Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
2
votes
1answer
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 ...
0
votes
0answers
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 ...
5
votes
1answer
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 ...
2
votes
1answer
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 ...
2
votes
1answer
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 ...
-4
votes
0answers
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}{...
8
votes
2answers
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 ...
3
votes
1answer
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
$$
f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\...
2
votes
0answers
76 views
Inequality concerning BV norm
Let $u(x) \in L^1( \mathbb{R}^n) \cap BV(\mathbb{R}^n)$ and let $\rho\ge 0$ be the standard mollifier on $\mathbb{R}^d$, supported in unit ball with $\int_{\mathbb{R}^d}\rho \,dx=1$ and define $\rho_\...
4
votes
3answers
133 views
Regular Borel measures and the measure of a singleton
I'm studying this paper: http://matwbn.icm.edu.pl/ksiazki/sm/sm73/sm7313.pdf
At the top of page 36, it states the following Proposition:
Let $S$ be a compact and $\mu$ a regular Borel measure on $...
1
vote
1answer
101 views
Characterization of state spaces of Boolean algebras
A state space of a Boolean algebra is a Choquet simplex but not all Choquet simplices can be viewed as state spaces of Boolean algebras. Is it known which Choquet simplices are precisely state spaces ...
12
votes
3answers
493 views
Which bounded sequence can be realized as the Fourier Series of a probability measure on the circle?
Given a finite Borel measure $\mu$ on $\mathbb{S}^1 = \mathbb{R}/\mathbb{Z}$, define its Fourier coefficients by
$$ \hat\mu(n) = \int e^{2i\pi nx} d\mu(x) \qquad\forall n\in \mathbb{Z}.$$
Clearly, $(...
3
votes
1answer
81 views
Every closed subspace $A$ of $C_0(K)$ can be regarded as a subspace of continuous functions on $A^*$?
We consider a locally compact Hausdorff space $X$ and the Banach space $C_0(X)$ of continuous functions on $X$ taking values at $\mathbb K = \mathbb R$ or $\mathbb C$, equipped with the supremum norm.
...
2
votes
0answers
62 views
The Poisson equation
I see the following theorem in Lihe Wang's A geometric approach to the Calderon--Zygmund estimates
$$\triangle u=f,in \> B_2 \>(1)$$
Lemma 7: There is a constant $N_1$ so that for any $ε > ...
2
votes
0answers
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$, ...
2
votes
0answers
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 ...
2
votes
0answers
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 $(...
-1
votes
0answers
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?
7
votes
0answers
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, ...
3
votes
1answer
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 $\...
-1
votes
0answers
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 ...
8
votes
2answers
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 ...
1
vote
0answers
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 $...
4
votes
1answer
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 $...
1
vote
0answers
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-...
3
votes
1answer
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 $...
1
vote
1answer
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 ...
3
votes
1answer
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....
2
votes
1answer
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}, ...
3
votes
2answers
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 $...
2
votes
1answer
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 ...
6
votes
1answer
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 ...
3
votes
1answer
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 \...
17
votes
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
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}...
6
votes
0answers
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 ...
0
votes
0answers
58 views
Reference Request: Egoroff Theorem for nets
Does there exist a generalization of Egoroff theorem for nets instead of sequences of functions?
1
vote
0answers
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.
...
0
votes
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
$$d((x,y),(x',y')) = \begin{cases}d_X(x,x'),&\mbox{ if }...
2
votes
0answers
32 views
Invariance of simple functions
Let $(A(s),D(A(s)))_{s\in\mathbb{R}}$ be a family of unbounded operators on a Banach space $X$ and $g:\mathbb{R}\rightarrow X$ be a simple function, i.e.,
\begin{align*}
g=\sum_{i=1}^n{x_i\textbf{1}_{...
1
vote
1answer
79 views
A distribution which is Wasserstein-close to a compactly supported distribution is almost compactly supported
I wonder whether this is true: If a distribution is very close (in the Wassertein sense) to another distribution with compact support, then the former must put only tiny amount of mass outside a ...
0
votes
1answer
71 views
Differences between the reduced Borel field and the category algebra of a space
Let $X$ be a topological space.
Halmos calls "reduced Borel field" the quotient $B(X)/M(X)$ where $B(X)$ is the Borel field of $X$ and $M(X)$ is the $\sigma$-ideal of meagre subsets of $X$.
Fremlin ...
1
vote
1answer
108 views
Operator power of another operator
I was reading a paper and encountered the following notation:
Let $\mathcal{H}=\ell^2(\mathbb{Z})$ and $\{e_p\}_{p\in \mathbb{Z}}$ be an orthonormal basis of $\mathcal{H}$.Define
$$ue_p=e_{p+1}\quad ...
18
votes
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
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 ...
1
vote
0answers
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?
2
votes
1answer
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,...
4
votes
0answers
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.
...
4
votes
1answer
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 ...
3
votes
0answers
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 $(\...
