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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.

3
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1answer
154 views

Minkowski sum of polytopes from their facet normals and volumes

By Minkowski's work in the early 1900s, every polytope $P\subset\mathbb R^n$ is determined up to translation by its unit facet normals $u_1,\dots,u_k$ and facet volumes $\alpha_1,\dots,\alpha_k$. ...
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0answers
163 views

Egorov's and Lusin's Theorem in the space with infinite measure

Both the fundamental Egorov's and Lusin's Theorem in measure theory are given on any measurable space $X$ whose measure is finite. On the measurable space whose measure is infinite, does there ...
1
vote
1answer
42 views

Monotonicity given an implicit function containing a Measure integral

The following question seems simple but I am not sure how to handle it correctly because of the integral with respect to a measure. I would be very thankful for any reply.Cheers! Knowing that $$f(\...
3
votes
2answers
238 views

Effect of perturbing the atoms of a measure on the Wasserstein distance

Let $(X,d)$ be a metric space, $x_1,\ldots,x_N\in X$ and $x_1',\ldots,x_N'\in X$ be atoms, and $G=\sum_{i=1}^Np_i\delta_{x_i}$, $G'=\sum_{i=1}^Np_i'\delta_{x_i}$, and $G''=\sum_{i=1}^Np_i'\delta_{x_i'}...
2
votes
1answer
183 views

continuity entropy with respect gibbs measures

Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider Bernoulli measures on $X$ only. Let $f:X\to \mathbb{R}$ be Holder continuous. The measure $\mu$ is a Gibbs measure with potential $f$ if there ...
3
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0answers
107 views

Completely I-non-measurable unions in Polish spaces

Problem. Let $X$ be a Polish space, $\mathcal I$ be a $\sigma$-ideal with Borel base, and $\mathcal A\subset\mathcal I$ be a point-finite cover of $X$. Is it true that $\mathcal A$ conatins a ...
2
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1answer
102 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 ...
6
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1answer
214 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 ...
3
votes
1answer
109 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 ...
2
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1answer
252 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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2answers
449 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 ...
2
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1answer
68 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})\...
2
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0answers
79 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_\...
5
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3answers
160 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 $...
1
vote
1answer
126 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 ...
12
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3answers
528 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, $(...
3
votes
1answer
103 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. ...
2
votes
1answer
192 views

The Poisson equation

I see the following theorem in Lihe Wang's A geometric approach to the Calderon--Zygmund estimates $$ \triangle u=f\quad in \quad \> B_2. \>\quad \quad \quad \quad (1) $$ Lemma 7: There is a ...
2
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0answers
46 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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0answers
100 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 ...
2
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0answers
115 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 $(...
7
votes
0answers
84 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, ...
3
votes
1answer
229 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 $\...
8
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2answers
513 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 ...
1
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0answers
43 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 $...
4
votes
1answer
69 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 $...
1
vote
0answers
53 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-...
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 $...
1
vote
1answer
140 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 ...
5
votes
1answer
214 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....
2
votes
1answer
100 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}, ...
3
votes
2answers
308 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 $...
2
votes
1answer
107 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 ...
6
votes
1answer
128 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 ...
3
votes
1answer
381 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 \...
17
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5answers
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 ...
3
votes
0answers
158 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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0answers
130 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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0answers
59 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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0answers
63 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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0answers
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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0answers
33 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}_{...
1
vote
1answer
83 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 ...
0
votes
1answer
84 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 ...
1
vote
1answer
111 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 ...
19
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6answers
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 $\...
2
votes
0answers
77 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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0answers
250 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?
2
votes
1answer
195 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,...
4
votes
0answers
105 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. ...