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

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-1
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1answer
61 views

A question about Lebesgue density [on hold]

Is there a set $ A \subseteq \mathbf{R} $ such that the upper lebesgue density of $ A $ and the upper lebesgue density of $ \mathbf{R} \setminus A $ are equal to $ 1 $ at a fixed point? I would say ...
2
votes
1answer
37 views

Is there a curve on a surface where an integrable function is pointwise bounded?

I consider $\Sigma:=S^2$, as a unit radius sphere in $\mathbb R^3$ so that it has an induced metric, measure etc on it. Is the following statement true? For a large constant $K$, there exist ...
2
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0answers
36 views

Are functions whose partial derivatives are simple functions dense in $W^{1,\infty}$?

In a 2D domain, are the functions whose partial derivatives are simple functions dense in $W^{1,\infty}$ ?
11
votes
1answer
482 views

A question of Erdős

In the following paper (pages 122-23), Erdős asks if there is a constant $c > 0$ such that every subset $A$ of plane of area more than $c$ contains the vertices of a triangle of unit area. Is this ...
5
votes
2answers
169 views

If $f : [-a,a] \rightarrow \mathbb{IR}$ is Scott continuous, why are $f^-$ and $f^+$ measurable?

In "A domain theoretic account of Picard's theorem" (http://www.doc.ic.ac.uk/~dirk/Publications/icalp2004.pdf), the authors assert the following. Let $\mathbb{IR}$ be the interval domain $\lbrace ...
1
vote
0answers
50 views

Fractional Sobolev space as a Cameron-Martin space

In their paper "On Fractional Brownian Processes" (link here to the working paper), Feyel and Pradelle (1997) write in the introduction: "we give very simple proofs of the existence of different ...
1
vote
0answers
81 views

Measurable selections of a finite familiy of measures

EDIT. I'm adding a missing hypothesis and a really TL;DR version of the core problem. Warning: This short statement is the strongest form of what I want, hence not as plausible as the original form. ...
0
votes
0answers
148 views

What does the following space look like?

Pick fixed $a=(a_1,a_2,\dots,a_d)\in\{\pm1\}^d$. Consider map $F_a:\underbrace{\Bbb R^n\times\dots\times \Bbb R^n}_d\rightarrow\Bbb R^n$ given by $F(x_1,\dots,x_d)=\sum_{i=1}^da_ix_i$. Denote ...
1
vote
1answer
82 views

Entropy, Convergence

imagine you have a sequence $\eta_{n}$ of (shift) invariant measures in the Bernoulli space $\{0,1\}^{\mathbb{N}}$ that satisfy the following: there are a $0<\delta <1$ and an $N$ such that $$n ...
0
votes
0answers
118 views

Question on product measure

Let $(\Omega_1,\Sigma_1,\mu_1)$ and $(\Omega_2,\Sigma_2,\mu_2)$ be two totally finite measure spaces (which implies that $\Sigma_1$ and $\Sigma_2$ are $\sigma$-algebras). (As usual ...
28
votes
2answers
1k views

Measurability and Axiom of choice

In some situations, you need to show Lebesgue-measurability of some function on $\mathbb{R}^n$ and the verification is kind of lengthy and annoying, and even more so because measurability is "obvious" ...
5
votes
0answers
39 views

Measure-minimizing simplex with fixed inradius

Let $\Delta^n$ be an $n$-simplex in $\mathbb{R}^n$. Let $V$ be the volume and $r$ the inradius (radius of the inscribed sphere) of $\Delta^n$. There is a well-known result that $$ V \geq ...
1
vote
1answer
111 views

Is it true that for dimension d=3, if $v\in H_0^1(\Omega)$ but $v\notin L_\infty(\Omega)$ then exponent of v or -v is not summable?

I have the following Question: 1) Is it true that if $\Omega\subset\mathbb R^3$, $\Omega$ - bounded, $v\in H_0^1(\Omega)$ but $v\notin L_\infty(\Omega)$ implies $\int\limits_{\Omega}{e^vdx}=+\infty$ ...
4
votes
0answers
118 views

Equidistribution of spheres in $\mathbb{R^2}/\mathbb{Z^2}$

Let $\mathbb{H^2}$ be the hyperbolic upper half place, and let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$ acting on $\mathbb{H^2}$. A proof of the equidistribution of spheres on $\mathbb{H^2/\Gamma}$ ...
4
votes
2answers
153 views

Is taking the product of signed measures weakly continuous?

For a Polish space $X$, let $C_b(X)$ denote the real Banach space of bounded continuous real-valued functions on $X$. Let $M(X)$ denote the space of all finite signed Borel measures on $X$, equipped ...
0
votes
0answers
60 views

What does the Plancherel theorem say about positive-definite distributions?

I'm trying to understand the answer to this MO question: Bochner's theorem for measures of positive type, which suggests a relationship between Bochner's theorem and the Plancherel theorem. The ...
2
votes
2answers
172 views

Commutative von Neumann algebras and localizable measure spaces

This is not my subject so I apologize if my question is too obvious or understood from other pages. I read some pages such as Reference for the Gelfand-Neumark theorem for commutative von Neumann ...
4
votes
2answers
312 views

Is a specific sequentially closed subset of $M([0,1])$ closed?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$ (with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all ...
0
votes
0answers
44 views

Does a positive-measure subset of the unit interval almost surely intersect a random translation of some countable subgroup of $\mathbb{R}$?

Please forgive me if this is a very easy question. Let $A \subset [0,1]$ be a Borel-measurable set with strictly positive Lebesgue measure. Does there necessarily exist a countable subgroup $G$ of ...
4
votes
1answer
188 views

continuity of the Boltzmann entropy in the Wasserstein metric

For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let $$ ...
0
votes
1answer
71 views

Approximation of general measurable maps by simple functions [closed]

Let $f : (\Omega, \mathcal F) \to (\mathbb R, \mathcal B(\mathbb R)$ be a measurable map, then it is well-known that $f$ could be approximated by a sequence $(f_n)$ of simple measurable functions, ...
2
votes
1answer
65 views

Sub-$\sigma$-algebras and conditional expectation

Is it true that any sub-$\sigma$-algebra of a Rokhlin-Lebesgue space is induced (up to completion) by a measurable map into another Rokhlin-Lebesgue space? In other words, is it true that conditional ...
5
votes
0answers
145 views

Sierpinski sets and extensions of Lebesgue measure

I am duplicating an old problem from stackexchange: Suppose that every countably generated sigma algebra extending the Borel sigma algebra on $[0, 1]$ admits a measure extending the Lebesgue measure ...
0
votes
1answer
181 views

Axiomatic explanation of why the volume of a parallelepiped is equal to the area of its base times its height [closed]

I asked this in MSE, it flashed and disappeared. Let $V_n$ be the volume on the set of polytopes in $\mathbb R^n$, defined axiomatically, i.e. a functional, that assigns to each polytope ...
0
votes
1answer
35 views

Convergence of generalized expectation under total variation norm

Consider the space of sub-distributions (i.e. positive measures of variation norm lower than 1) over a discrete subset $S$ of $\mathbb{R}$ (this set is measured by its powerset). Let $(\mu_n)_n$ be a ...
8
votes
2answers
627 views

Is $f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}$ essentially bounded?

Let $$f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)} $$ Is it true that $\|f\|_{L^{\infty}(\mathbb{R}^2)}<\infty$? i.e. is $f$ essentially bounded?
1
vote
1answer
54 views

When is the hitting time of an open set a stopping time?

Let $(\Omega, \mathcal{F},P)$ be a probability space and $(\mathcal{F}_t)_{t \in [0,T]}$ a filtration. Consider an adapted, right-continuous process $X$ taking values in $\mathcal{X}$ and let $B$ be ...
4
votes
1answer
203 views

Measure algebra of a total extension of Lebesgue measure

Solovay shows that the existence of a measurable cardinal is equiconsistent with the existence of a countably additive extension of Lebesgue measure that is defined on all sets of real numbers. Given ...
0
votes
0answers
36 views

The use of Haar measure in the Blichfeldt-Minkowski Lemma

I'm trying to understand a proof of the following result Theorem: Let $K$ be a number field, and $|| \cdot ||$ the idelic norm (product of the normalized absolute values at each place). There ...
4
votes
2answers
205 views

Is the space of signed finite measures on a compact set $M([0,1])$ a sequential space?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$ (with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all ...
8
votes
1answer
186 views

Is the Jordan decomposition of a self-adjoint functional constructive?

Let $A$ be an abstract C*-algebra, and let $\varphi\colon A \rightarrow \mathbb C$ be a bounded linear function. Assuming the axiom of choice, there exist unique positive bounded linear functions ...
0
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0answers
114 views

Finitely additive measure over integers [duplicate]

We know that, with Axiom of Choice (AC), it can be shown that there exists a finitely additive uniform distribution defined for all subsets of the integers (see, e.g., Hrbacek and Jech 1999, Ch. 11). ...
1
vote
1answer
75 views

First mean value theorem for integration and Lebesgue measureability

According to first mean value theorem for integration, if $G \ : \ [a,b] \to \mathbb{R}$ is a continuous function, there exists $x \in (a,b)$ such that $$\int_a^b G(t) dt = G(x)(b-a)$$ Assume $G$ is ...
1
vote
0answers
108 views

Is the implication ($f$ is Riemann integrable over $D_1$ and $D_2$) $\Rightarrow $ ($f$ is Riemann integrable over $D=D_1\cup D_2$) true?

Let $D_1,D_2$ be a bounded subset of $\mathbb{R}^n$ and $\partial D_1,\partial D_2$ are both of Lebesgue measure zero (that is to say: $D_1,D_2$ are Jordan measurable). Also, let $f:D_1\cup ...
0
votes
0answers
38 views

Interchange limit with integral for subsequence in subsequence of general distribution functions

I asked this question on math.stackexchange a few days ago but didn't get any response, so I thought I would try here. I'm trying to find a solution for the following problem: Let ...
0
votes
2answers
115 views

Continuity of an integral

Let $f \in L^1(\Omega)$, where $\Omega$ is a bounded set in $\mathbb{R}^n$ Let $Z = (z_1,z_2,\dots,z_k)$ denote a k-tuple where each $z_i \in \Omega$ Consider $$F(Z) = \int_\Omega f(y)\min_{1\leq i ...
0
votes
0answers
38 views

convex function with distributional Hessian $D^2 f \leqslant \lambda$, $\lambda$-concave?

Let $f:R^n \to R$ be convex (may not $C^1$), $$[D^2f]=[D^2f]_{ac}+[D^2f]_s=[h_{ij}] L^n+[D^2f]_s$$ is the Lebesgue decomposition of the Hessian matrix. Where $[h_{ij}]$ is the density w.r.t the ...
2
votes
2answers
292 views

Can one estimate the distribution of eigenvalues of a matrix by its Cauchy/Stieltje transform?

Given a real symmetric $n$ dimensional matrix $A$, with eigenvalues $\lambda_i$ I am defining its Cauchy transform as the function, $f_A(z) = \sum_i \frac{1}{z-\lambda_i}\,$ Is there any information ...
6
votes
1answer
220 views

Can ITTM recognize a non-measurable set?

Throughout the question ITTM refers to Hamkins' infinite Turing machines, though I will be interested in results related to stronger models. Recently I was wondering, is it consistent that there is ...
2
votes
0answers
48 views

Sequences of transition probability measures

Suppose that $X$ and $Y$ are compact metric spaces. A Borel probability measure $\mu$ on $X\times Y$ satisfies $$ \mu(A\times B)=\int_A\mu(B|x)\mu_X(dx), $$ for $A$ and $B$ Borel sets in $X$ and $Y$ ...
4
votes
2answers
635 views

Polish by compact is Polish?

Let $X,Y$ be separable and metrizable, with $Y$ Polish, and suppose there is a topological quotient map $f:X\to Y$ with compact fibers. Is $X$ Polish? I have a specific space in mind, so if the ...
5
votes
1answer
116 views

Simultaneous approximation of arbitrary functions in Hölder space and in $L^2(\mu)$ by a smooth function and its derivative

Let $\mu$ be a probability measure on the circle $S^1=\mathbb{R}/\mathbb{Z}$ which is singular with respect to the Lebesgue measure $\lambda$. Consider the functions spaces $L^2(\mu)$ on the one hand, ...
5
votes
2answers
78 views

Accuracy of the truncated Hausdorff moment problem

For a sequence of real numbers $s = (s_i)_{i \in n}$ let $M_s$ be the collection of functions $f:[0,1] \to [0,1]$ such that $$(\forall i \leq n) \int_0^1 x^i f(x) dx = s_i$$ In other words, $M_s$ ...
0
votes
1answer
64 views

Convergence of measures to an absolutely continuous measure

Suppose that $\{\mu_n\}$ is a sequence of Borel probability measures on a compact metric space $X$ and suppose that $\{\mu_n\}$ converges weakly to a Borel probability measure $\mu$ on $X$. If $\mu$ ...
7
votes
3answers
270 views

Borel coloring of a graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$

The following question was asked in a comment by Joel David Hamkins in Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$. Let $V$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$. ...
0
votes
1answer
121 views

topology of setwise convergence of measures

It is well known that if $X$ is, say, compact and metric, then the set of probability measures on the Borel subsets of $X$ endowed with the usual topology of weak convergence of measures has as a ...
3
votes
1answer
97 views

Measurability and continuity for general topological spaces

Let $(X,\tau)$ be a topological space. We call $S\subseteq X$ saturated if $S=\bigcap\{U\in\tau: U\supseteq S\}$. Let $\sigma(X,\tau)$ be the $\sigma$-algebra generated by $\tau\cup\{K\subseteq X: K ...
8
votes
1answer
132 views

Specifying $L^p$ norms of derivatives

Given a sequence of positive numbers $\{a_n\}$ and $1 < p < \infty$, $p\neq 2$, is it possible to build a function $f\in C^\infty(\mathbb R)$ so that $\|f^{(n)}\|_{L^p(\mathbb R)} = a_n$? For ...
0
votes
3answers
146 views

When Banach indicatrix is measurable?

Let $f:X\to Y$ is a measurable function. Banach indicatrix $$ N(y,f) = \#\{x\in X \mid f(x) = y\} $$ is the number of the pre-images of $y$ under $f$. If there are infinitely many pre-images then ...
6
votes
1answer
280 views

An indicator of a planar subset as an element of a tensor product

Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that $$ f^2=f $$ (that ...