# 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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54 views

### A question concerning Lusin’s Theorem

We consider only the set $M$ of a.e. essentially locally bounded measurable functions $[0, 1] \to \mathbb R$. Here $m(S)$ denotes the Lebesgue measure of $S$.
Let $f$ be measurable. For every $e$ in $...

**0**

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**0**answers

23 views

### If the finitely additive measure of an open set is approximable by clopen sets, is it approximable from within?

Let $F$ be a finite set equipped the discrete topology. Let $X = F \times F \times ...$ be equipped with the product topology. Let $\mathcal A$ be any field of subsets of $X$ that contains the open ...

**3**

votes

**1**answer

73 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_{...

**0**

votes

**1**answer

78 views

### Finitely additive measure on Cartesian square of countable set

Let $\mu$ be a probability measure on $(\omega, 2^\omega, \mu)$ measure space which is finitely additive and $\mu(A)=0$ for finite sets.
We can define as usual $\mu^2$ on semiring $\mathcal{G}=\{A\...

**1**

vote

**0**answers

32 views

### Formal justification of the Chaos game in the Sierpinsky triangle

I want to justify why the Chaos game works to produce Sierpinsky triangle. I use a theorem taken from Massopoust Interpolation and Approximation with Splines and Fractals.
Suppose that $(X,d)$ is a ...

**4**

votes

**2**answers

116 views

### Box dimension of the graph of an increasing function

This Hausdorff dimension of the graph of an increasing function shows that:
Let $f$ be a continuous, strictly increasing function from $[0,1]$ to
itself with $f(0)=0, f(1)=1$. Then $dim_H \; G = ...

**3**

votes

**0**answers

51 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 ...

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50 views

### Relation between the set measures of real number and the real coordinate space of finite dimensions [on hold]

Consider a compact interval $[a,b]\subset \mathbf{R}$.
Let $\Delta_1=\Delta([a,b])$ be the set of all Borel probability measures over $[a,b]$.
Consider a Natural number $N\in \mathbf{N}$.
What is ...

**3**

votes

**0**answers

126 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 ...

**6**

votes

**1**answer

181 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 ...

**3**

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60 views

### Is there a T3½ category analogue of the density topology?

Motivation: I understand that various attempts have been made at defining a topology on $\mathbb{R}$ that is an analogue of the density topology ([1]) but for category (and meager sets) instead of ...

**2**

votes

**1**answer

174 views

### Can the characteristic function of a Borel set be approached by a sequence of continuous function through a certain convergence in $L^\infty$?

Want to find $f_n$ a sequence of continuous functions, so that for all Borel regular measure $\mu$, we have
$\int f_n d\mu\rightarrow\int \chi_\Delta d\mu$, as $n$ goes to infinity, where $\Delta$ ...

**1**

vote

**1**answer

97 views

### Measurable function

Let $S$ be a countable set. Consider $X=S^{\mathbb{N}\cup\{0\}}$ the topological Markov shift equipped with the topology generated by the collection of cylinders.
Denoted $\mathcal{B}$ as the Borel $\...

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43 views

### Identical push-forward but not stationary

I'm having some trouble coming up with a counter-example for this problem:
Give an example of a stochastic process $\{X_n : n \in \mathbb{Z}^+\}$ on $(\Omega, \mathcal{F}, P)$ such that $P_{X_n} = P_{...

**1**

vote

**0**answers

68 views

### Extrinsic applications of Haar measure

I am looking for examples of (not necessarily deep) results whose proofs rely on the Haar measure (or rather such that there exists an "elegant" proof involving it). Further, the formulation of such ...

**2**

votes

**1**answer

73 views

### Measure on union of measure spaces and on quotient space

There are two questions about measures bothered me a lot.
Given a set X and a countable covering ${U_i}$ of $X$. Suppose that for each i, there is a measure $m_i$ on $U_i$. Is there a very general ...

**1**

vote

**1**answer

54 views

### Convergence of probability density function

There are various kinds of (convergence of random variables) but I have never read about convergence of density functions.
Let $X_1, X_2, \dots, X$ be random variables $\Omega \to \mathbb{R}$ and $...

**2**

votes

**2**answers

134 views

### The space of Borel function modulo comeager sets is Dedekind complete

Let $(X,\tau)$ be a topological space. Denote by $Bor(X)$ the space of Borel functions $f:X\to\mathbb{R}$ where we identify two functions whenever they agree on the complement of a meager set. We ...

**3**

votes

**0**answers

64 views

### Measures on formal power series over a finite field

For various reasons not important for this question, I'd like to show that certain subsets of $F_p[[t]]$, the ring of formal powers over the finite (prime) field $F_p$ in the variable ...

**2**

votes

**1**answer

158 views

### Generalised raindrop function

Given a sequence of reals $(a_n)_{n > 0}$, let $f: [0, 1] \to R$ be the generalised raindrop function defined:
$f(x) = a_q$ if $x$ is rational, with denominator $q$ in lowest form; $0$ otherwise.
...

**7**

votes

**1**answer

530 views

### Digits in an algebraic irrational number

I am trying to solve a problem and I got a conditional result related to normality of algebraic irrational numbers (Borel conjecture).
I know that by using Ridout theorem or Schmidt subspace theorem ...

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**0**answers

80 views

### Estimation of the Gromov–Wasserstein distance of spheres

Let $(X,d_X,\mu_X)$ and $(Y,d_Y,\mu_Y)$ be two metric measure spaces. A probability measure $\mu$ over $X\times Y$ is called a coupling if $(\pi_1)_\sharp \mu=\mu_X$ and $(\pi_2)_\sharp \mu=\mu_Y$. We ...

**4**

votes

**1**answer

119 views

### Proving Conditional Independence

Each of the scalar random variables, $ Y $, $ X $, $ U $, and $ V $, is continuous and possibly has $ \mathbb{R} $ as its support. The random variable, $Z$, could be vector valued, but continuous.
I ...

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vote

**1**answer

85 views

### Finitely Additive Homogeneous Translation Invariant Measure on $\mathcal{P}(\mathbb{R})$

It is known that there exists a finitely additive translation invariant measure on $\mathbb{R}$ that extends the Lebesgue measure. I.e. a function $m:\mathcal{P}(\mathbb{R}) \rightarrow [0,\infty]$ ...

**5**

votes

**1**answer

322 views

### Existence of a strange function

Inspired by A discontinuous construction:
Does there exist a function $a \colon [0,1] \to (0,\infty)$ and a family $\{D_x \colon x \in [0,1]\}$ of countable, dense subsets of $[0,1]$ with $\bigcup_{x \...

**0**

votes

**0**answers

39 views

### Kantorovich-Rubinstein like distance

Fix the closed unit disk (for simplicity) $\mathbb{D}^n\subset \mathbb{R}^n$. Given two probability measures $\mu,\nu$ on $\mathbb{D}^n$ we define the Kantorovich-Rubinstein distance between them to ...

**7**

votes

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356 views

### A discontinuous construction

Suppose we have an uncountable family of functions $f_r: [0, 1] \to R$ indexed by $r \in [0, 1]$ such that for each $r$, there exists a unique $x$ in $[0, 1]$ such that $f_{r}$ is positive on $x$ and $...

**3**

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73 views

### Transport Distance between Level Sets of a Convex Function

Suppose I have a well-behaved, strictly convex function $f : \mathbf{R}^d \to [0, \infty)$, and assume that $f$ has its unique minimiser at $x = 0$, with $f(0) = 0$.
For $y > 0$, I define the ...

**12**

votes

**1**answer

392 views

### Are Hausdorff measures on the real line Haar measures for some locally compact topology?

For $0\leq d\leq 1$, let $\lambda_d$ be the $d$-dimensional Hausdorff measure on $\mathbb{R}$. Note that it is translation-invariant. Does there exist a locally compact topology $\mathscr{T}_d$ on $\...

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109 views

### Definition of the surface measure in some books

I am studying PDEs and in some books (Folland, Introduction to Partial Differential Equations and Evans, Partial Differential Equations), I found an integral integrated by the surface measure on a $C^...

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votes

**2**answers

200 views

### Measure of the numbers with length of $n$ for a nonstandard number $n$

Is there any nonstandard model of $PA$ with the following properties?
There exists a nonstandard number $n\in M$ such that $M\upharpoonright n$ is countable,
Let $|x|=\lceil\log_2x\rceil$, ...

**4**

votes

**1**answer

156 views

### ODE with a measurable vector field

Suppose we have a bounded Borel measurable vector field $F:\mathbb{R}^n\to\mathbb{R}^n$. To make the question non-trivial, assume that $F\neq 0$ eveywhere.
Question. Does there exist at least one ...

**2**

votes

**1**answer

74 views

### optimal transport, measurable selection

Let $X = Y = \mathbb{R}^d$ and let $\nu$ be a probability measure on $\mathbb{R}^d$. Consider the collection of probability measure $\pi$ on $X\times Y$ such that $\pi$ has $y$-marginal $\nu$:
$$
\Pi(...

**0**

votes

**0**answers

59 views

### Uniform lower bound on integral over sets of the form $A \times A^c$

Consider a function $f(x,y): [0,1]^2 \to [0,\infty)$ continuous almost everywhere, for which there is no $A \subset [0,1]$ such that $0<\mu(A)<1$ and $\int_{A \times A^c} f(x,y)dxdy=0$.
Is it ...

**10**

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**0**answers

119 views

### Maximizing an integral w.r.t. a measure on the unit sphere

I would like to know if the answer to the following question is known.
Let $d \ge 3$. What is the value of
$$
\theta(d) := \max_{\mu} \int_{S^{d-1}} \int_{S^{d-1}} \cdots \int_{S^{d-1}} |x_1 \...

**2**

votes

**1**answer

145 views

### Research-type questions in probability illustrating measure-theoretical techniques for students

In short, in the perspective of preparing a new course, I am looking for examples of "concrete" (hopefully research-type) questions concerning various models in probability theory which give the ...

**2**

votes

**2**answers

127 views

### Questions about some properties of random probabilities and random expectations

Let $(\Omega, \mathcal{A}, \mathbb P)$ be a probability space with $\mathcal{A}$ countably generated, and let $P: \mathcal{A} \times \Omega \to [0,1]$ be a random probability measure. By that I mean $...

**2**

votes

**0**answers

50 views

### The dual of the space of continuous sections in a vector bundle

If $X$ is a compact Hausdorff space, one may view the space of complex, continuous functions on it as the space of continuous sections in the trivial Hermitian bundle $X \times \mathbb C$. By the ...

**8**

votes

**1**answer

182 views

### Measure support decomposition that “tends to infinity”

I would like to know the answers to the following two questions.
Let $S$ be a locally compact Hausdorff space, $\mu$ be a regular Borel measure with non-compact support $M$. Denote
$$
\mathscr{H}=\{\...

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**0**answers

59 views

### Can Gaussian measure be characterized by unitary representations?

It is well known that Fourier transform switches positive-definite functions with positive measures on a (locally compact topological) group. Further, the positive definite functions can be ...

**27**

votes

**2**answers

815 views

### Rademacher theorem

If $f:\mathbb{R}^n\to\mathbb{R}^m$ is of class $C^1$ and $\operatorname{rank} Df(x_o)=k$, then clearly $\operatorname{rank} Df\geq k$ in a neighborhood of $x_o$. It is not particularly difficult to ...

**2**

votes

**0**answers

44 views

### Lower and upper (combinatorial) discrepancy

(I will state my question in terms of combinatorial discrepancy, but the same could be also asked about measure-theoretic discrepancy as well.)
The combinatorial discrepancy of a family $\mathcal F$ ...

**12**

votes

**2**answers

389 views

### Category theory & geometric measure theory?

My background is essentially Geometric Measure Theory and its application to partial differential equations (e.g. linear and non-linear hyperbolic conservation laws). These are currently my research ...

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votes

**0**answers

65 views

### Bounding the total variation distance between two measures from a given set

I have a distance on the space of probability measures on $[0,2]$. It is defined as such for two probability measures $\mu_1$ and $\mu_2$ :
$d_p(\mu_1,\mu_2) := \sum_{k=0}^p ( \mathbb{E}[X_1 ^k]- \...

**47**

votes

**4**answers

7k views

### Why isn't integral defined as the area under the graph of function?

In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also ...

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**0**answers

132 views

### Baker map-like problem

Let $S =[0, 1]^2$ denote the unit square in $\mathbb R^{2}$. For any subset $A$ of $S$ let $A^{c}$ denote its complement in $S$, and $\overline{A}$ its closure in $S$. Given a measurable map $g: W \...

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votes

**2**answers

186 views

### Is there a second countable topological space, which can not be equipped with a finite borel measure of full support?

If I have a second countable topological space X, can i Always find a finite borel measure, such that every non-empty open set has positive measure?
without second countability, the discrete topology ...

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51 views

### If $(κ_t)_{t≥0}$ is the transition semigroup of a continuous Markov process, is $t↦(κ_tf)(x)$ continuous for all bounded continuous $f$ and fixed $x$?

Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(E,\mathcal E)$ be a measurable space
$(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$ and $$\kappa_tf:=\int\kappa_t(\;\...

**3**

votes

**1**answer

101 views

### Hausdorff measure of intersection of a ball and a set in $\mathbb {R} ^ n$

Let $A$ a subset of $\mathbb R ^n$, $B=B(x,r) \subset \mathbb {R} ^n$ an open ball, and denote the $(n-1)$-dimensional Hausdorff measure in $\mathbb R ^n$ by $\mathcal H^{n-1}$. Also assume that $\...

**2**

votes

**0**answers

135 views

### 3 questions around the Stone space of the free $\sigma$-algebra on $\omega_1$ free generators

During my studies, I came across several different Stone spaces, e.g.:
(i) The Cantor cube $X=\{0,1\}^{\omega_1}$, which is the Stone space of the free Boolean algebra on $\omega_1$ free generators;
...