get help. Suppose I have ...

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**1**answer

141 views

### Equivalent definitions of strongly proximal action

Consider the following fragment from the paper "C*-simplicity and the unique trace property for discrete groups" by Breuillard, Kalantar,
Kennedy and Ozawa:
I have two questions:
(1) What ...