# Questions tagged [measure-theory]

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

2,966
questions

4
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291
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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 ...

4
votes

0
answers

214
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### 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 ...

2
votes

1
answer

139
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### 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 ...

2
votes

1
answer

82
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### 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 ...

2
votes

1
answer

139
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### 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 $\...

0
votes

0
answers

80
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+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 \...

39
votes

2
answers

2k
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### 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\...

0
votes

0
answers

18
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### 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 ...

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....

2
votes

1
answer

134
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### 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 $E\,$? ...

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 $$\...

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(...

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 ...

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 ...

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{...

5
votes

0
answers

73
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### 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, ...

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 ...

2
votes

0
answers

94
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### 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 ...

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}{\...

4
votes

1
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201
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### 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 ...

3
votes

1
answer

88
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### 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{|...

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{...

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 ...

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 ...

7
votes

2
answers

424
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### 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 $...

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 ...

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 ...

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 ...

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, ...

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}{\...

1
vote

2
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109
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### 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{...

0
votes

0
answers

116
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### 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 $$...

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 - \...

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 $A$ ...

1
vote

1
answer

94
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### 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 $\...

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}{\...

2
votes

0
answers

59
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### 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}{\...

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 ...

3
votes

0
answers

72
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### 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 ...

6
votes

1
answer

287
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### 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 ...

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 ...

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} \...

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 ...

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: ...

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}(\...

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 ...

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 $k\in M$,...

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 ...

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 ...

0
votes

2
answers

129
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### 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):=\...