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Questions tagged [measure-theory]

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

4
votes
1answer
385 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 \...
6
votes
0answers
226 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 $...
3
votes
1answer
116 views

Volume form under holomorphic automorphisms

$(M,\omega)$ is a compact Kaehler manifold and $f_{t,s}$ are 1-parameter group generated by holomorphic vector fields $V_s$. My question is whether the function $\frac{f_{t,s}^* \omega^n}{\omega^n}$ ...
0
votes
0answers
61 views

Duality mapping of a the space of continuous functions? [on hold]

The duality map $J$ from a Banach space $Y$ to its dual $Y^{*}$ is the multi-valued operator defined by: $J(y)=\{\phi\in Y^{*}:\, \left< y,\phi\right >=\Vert y\Vert^{2}=\Vert \phi \Vert^{2}\},...
3
votes
1answer
93 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 the countably infinite product space equipped with the product topology. Let $\mathcal A$ be any field of ...
3
votes
1answer
76 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
91 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\...
4
votes
2answers
120 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 = ...
1
vote
0answers
33 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 ...
13
votes
6answers
2k views

Applications of Rademacher's Theorem

Rademacher's Theorem (that every Lipschitz function on $\mathbb{R}^{n}$ is almost everywhere differentiable) is a remarkable result on the structure of the space of Lipschitz functions, but I was ...
3
votes
0answers
59 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 [closed]

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
127 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
186 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
63 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
180 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 $\Delta$ ...
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 ...
5
votes
1answer
377 views

Base zero-dimensional spaces

Definition. A zero-dimensional topological space $X$ is called base zero-dimensional if for any base $\mathcal B$ of the topology that consists of closed-and-open sets in $X$, any open cover $\mathcal ...
27
votes
1answer
849 views

Are the Sierpiński cardinal $\acute{\mathfrak n}$ and its measure modification $\acute{\mathfrak m}$ equal to some known small uncountable cardinals?

This question was motivated by an answer to this question of Dominic van der Zypen. It relates to the following classical theorem of Sierpiński. Theorem (Sierpiński, 1921). For any countable ...
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 $\...
2
votes
2answers
136 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 ...
1
vote
0answers
127 views

References about distances between singular probability measures

I would be interested in references on the topic of distances between probability measures that are singular with one another and not reduced to trivial ones. For example from here we know that total ...
-1
votes
0answers
44 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_{...
2
votes
1answer
76 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
0answers
69 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 ...
1
vote
1answer
56 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 $...
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 ...
6
votes
4answers
742 views

Haar Measure on Locally Compact Semigroups

I'm reading on Haar measure and we know that every locally compact group admits a Haar measure, is the same true for semigroups? if not, is there a class of semigroups that admits a Haar measure? ...
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
159 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. ...
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 \...
0
votes
2answers
239 views

Uncountable infimum of measurable functions

I know that in general it does not hold, but there exists a positive result under some conditions for which the following claim holds? Claim: given an uncountable set of measurable functions $f_i : U ...
7
votes
1answer
534 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 ...
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}=\{\...
38
votes
18answers
60k views

Suggestions for a good Measure Theory book

I have taken analysis and have looked at different measures, but I am currently looking at realizing a certain problem in a different light and feel that I need a better background in various measures ...
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 ...
7
votes
0answers
363 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 $...
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 ...
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 $m:\mathcal{P}(\mathbb{R}) \rightarrow [0,\infty]$ ...
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 ...
5
votes
2answers
206 views

Given any sequence of interpolating nodes, can we find a continuous function $f$ whose interpolating polynomials doesn't converge to $f$ point-wise

Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
3
votes
0answers
74 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 ...
27
votes
2answers
818 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 ...
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 ...
7
votes
2answers
201 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 $|x|=\lceil\log_2x\rceil$, ...
0
votes
0answers
110 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^...
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
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 ...
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(...
8
votes
2answers
736 views

Patching together homeomorphisms: how badly can it fail?

Suppose we have a set $X$ with $X=U \cup V$. If we pick a permutation $f$ of $U$ and a permutation $g$ of $V$ which agree on the intersection $U \cap V$, we can coalesce them into one big endo-map $F$ ...