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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.

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Golomb subsets of $\mathbb{N}$

A set $A\subseteq\newcommand{\N}{\mathbb{N}}\N$ is said to be Golomb if whenever $a<b \in A$ and $a'<b' \in A$ with $(b-a) = (b' - a')$, then $a=a'$ and $b=b'$. If $A\subseteq \N$ is Golomb, we ...
Dominic van der Zypen's user avatar
4 votes
0 answers
214 views

Intuition for why Kelly criterion is 'so simple'

The Kelly criterion gives a simple formula to calculate the fraction of one's current wealth/bankroll. On the page above it says Assuming that the expected returns are known, the Kelly criterion ...
wood's user avatar
  • 2,770
2 votes
1 answer
139 views

Law of large numbers for a continuum of Bernoullis

Suppose I have a family of $n$ independent Bernoulli random variables described by a vector of parameters $(p_i)_{i=1}^n$. As it is well known, the number of successes within this family is a random ...
Francesco Bilotta's user avatar
2 votes
1 answer
82 views

Signed measures on algebras (fields) and their boundedness properties

I asked this question here on math.StackEchange, but it might be too technical so I re-post it here. Let $X$ be a compact Hausdorff second countable topological space. Let $\mathcal{B}$ a countable ...
Ennio's user avatar
  • 21
2 votes
1 answer
139 views

Lower bound on the Lévy-Prokhorov metric for normal distributions

Let $\mathfrak M(\mathbb R^n)$ denote the metric space of probability measures (over $\mathbb R^n)$ equipped with the Lévy-Prokhorov metric $\rho$. Consider two $n$-variate normal distributions $\...
Syd Amerikaner's user avatar
0 votes
0 answers
80 views
+50

Does the sequence of bounded symmetric square integrable holomorphic functions have a convergent subsequence?

Let $f$ be a bounded holomorphic function on $\mathbb D^2$ and $s : \mathbb C^2 \longrightarrow \mathbb C^2$ be the symmetrization map given by $s(z) = (z_1 + z_2, z_1 z_2),$ for $z = (z_1, z_2) \in \...
Anacardium's user avatar
39 votes
2 answers
2k views

How decreasing can a bijection $f:\mathbb{N}\to\mathbb{N}$ be?

This is a follow-up to this question by Dominic van der Zypen. For each bijection $f:\mathbb{N}\to\mathbb{N}$, let $$\operatorname{rc}(f) := \liminf_{N\to\infty} \frac{\left|\left\{(m,n)\in\{1,\dots,N\...
Saúl RM's user avatar
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0 votes
0 answers
18 views

Parameter dependent functions integral [migrated]

I got this question on a measure theory exam today, and after hours of discussing with my colleagues, im still quite confused. I have been able to prove the first point, but I am having trouble with ...
TNTPablo's user avatar
0 votes
0 answers
40 views

Existence of sequence of regular projections

Reading the book :Krasnosel'skii, M.A.; Pustylnik, E.I.; Sobolevskii, P.E.; Zabreiko, P.P. (1976), Integral Operators in Spaces of Summable Functions, Leyden: Noordhoff International Publishing, 520 p....
Guillermo García Sáez's user avatar
2 votes
1 answer
134 views

Is projection of a closed subspace Borel?

Specifically, letting $E$ be a separable infinite-dimensional real Banach space, and $D_2$ in $E\times E$ a closed linear subspace, is then $\{\,x:\exists\,y\,;(x,y)\in D_2\}$ a Borel set in $E\,$? ...
TaQ's user avatar
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9 votes
1 answer
371 views

Min–max reversing bijections $f:\mathbb{N}\to\mathbb{N}$

For any set $X$, let $\newcommand{\N}{\mathbb{N}}[X]^2 = \big\{\{x,y\}:x\neq y \in X\big\}$ and set $[n]^2 = [\{0,\dotsc,n-1\}]^2$ for any positive integer $n$. For $A\subseteq [\N]^2$ we set $$\...
Dominic van der Zypen's user avatar
1 vote
1 answer
46 views

Sufficient condition for uniform convergence of the Stieltjes transform

Let $\mu$ be a probability measure and $\mu_N$ be a sequence of probability measures. For simplicity we may assume them to have compact supports contained in $[-1,1]$. Define $$G_\mu(z):=\int\frac{\mu(...
Jiyuan Zhang's user avatar
1 vote
0 answers
41 views

Modifiying a sequence of measures to assign a certain value when integrating a fixed function?

Let $f:\mathbb R ^d\to \mathbb R$ be some continuous function, $|f|\leq A(1+|x|)$, where $|\cdot|$ denotes the usual Euclidean norm. Fix a measure $\mu$ and constant $C$. Assume that $\mu_n$ is a ...
J.R.'s user avatar
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3 votes
1 answer
119 views

Sparse "bijection-proof" subsets of $[\mathbb{N}]^2$

We call a collection ${\cal S}\subseteq {\cal P}(\newcommand{\N}{\mathbb{N}}\N)$ bijection-proof if for any bijection $\varphi:\N\to\N$ there is $T\in{\cal S}$ with $\varphi(T) \in {\cal S}$. For any ...
Dominic van der Zypen's user avatar
0 votes
1 answer
81 views

Sequential compactness of a sequence of curves of Borel probability measures

$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \DeclareMathOperator*{\supp}{supp} \newcommand{\bR}{\mathbb{R}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bP}{\mathbb{...
Akira's user avatar
  • 1,085
5 votes
0 answers
73 views

Is there an equivalent condition for Borel projections being Borel?

Let $X,Y$ be Polish spaces, and $B\subseteq X \times Y$ a Borel subset. The projection $B_X$ is not necessarily Borel in $X$. I have seen a few sufficient conditions for the projection to be Borel, ...
J.R.'s user avatar
  • 271
2 votes
1 answer
114 views

Show that $\|P(f\circ\varphi_{\lambda})-\widetilde{f}(\lambda)\|_p=\|P(f\circ\varphi_{\lambda}-\overline{P(\overline{f}\circ\varphi_{\lambda}}))\|_p.$

Let $\Omega = \mathbb B_n,$ the unit ball in $\mathbb C^n$ and $L^2_a(\Omega)$ be the Bergman space endowed with the normalized volume measure on $\Omega.$ Let $k_{\lambda}$ be the associated Bergman ...
Anacardium's user avatar
2 votes
0 answers
94 views

Majorization theory on $\sigma$-finite measure spaces

I want to learn about majorization and submajorization theory on $\sigma$-finite measure spaces. I know things get a bit more complicated compared with the case of a finite measure spaces but I'm ...
Lau's user avatar
  • 749
2 votes
0 answers
70 views

Stability of Hölder constants of frozen Itô stochastic integrals

$ \newcommand{\RR}{\mathbb{R}} \newcommand{\TT}{\mathbb{T}} \newcommand{\NN}{\mathbb{N}} \newcommand{\PP}{\mathbb{P}} \newcommand{\EE}{\mathbb{E}} \newcommand{\FF}{\mathbb{F}} \newcommand{\PPP}{\...
Akira's user avatar
  • 1,085
4 votes
1 answer
201 views

Shrinking and expanding pairs in bijections $\varphi:\mathbb{N}\to\mathbb{N}$

Motivation. If we consider any bijection $\varphi:\newcommand{\N}{\mathbb{N}} \N \to \N$, we say integers $m\neq n$ are shrinking with respect to $\varphi$ if $|m-n|>|\varphi(m) - \varphi(n)|$, and ...
Dominic van der Zypen's user avatar
3 votes
1 answer
88 views

How to check that the surface measure is the weak limit of $\delta^{-1}\mathcal{L}^n|_{B(0,1+\delta)\setminus B(0,1)}$?

We denote the unit sphere $\{x\in\mathbb{R}^n:|x|=1\}$ by $S^{n-1}.$ If $x\in\mathbb{R}^n\setminus\{0\}$, the polar coordinates of $x$ are \begin{align*} r=|x|\in(0,\infty),\quad \gamma=\dfrac x{|...
ljjpfx's user avatar
  • 207
1 vote
1 answer
102 views

Is there $\varepsilon \in (0, 1)$ such that $\sup_{t \in [0, \varepsilon]} [\ell_t]_\beta < \infty$?

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bD}{\mathbb{...
Akira's user avatar
  • 1,085
2 votes
0 answers
173 views

How can the maximal ideal space of the Fourier Stieltjes algebra be non-separable?

I have been asking a fair few (probably elementary) questions about abstract harmonic analysis lately. By means of explanation, I am just feeling around the subject at the moment and trying to build ...
Daron's user avatar
  • 1,915
1 vote
0 answers
43 views

Gaussian Hypercontractivity of Chaos based on Gaussian with value in Hilbert spaces?

The classical Gaussian hypercontractivity is stated as following: Suppose $\xi$ is a Gaussian variable and $H_n(\xi)$ is the space of n-th homogeneous Wiener chaos constructed from $\xi$, then for any ...
Inuyasha's user avatar
  • 253
7 votes
2 answers
424 views

Uncountable collections of distinct subsets of an interval (existence)

Throughout, $\mu$ is just the Lebesgue measure. Question: does there exist an uncountable family of distinct subsets of $[-1, 1]$, denoted by $(U_j)_{j \in [-1, 1]}$, with $\mu(U_j) > 0$ for each $...
Stepan Plyushkin's user avatar
11 votes
4 answers
791 views

Is there a name for finite unions of intervals?

Finite unions of intervals are simple sets that are used quite often, e.g. in measure theory. (The construction of the Cantor set is a noble example). I realised that I do not have a name for them. Is ...
Pietro Majer's user avatar
  • 57.8k
2 votes
1 answer
86 views

Sufficient conditions for the space of Radon measure to be a Banach space

Let $\mathcal{X}$ be a Hausdorff space and consider the space of Radon measures with bounded total variation $M(\mathcal{X})$ on $\mathcal{X}$. Usually, the additional assumptions on $\mathcal{X}$ are ...
ChocolateRain's user avatar
3 votes
0 answers
101 views

How wild is the maximal ideal space of the Fourier-Stieltjes algebra of the real line?

The Fourier-Stieltjes algebra of $\mathbb R$ is the set of all sufficiently nice measures on $\mathbb R$. The vector product is convolution of measures. By identifying each measure with its Fourier ...
Daron's user avatar
  • 1,915
2 votes
0 answers
61 views

Topological measure theory on spaces that are not completely regular

In the usual discourse regarding approaches to measure theory, it is often pointed out that the restriction of topological measure theory to locally compact Hausdorff spaces is insufficient. However, ...
Cameron Zwarich's user avatar
2 votes
1 answer
51 views

Approximate a non-negative function which is measurable in product $\sigma$-algebra

$ \DeclareMathOperator*{\supp}{supp} \newcommand{\bR}{\mathbb{R}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\...
Akira's user avatar
  • 1,085
1 vote
2 answers
109 views

Let $D$ be the set of those $\omega \in \Omega$ such that $f(\omega, \cdot)$ is $\mu$-integrable. Is $D$ measurable?

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bD}{\mathbb{...
Akira's user avatar
  • 1,085
0 votes
0 answers
116 views

Equality of two measures on functional spaces

It is well known that if $\mu$ and $\nu$ are two measures on the space $C^0([0,1],\mathbb{R}^n)$ of continuous mappings from $[0,1]$ to $\mathbb{R^n}$ endowed with its Borel $\sigma$-algebra satisfy $$...
Anico's user avatar
  • 1
0 votes
1 answer
94 views

Approximation on $H^1_0(B)$ and cut-off functions

Let $u \in H^1_0(B)$, where $B$ is the unit ball in $\mathbb{R}^N$. Given $\epsilon > 0$, I need to show there exists a function $\chi_\epsilon \in C^\infty_0(\mathbb{R}^N)$ such that $$ \| u - \...
Lucas Linhares's user avatar
1 vote
0 answers
92 views

Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$. Schwartz space is dense in $A$ wrt $\|f\|:= \|\hat{f}\|_1+\|\hat{f'}\|_1$?

Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$, where $\hat{f}$ is the Fourier transform of $f$. Then is it true that Schwartz space $\mathcal{S}(\mathbb{R})$ is dense in $A$ ...
mathlover's user avatar
1 vote
1 answer
94 views

Is a $\sigma$-algebra generated by complete independent $\sigma$-algebras also complete?

$ \newcommand{\cA}{\mathcal{A}} \newcommand{\cB}{\mathcal{B}} \newcommand{\sP}{\mathscr{P}} $ Let $(\Omega, \cA, \mu)$ be a probability space and $\cA_1, \cA_2$ sub $\sigma$-algebras of $\cA$. Let $\...
Akira's user avatar
  • 1,085
1 vote
1 answer
211 views

Is it true that $F(X_0, \cdot) = X_0 + \int_0^T \sigma(s, X_0) \, \mathrm d B_s$ a.s.?

$ \newcommand{\RR}{\mathbb{R}} \newcommand{\TT}{\mathbb{T}} \newcommand{\NN}{\mathbb{N}} \newcommand{\PP}{\mathbb{P}} \newcommand{\EE}{\mathbb{E}} \newcommand{\FF}{\mathbb{F}} \newcommand{\PPP}{\...
Akira's user avatar
  • 1,085
2 votes
0 answers
59 views

Is $F: \mathbb T \times \mathbb R^d \times \Omega \to \mathbb R^d$ (constructed from Itô integral) Borel measurable in the product $\sigma$-algebra?

$ \newcommand{\RR}{\mathbb{R}} \newcommand{\TT}{\mathbb{T}} \newcommand{\NN}{\mathbb{N}} \newcommand{\PP}{\mathbb{P}} \newcommand{\EE}{\mathbb{E}} \newcommand{\FF}{\mathbb{F}} \newcommand{\PPP}{\...
Akira's user avatar
  • 1,085
4 votes
1 answer
252 views

If $f=h\circ g$, then there's a measurable function $\tilde h$ such that $f=\tilde h\circ g$

Let $X,Y,Z$ be three standard measurable spaces and $f:X\to Z$ and $g:X\to Y$ two measurable functions. Suppose that there's a function $h:Y\to Z$ such that $f=h\circ g$. How can I show that there's a ...
rfloc's user avatar
  • 533
3 votes
0 answers
72 views

Continuity of disintegrations in non locally compact spaces

Let $X$ and $Y$ be Radon spaces, $\mu$ a Borel probability measure on $X$, $F\colon X\to Y$ measurable. Then the disintegration theorem gives us a disintegration $\{\mu^y\}_{y\in Y}$ of $\mu$ with ...
Nathaël's user avatar
6 votes
1 answer
287 views

Well distributed sets

Note: All integrals are taken with respect to Lebesgue measure. The symbol $\def\avint{\mathop{\rlap{\raise.15em{\scriptstyle -}}\kern-.2em\int}\nolimits} \avint$ denotes the average integral. We say ...
Nate River's user avatar
  • 5,501
4 votes
1 answer
178 views

How probability-rich is the $\sigma$-algebra generated by a sequence of sets? (Sierpiński's theorem on non-atomic measures without using the AoC.)

$\newcommand\F{\mathcal F}\newcommand\si{\sigma}\newcommand\Om{\Omega}\newcommand\ep{\varepsilon}$Let $p\in(0,1)$ and let $(\Om,\F,P)$ be a probability space. Let $(A_n)$ be a sequence in $\F$ such ...
Iosif Pinelis's user avatar
4 votes
1 answer
167 views

Extracting a subsequence Cesàro converging to the limsup of the Cesàro sums

Let $X_n$ be a sequence of uniformly bounded random variables — that is, there exists some $K > 0$ such that $|X_n| \leq K$ almost surely for all $n \in \mathbb N$. Write $\bar X_N := \frac{1}{N} \...
Nate River's user avatar
  • 5,501
3 votes
0 answers
89 views

When are increasing functions on posets (specifically, lattices) the CDF of a probability measure?

This is perhaps a basic question, but I couldn't find a reference. Let $P = (X,\leq)$ be a poset. Given a probability measure $\mu$ on $P$ (with respect to the Borel $\sigma$-algebra generated by sets ...
Roger Van Peski's user avatar
2 votes
1 answer
162 views

De la Vallée Poussin criterion on uniform integrability for infinite measures

The de la Vallée Poussin criterion (which is often used in combination with the Dunford-Pettis theorem) is usually formulated for probability measures/finite measures, for example in [Bogachev: ...
kumquat's user avatar
  • 85
4 votes
1 answer
104 views

Dual spaces of Banach-valued $L^{p}$-spaces

Let $(\Omega,\mathcal{F},\mu)$ be a measure space (say complete and $\sigma$-finite, for simplicity). Furthermore, let $(X,\Vert\cdot\Vert_{X})$ be an arbitrary Banach space. I denote by $(L^{p}(\...
G. Blaickner's user avatar
  • 1,209
2 votes
1 answer
116 views

Domain of the infinitesimal generator of a composition $C_0$-semigroup

In the paper [1] the following $C_0$-group is presented, $$ T(t)f(x) = f(e^{-t} x) , \quad x \in (0,\infty) \quad f \in E $$ where $E$ is an ($L^1,L^\infty$)-interpolation space. In mi case, I'm just ...
Scottish Questions's user avatar
0 votes
0 answers
26 views

Why $d\mu (q)\delta (k,q)$ is $G$-invariant?

Let $G$ be a Lie group acting transitively on a smooth manifold $M$ endowed with a quasi-invariant measure $\mu$ (then there exists Radon-Nikodym derivative $\rho_f$ for every $f\in G$). For $k\in M$,...
Mahtab's user avatar
  • 277
0 votes
0 answers
56 views

Measurable Extension

Let $(\Omega, \mathcal{F})$ be a measurable space and $X$ some metric space (probably Polish) with the Borel $\sigma$-algebra and a function $f: \Omega \times X \to \mathbb{R}$. Usually, functions ...
Mrcrg's user avatar
  • 136
0 votes
1 answer
68 views

Projection on a countable union of linear subspace

For any natural number $n$, $V_n$ denotes a closed linear subspace of a $L_2(m)$ space, which is an Hilbert Space, where $m$ denotes a finite measure. Moreover $(V_n)$ is increasing, that is $V_n$ is ...
Guest2024bis's user avatar
0 votes
2 answers
129 views

Is a signed measure $\mu$ on $\mathbb{R}^d$ characterized by the transform $\mathcal{L}_\mu (\lambda ):=\int e^{\langle \lambda,x\rangle }\mu (dx)$?

In the book "Probability Theory" by Achim Klenke there's the following theorem: a finite measure $\mu$ on $[0,\infty )$ is characterized by its Laplace transform $\mathcal{L}_\mu(\lambda):=\...
noob's user avatar
  • 15

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