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

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86 views

### Measure preserving coordinates of $S^2$ from $[0,1]^2$

Consider the unit sphere $S^2 = \left\{x\in\mathbb{R}^3 ~ {\large|} ~ |x|=1 \right\}$ and denote the uniform (Lebesgue) measures on the $S^2$ and $[0,1]^2$ by $m_S$ and $m_2$, respectively.
Question ...

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**0**answers

17 views

### Weak* convergence in a dual Banach lattice vs norm convergence of moduli

Let $E$ be a dual Banach lattice, that is, $E = E_*^*$ for some Banach lattice $E_*$ (I have $M(X)=C(X)^*$ specifically in mind for a compact space $X$).
Suppose that $(x_n)$ is a weak*-null ...

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votes

**1**answer

117 views

### Weak*-convergence of signed measures

Let $X$ be a compact Hausdorff space and let $M(X)$ denote the space of signed measures that is naturally dual to $C(X)$, the space of continuous functions on $X$. I am interested whether the ...

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votes

**1**answer

65 views

### Borel $\sigma$-algebra in $\beta \mathbb N \times \beta \mathbb N$

For a second-countable space $X$, we have $${\rm Bor}(X\times X) = {\rm Bor}\, X \otimes {\rm Bor}\, X,$$ that is the Borel $\sigma$-algebra of the product is the product $\sigma$-algebra. Some ...

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votes

**1**answer

178 views

### Is there a finitely additive measure on R which is not sigma-additive?

Consider the usual measurable space of real number $( \mathbb{R}, \mathcal{B}(\mathbb{R}))$.
My question is:
Is there an application $\mu$ on $\mathcal{B}( \mathbb{R}) \rightarrow [0,+\infty]$ such ...

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votes

**0**answers

59 views

### the limits of series [on hold]

let $S_n=\sum^n_{i=1} \frac{3 i^2}{4^i-1}$
$$\frac{3 i^2}{4^i-1} \leq \frac{3 i^2}{3^i}=\frac{i^2}{3^{i-1}}$$
put $A_i=\ln \frac{i^4}{3^{i-1}}$
$$A_i=4 \ln i- (i-1)\ln 3 $$
for $i>1$ $A_i=\frac{1}{...

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votes

**2**answers

416 views

### Is taking the positive part of a measure a continuous operation?

Here is question I tried to answer for some time - it seems to be straightforward, but I have trouble figuring it out.
Let $\Omega$ be a compact domain in $\mathbb{R}^n$. For any signed Borel measure ...

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votes

**1**answer

57 views

### Echange of Infimum Integral with Pointwise Infimum

Setup
Suppose that $U$ is a subset of $L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F})$ defined by
$$
f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\...

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votes

**0**answers

76 views

### Inequality concerning BV norm

Let $u(x) \in L^1( \mathbb{R}^n) \cap BV(\mathbb{R}^n)$ and let $\rho\ge 0$ be the standard mollifier on $\mathbb{R}^d$, supported in unit ball with $\int_{\mathbb{R}^d}\rho \,dx=1$ and define $\rho_\...

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votes

**3**answers

133 views

### Regular Borel measures and the measure of a singleton

I'm studying this paper: http://matwbn.icm.edu.pl/ksiazki/sm/sm73/sm7313.pdf
At the top of page 36, it states the following Proposition:
Let $S$ be a compact and $\mu$ a regular Borel measure on $...

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vote

**1**answer

101 views

### Characterization of state spaces of Boolean algebras

A state space of a Boolean algebra is a Choquet simplex but not all Choquet simplices can be viewed as state spaces of Boolean algebras. Is it known which Choquet simplices are precisely state spaces ...

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votes

**3**answers

493 views

### Which bounded sequence can be realized as the Fourier Series of a probability measure on the circle?

Given a finite Borel measure $\mu$ on $\mathbb{S}^1 = \mathbb{R}/\mathbb{Z}$, define its Fourier coefficients by
$$ \hat\mu(n) = \int e^{2i\pi nx} d\mu(x) \qquad\forall n\in \mathbb{Z}.$$
Clearly, $(...

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votes

**1**answer

81 views

### Every closed subspace $A$ of $C_0(K)$ can be regarded as a subspace of continuous functions on $A^*$?

We consider a locally compact Hausdorff space $X$ and the Banach space $C_0(X)$ of continuous functions on $X$ taking values at $\mathbb K = \mathbb R$ or $\mathbb C$, equipped with the supremum norm.
...

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**0**answers

62 views

### The Poisson equation

I see the following theorem in Lihe Wang's A geometric approach to the Calderon--Zygmund estimates
$$\triangle u=f,in \> B_2 \>(1)$$
Lemma 7: There is a constant $N_1$ so that for any $ε > ...

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**0**answers

39 views

### Convergence to the probability generating function of a Poisson process

I'm working currently with a Poisson process trying to proove Renyi's Theorem, so far I want to show that
$\prod_{i=1}^{k_n}[z + (1-z)e^{-\mu(A_{n_i})}] \to e^{-(1-z)\mu(A)}$ as $\mu(A_{n_i}) \to 0$, ...

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votes

**0**answers

86 views

### Can computers take uniform samples from a polytope?

For each $r \in \mathbb N $ write $\mathbb Z/ 10^r = \{a/10^r: a \in \mathbb Z\}$ and $P(r)$ for the lattice $(\mathbb Z/10^r)^N \subset \mathbb R^N$.
Suppose the plane $P \subset \mathbb R^N$ is ...

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**0**answers

76 views

### Lebesgue density theorem for “doubling uniformly covering collections of subsets”

I am looking for a version of Lebesgue density theorem that works when restricting to "good" collections of balls with respect to (not necessarily doubling) metric measure spaces. Specifically
Let $(...

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**0**answers

22 views

### Q: The relationship between convergence of sequence (fn) almost everywhere, almost uniformly and convergence in measure [migrated]

Q:how we can discribe relationship between convergence of sequence (fn) almost everywhere, almost uniformly and convergence in measure?

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**0**answers

79 views

### Volume of a neighborhood of singular matrices

Suppose we take the set of all $n\times n$ real matrices with entries in $[0,1]$ in Euclidean space. Let $N_\epsilon$ be the $\epsilon$ neighborhood of the set of all singular matrices in this space, ...

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votes

**1**answer

221 views

### Measures on sites

Let $\mathcal{C}$ be a category equipped with a Grothendieck topology $\tau$, i.e., a site.
Under which conditions on $\mathcal{C}$ can one construct a Borel $\sigma$-algebra, $\sigma_\tau$, for $\...

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**0**answers

45 views

### A usage of mean value theorem for Nemytskii operators

Let $F$ be a real-valued continuously differentiable function over $\mathbb{R}$. Let $\Omega$ be a bounded set in $\mathbb{R}^2$; and $w_1$ and $w_2$ be in $H^1_0(\Omega)$. I am doing this calculation ...

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**2**answers

378 views

### Measures and differential forms on manifolds

Let $M$ be a differentiable manifold. Let $\mu$ be a (probability) measure on $M$.
What are the conditions under which $\mu$ is given by a differential form on $M$? I imagine some sort of ...

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**0**answers

34 views

### On different norms of the interpolating operator

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 $...

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**1**answer

64 views

### Find $p$ s.t. there is a sequence of nodes in $[0,1]$ s.t. sequence of interpolating polynomials of every continuous function converges in $p$-norm

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 $...

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vote

**0**answers

46 views

### Existence of Time-Reversed Markov Kernels

Suppose I have a probability measure $\pi$ and a Markov kernel $q$ which leaves $\pi$ invariant, in the sense that
\begin{align}
\int_x \pi(dx) q(x \to dy) = \pi(dy)
\end{align}
Then, a (the) time-...

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votes

**1**answer

158 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 $...

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vote

**1**answer

98 views

### Is the boundary of an open set in a $\sigma$-space empty?

Recall that a Boolean space is a $\sigma$-space in case the closure of every open Borel set is open.
Let $\{B_i\}$ be a denumerable family of open-closed sets in a $\sigma$-space $X$. Then $\bigcup_i ...

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votes

**1**answer

167 views

### What is to Stone space of the free sigma-algebra on countably many generators?

I asked the question on MSE.
https://math.stackexchange.com/questions/2898377/what-is-the-stone-space-of-the-free-sigma-algebra-on-countably-many-generators
The answer I got, however, seems disputed....

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votes

**1**answer

94 views

### time delay ergodic theorem

given dynamic system $(X, \mathcal{B}, F, \mu), \mu \circ F^{-1}=\mu, F $ is mixing, $ A \in \mathcal{B}, s.t. \mu(A) >0 $.
consider dynamic system $(X\times X, \mathcal{B}\otimes \mathcal{B}, ...

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**2**answers

285 views

### Important examples or applications of lattices in locally compact groups

If $G$ is a locally compact group and $\Gamma $ is a discrete subgroup such that the quotient $G / \Gamma$ carries a finite left $G$-invariant Haar measure, then we say that $\Gamma$ is a lattice in $...

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**1**answer

84 views

### Direct proof a property of hyperstonean spaces

First, let me state some basic facts and definitions for my question. I believe these are well-known among experts working on von Neumann algebras, but let me state them anyway since my question is ...

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votes

**1**answer

110 views

### Transportation-cost inequality for pushforward measure

Let $X=(X,d_X)$ and $Y=(X,d_Y)$ be metric spaces and $\varphi: X\rightarrow Y$ be an $L$-Lipschitz map, with $0 \le L < \infty$. Suppose $\mu$ is a probability measure on $X$ which satisfies ...

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votes

**1**answer

159 views

### wasserstein distance between distributions with bounded ratio

Let $p$ and $q$ be probability distributions on a metric space $X=(X, d)$ with densities $dp$ and $dq$, such that there exists $0 < \alpha < \beta < \infty$ satisfying
$$
\alpha d p \le dq \...

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votes

**5**answers

1k views

### Why is Lebesgue measure theory asymmetric?

A set $E\subseteq \mathbb{R}^d$ is said to be Jordan measurable if its inner measure $m_{*}(E)$ and outer measure $m^{*}(E)$ are equal.However, Lebesgue mesure theory is developed with only outer ...

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votes

**0**answers

133 views

### Cardinality of extreme points of finitely additive probabilistic extensions

Let $\Omega = \{0,1\}^\mathbb{N}$, let $\mathcal{A}$ be the algebra generated by the open subsets of $\Omega$, where we use the product of discrete topologies, and let $\mathcal{F} = \sigma(\mathcal{A}...

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**0**answers

112 views

### What (if any) was known about null sets before Lebesgue?

The notion af a null set, i. e., a set of Lebesgue measure zero, does not require a full blown construction of Lebesgue measure:
A set is $E\subset \mathbb{R}$ is called a null-set if it can be ...

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**0**answers

58 views

### Reference Request: Egoroff Theorem for nets

Does there exist a generalization of Egoroff theorem for nets instead of sequences of functions?

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54 views

### Dense Egoroff theorem

Suppose that $f_n:X\rightarrow V$ is a sequence of continuous functions from a compact metric space $X$ to a Banach space $V$ and let $\mu$ be a Radon measure on $X$ and $\epsilon>0$ be given.
...

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37 views

### Reformulate Wasserstein constraint optimization on product space in terms of marginal

Let $X = (X,d_X)$ be a metric space and $Y$ be an abstract set (with at least two elements). Consider the metric on $X \times Y$ defined by
$$d((x,y),(x',y')) = \begin{cases}d_X(x,x'),&\mbox{ if }...

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32 views

### Invariance of simple functions

Let $(A(s),D(A(s)))_{s\in\mathbb{R}}$ be a family of unbounded operators on a Banach space $X$ and $g:\mathbb{R}\rightarrow X$ be a simple function, i.e.,
\begin{align*}
g=\sum_{i=1}^n{x_i\textbf{1}_{...

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vote

**1**answer

79 views

### A distribution which is Wasserstein-close to a compactly supported distribution is almost compactly supported

I wonder whether this is true: If a distribution is very close (in the Wassertein sense) to another distribution with compact support, then the former must put only tiny amount of mass outside a ...

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**1**answer

71 views

### Differences between the reduced Borel field and the category algebra of a space

Let $X$ be a topological space.
Halmos calls "reduced Borel field" the quotient $B(X)/M(X)$ where $B(X)$ is the Borel field of $X$ and $M(X)$ is the $\sigma$-ideal of meagre subsets of $X$.
Fremlin ...

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**1**answer

108 views

### Operator power of another operator

I was reading a paper and encountered the following notation:
Let $\mathcal{H}=\ell^2(\mathbb{Z})$ and $\{e_p\}_{p\in \mathbb{Z}}$ be an orthonormal basis of $\mathcal{H}$.Define
$$ue_p=e_{p+1}\quad ...

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votes

**6**answers

1k views

### Lebesgue measure theory applications

I'm looking for reasonably simple examples of applications of Lebesgue measure theory outside the measure theory setting. I give an example.
Theorem: Let $X$ be a differentiable submanifold of $\...

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**0**answers

65 views

### A variant of the optimal transport

Let $\mu$, $\nu$ and $\gamma$ be three probability measures on $\mathbb R$. Consider the optimisation problem as follows:
$$\inf_{(X,Y,Z)}~ \mathbb E\big[|Y-Z|^2\big],$$
where the inf is taken ...

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**0**answers

241 views

### Is that correct $\mathbb R^2\cong\mathbb R$ as measurable spaces? [closed]

Is that correct $R^2\cong R$ as measurable spaces?
If we consider $R$ and $R^2$ with Borel $\sigma$-algebras, is there measurable map from $R$ to $R^2$ with measurable inverse?

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**1**answer

190 views

### Schwartz space on $\bigcup_{n=1}^CR^n$

I have an application where I need to work with the following idea.
Let the space $\bigcup_{n=1}^C \mathbb{R}^n$ be associated with the metric $d$ such that for $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,...

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72 views

### Cadlag and adapted (usual conditions assumed) imply progressively measurable (related to Protter's Stochastic Calculus theorem 6)

Hi maybe someone on here can help me. I have been stuck on showing this fact for several months. I asked this question in the stack exchange and it has floated around for a while but to no avail.
...

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votes

**1**answer

106 views

### Improving equi-integrability for a family $\mathcal F$ in $L^1(\Omega)$

Let $\mathcal F$ be a weakly compact subset of $L^1(\Omega)$. Dunford–Pettis theorem says that $\mathcal F$ is uniformly integrable. Also, by de la Vallée-Poussin theorem we can find an increasing ...

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**0**answers

55 views

### A conjecture characterizing almost uniform convergence of finitely additive conditional probabilities

This question is a continuation of a question I asked a couple weeks ago.
Let $(\Omega, \mathcal{C})$ be the Cantor space of binary sequences equipped with the usual product topology, and let $(\...