Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange
Join us in building a kind, collaborative learning community via our updated Code of Conduct.

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

-1
votes
0answers
41 views

Approximation by regular sets

Let $(X,\Sigma,\mu)$ be a Radon space and $X$ be a compact metric space. By definition of $\mu$, every Borel set $A\in \Sigma$ can be approximated from within by compact sets. However, if $A$ is ...
0
votes
0answers
28 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 }...
1
vote
0answers
27 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
72 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
47 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 ...
0
votes
1answer
101 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 ...
17
votes
6answers
963 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 $\...
0
votes
0answers
35 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
236 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
186 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,...
3
votes
0answers
51 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. ...
2
votes
1answer
62 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
54 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 $(\...
1
vote
0answers
55 views

Nonlinear maps in Riesz Thorin theorem

The Riesz Thorin theorem allows us to interpolate between $L^p$ spaces and the usual assumption is that the map $T$ is linear. What I was wondering about is whether this is because otherwise you do ...
2
votes
0answers
40 views

Dependency of the Wasserstein distance on the parameter: a differential perspective

Let $\mu(dx)=\sum_{i=1}^np_i\delta_{x_i}(dx)$ and $\nu(dy)=\rho(y)dy$ be two probability measures on $\mathbb R^d$. Consider the $2-$Wasserstein distance below: $$W_2(\mu,\nu)^2 \quad := \quad \inf_{\...
0
votes
0answers
55 views

On measurability in Wiener space

Let $f$ be a complex-valued continuous function on Wiener space such that $|f|$ is measurable. Is $f$ then measurable, too? I am looking for a proof or a counterexample.
7
votes
2answers
195 views

Non-separable metric probability space

Let us say a metric probability space $(X,\rho,\mu)$ has property (*) if: the support of $\mu$ is contained in a separable subspace of $X$. Questions: 1. Is there a standard name for this property? ...
3
votes
0answers
38 views

Measure of set of vectors whose outer product are bounded

Let consider the canonical Euclidean space $E = \mathbb{R}^n$, endowed with the Lebesgue measure $\mu$. Define the map $v_k: E^k \rightarrow \mathbb{R}$ that sends a $k$-uple $x_1,\cdots, x_k$ of ...
1
vote
1answer
81 views

are there measure preserving mapping in this case?

Suppose f and g are two Borel function on [0, 1]. The push-forward of the Lebesgue measure on [0,1] by f and by g are the same. Then are there some Borel measurable function from [0,1] to [0,1], ...
3
votes
1answer
230 views

Function square-integrable

Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function $$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$ where $x_0$ is an ...
0
votes
1answer
91 views

Questions on a new definition of continuous multivariate distribution

For a univariate distribution or a univariate random variable, we call it continuous/absolutely continuous if its cumulative distribution function (CDF) is continuous/absolutely continuous. Now I am ...
1
vote
2answers
164 views

Number theory on Banach space $L^2(\mathbb R)$ meets linear independence?

Consider an orthonormal basis $(\varphi_k)$ of $L^2(\mathbb R)$ with Lebesgue measure. I came along a nice number theoretic question in analysis: Write $$f_k(x):=\int_{\left\lvert y \right\rvert \...
2
votes
0answers
111 views

Volume of critical points decrease under symmetric decreasing rearrangements?

If, $u: \mathbb{R}^n \to \mathbb{R}$ be a non-negative test function, i.e., $u \in \mathcal{D}(\mathbb{R}^n)$ and $u \ge 0$, then does it follow that, $$\mathcal{H}^{n}(\{s < u \le t \} \cap \{ \...
2
votes
0answers
58 views

Can countable additivity be removed from this elementary proof of martingale convergence of conditional probabilities?

This question is cross-posted at MSE. Let $(\Omega, \mathcal{C})$ be the Cantor space of binary sequences equipped with the usual product topology, and let $(\Omega, \mathcal{F})$ be the associated ...
9
votes
1answer
246 views

Can a big set always look small?

For a set $C\subset \mathbb R^2$, define its visibility from a point $x$ as $vis_C(x)=\{\varphi\in \mathbb S^1\mid \exists t>0~~x+t*\varphi\in C\}$, where $\mathbb S^1$ denotes the unit circle. Say ...
2
votes
1answer
82 views

Closedness of the set of probability measures anihilating a measurable function

Let $X$ be a compact metrizable topological space and let $f$ be a bounded, real valued, Borel function on $X$. Denoting by $P(X)$ the collection of all probability measures on $X$, consider the ...
1
vote
1answer
85 views

Convergence of measurable functions in a locally compact space

Set $(X,\mathcal{B})$ a measurable space. If $f:X\rightarrow[0,\infty)$ is a measurable function then exists a sequence of simple functions $\{s_n\}_{n\geq1}$ such that $$0\leq s_1 \leq s_2\leq \...
2
votes
1answer
101 views

Optimal-score partitions

The question about throwing darts asked on the MathOverflow page Sacred Geometry of Chance was not well received, apparently because of "[t]oo much noise around the actual math", as stated in a well-...
4
votes
0answers
85 views

A quantity that distinguishes finer than Hausdorff dimension

Consider sets $A\subseteq \mathbb{R}$ with Lebesgue measure zero and Hausdorff dimension one. For instance the set of real numbers with bounded entries in their continued fraction expansion have ...
2
votes
0answers
96 views

Probability bound involving random, convex sets

Let $X,Y$ and $Z$ be random vectors in $\mathbb{R}^d$ defined on $(\Omega,\mathcal{H},\mathsf{P})$ s.t. $Z$ is $\mathcal{F}$-measurable for some $\mathcal{F}\subset\mathcal{H}$. Define a family of ...
2
votes
1answer
98 views

Support of functions in Fourier domain

Let $\mathcal F$ be the Fourier transform. I would like to understand whether being in a Sobolev space implies that the Fourier transform of a function is necessarily supported on a compact ball up to ...
4
votes
2answers
121 views

Can the differential entropy of a continuous distribution with lebesgue integrable density be negative infinity?

Define the (differential) entropy for a density $f$ as $$ H(f) :=-\int_{0}^{1} f(x) \log_{2}(f(x)) dx \, .$$ I am trying to find a $f \in L_{1}([0,1])$ such that $f\geq 0, \int_{0}^{1} f(x) dx = 1$...
7
votes
0answers
64 views

When is Radon-Nikodym derivative induced by a proper map of manifolds bounded?

Let $X,Y$, be compact complex manifolds, and let $f:X\to Y$ be a smooth, proper (i.e. for each $y\in Y$, $f^{-1}(y)$ is a compact set) and surjective map. Choose metrics on $X,Y$ and let $\mu_X, \mu_Y$...
5
votes
2answers
151 views

Von Neumann's theorem on realizing automorphisms of the measure algebra

I'm looking for a proof, in English, of the following theorem due to von Neumann (which apparently originates in the paper Einige Sätze über Messbare Abbildungen, Ann. of Math, 1932): Every ...
1
vote
1answer
60 views

Borel $\sigma$-algebra on the space of Hölder continuous functions

Let $(M,d)$ be a separable metric space $E$ be a $\mathbb R$-Banach space $\alpha\in(0,1]$ Moreover, let $$\left\|f\right\|_{C^{0+\alpha}(K,\:E)}:=\sup_{x\in K}\left\|f(x)\right\|_E+\sup_{\substack{...
3
votes
0answers
54 views

Antisymmetry of the stochastic order

An ordered topological space is a topological space $X$ equipped with a partial order $\leq$ which is closed as a subset of $X\times X$. By antisymmetry of $\leq$, it follows that the diagonal of $X$ ...
2
votes
2answers
218 views

Non-probabilist term for conditional expectation?

When writing an article I encounter what is essentially a conditional expectation - function defined on a bounded interval (not necessarily of unit length) with Lebesgue measure, but information about ...
2
votes
0answers
94 views

Parametric distances on product spaces of measures

Disclaimer: Please excuse my loose language. I'm neither an expert in geometry nor probability. Please ask for clarification if something appears unclear or awkward to you. Let $X$ be a topological ...
1
vote
0answers
49 views

Are the sets whose convex hull surface admits multiple representations a shy set of sets?

Consider a compact subset $A$ of $R^n$. Let me call $A$ special if any point $x$ that is on the boundary of $\textsf{conv.hull}(A)$ admits a unique representation as a convex combination of points in $...
9
votes
0answers
204 views

Covering inequality for sets of intervals

Let $I$ and $J$ be finite sets of open intervals $(a,b)\subset\mathbb R$. For a finite set of points $P\subset \mathbb R$ we denote those subsets of intervals from $I$ and $J$ containing some point ...
3
votes
1answer
72 views

automorphisms of a measurable space can be approximated by continuous measure preserving maps?

Suppose first that $D=[0,1]$ is equipped with the usual Lebesgue measure, and that $\varphi$ is a measure-preserving transformation $\varphi:D\to D$ that is bijective and whose inverse is also measure ...
1
vote
1answer
116 views

How small can a set admitting a nonatomic finite measure be?

Is it consistent that there exists a nonzero atomless finite measure on some $\sigma$-algebra on a cardinal $\kappa$ satisfying $\kappa<\mathfrak{c}$? Can there be such a measure on $\omega_1$ ...
3
votes
0answers
44 views

$L^p$-spaces for locally convex spaces

Let $(X,\sigma)$ be a locally convex space, say generated by a family of seminorms $\mathfrak{P}$. I know that there is the notion of the space of integrable functions $f:\Omega\rightarrow(X,\sigma)$ ...
1
vote
1answer
42 views

Infinitely many independent functions that are only frequency localized?

A function $f \in L^2(\mathbb R^d)$ will be called $K$-frequency localized if the following inequality holds $$\int_{\mathbb R^d} \lvert \widehat{f}(x) \rvert^2 x^2 \ dx \le K \int_{\mathbb R^d} \...
4
votes
1answer
77 views

Non-linear translation invariant functionals on $L^1$

I have recently come across a class of (possibly non-linear) operators $F$ defined on $L^1$ such that $F \colon L^1(\mathbb R^d) \to \mathbb [0,+\infty]$; $F(u(\cdot - z)) = F(u(\cdot))$ for every $...
8
votes
1answer
180 views

Is the measurable space $(\omega_1,\mathcal{P}(\omega_1))$ separable?

Here $\omega_1$ is the first uncountable ordinal, and $\mathcal{P}(\omega_1)$ denotes the power set of $\omega_1$. Separable means countably generated as a $\sigma$-algebra.
2
votes
1answer
163 views

How “compact” are sets of finite measure?

Let $K$ be a compact set of $\mathbb R^n$, then every open cover of $K$ will have a finite subcover. Now consider the following situation: Everything I say in the following is with respect to the ...
1
vote
0answers
922 views

The Rise and Fall of Dictators & How it Depends on Our Choice

This question is loosely inspired by the following paper of Shelah on Arrow property in which he answered a question of Gil Kalai affirmatively. Shelah, Saharon, On the Arrow property. Adv. in Appl. ...
3
votes
2answers
145 views

Existence of a separating affine functional

Let be $S$ a separable(non compact) metric space and $X=C_b(S)$ the set of all bounded continuous functions, then it's topological dual $X^{\star}=rba(S)$ is the set of all regular Borel additive ...
3
votes
0answers
97 views

Tangent space and gradient on subspace of Wasserstein space given by finitely supported measures

Let $\mathcal{P}_2(M)$ be the 2-Wasserstein space over some Riemannian manifold $(M,g)$ (connected, complete, and without boundary). Let $\mathcal{FP}_2(M,n)$ be the subspace of probability measures ...