Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
4
votes
0answers
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
votes
0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
0answers
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 ...
-1
votes
0answers
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
0answers
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
1answer
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
votes
0answers
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
1answer
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
1answer
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 $\...
-1
votes
0answers
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
0answers
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
1answer
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
1answer
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
2answers
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
0answers
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
1answer
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
1answer
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 ...
9
votes
0answers
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
1answer
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 ...
1
vote
1answer
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
1answer
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
0answers
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
0answers
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
votes
0answers
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
1answer
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 $\...
0
votes
0answers
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^...
7
votes
2answers
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
1answer
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
1answer
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
0answers
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
votes
0answers
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
$$
\theta(d) := \max_{\mu} \int_{S^{d-1}} \int_{S^{d-1}} \cdots \int_{S^{d-1}} |x_1 \...
2
votes
1answer
145 views
Research-type questions in probability illustrating measure-theoretical techniques for students
In short, in the perspective of preparing a new course, I am looking for examples of "concrete" (hopefully research-type) questions concerning various models in probability theory which give the ...
2
votes
2answers
127 views
Questions about some properties of random probabilities and random expectations
Let $(\Omega, \mathcal{A}, \mathbb P)$ be a probability space with $\mathcal{A}$ countably generated, and let $P: \mathcal{A} \times \Omega \to [0,1]$ be a random probability measure. By that I mean $...
2
votes
0answers
50 views
The dual of the space of continuous sections in a vector bundle
If $X$ is a compact Hausdorff space, one may view the space of complex, continuous functions on it as the space of continuous sections in the trivial Hermitian bundle $X \times \mathbb C$. By the ...
8
votes
1answer
182 views
Measure support decomposition that “tends to infinity”
I would like to know the answers to the following two questions.
Let $S$ be a locally compact Hausdorff space, $\mu$ be a regular Borel measure with non-compact support $M$. Denote
$$
\mathscr{H}=\{\...
1
vote
0answers
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
2answers
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
0answers
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
2answers
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 ...
0
votes
0answers
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
4answers
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 ...
2
votes
0answers
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 \...
3
votes
2answers
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 ...
0
votes
0answers
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
1answer
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
0answers
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;
...
