# Questions tagged [measure-theory]

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

2,699
questions

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### Consistency of a strong Fubini type theorem for measure zero sets

Is the following statement (†) consistent with ZFC?
If $E \subseteq [0,1]^2$ is such that $E_x := \{y\in[0,1] : (x,y)\in E\}$ has measure zero for almost all $x$, then $E^y := \{x\in[0,1] : (x,y)\in ...

12
votes

2
answers

364
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### If $A, B$ is a non-trivial partition of $S^1$, is it possible that $R_\theta(A) \cap B$ has measure zero for all rotations $R_\theta$?

This was previously posted to Math StackExchange. I was originally unsure whether it is suitable for posting here, but I've yet to get an answer there, so I'm just trying to see if people here can ...

0
votes

1
answer

60
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### Difference between $P(f(x,w)>0)→1$ at any $x$ and $P(\inf(f(x,w))>0)\to1$ when dimension grows

Let $T:=[-1,1]^{n-1}\times (0,1]$. Let
$$f_n(x_1,\cdots,x_n,w_1,\cdots,w_n):=g(x_1,w_1)+\cdots+g(x_n,w_n)=\sum_{i=1}^ng(x_i,w_i),$$
where
(i) $w_1,\cdots,w_n$ are i.i.d. Gaussian random variables
(ii) ...

1
vote

0
answers

42
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### Time-inhomogeneous Krylov-Bogoliubov Existence Theorem

I am interested in what is known about the application of the Krylov-Bogoliubov existence theorem to the time-inhomogeneous case, especially as it relates to an underlying random dynamical system (...

2
votes

0
answers

57
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### Is the product of two outer regular Radon measures outer regular?

Everything is nice on second countable spaces: the product of two outer regular Radon measure is still an outer regular Radon measure. But what happens without the assumption of second countability?
...

8
votes

1
answer

532
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### Does the family of fat Cantor sets contain a measurable rectangle?

Let $S \subset (0, \frac{1}{3}) \times [0, 1]$, be the set such that for each $0 < t < \frac{1}{3}$, $S \cap (\{ t \} \times [0, 1])$ is the standard Smith-Volterra Cantor set of parameter $t$.
...

2
votes

2
answers

135
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### Monotone class theorem for pre-Dynkin system ("finitely additive Dynkin system/λ system)

A pre-Dynkin system is a set system $\mathcal D \subset \wp(\Omega)$ which contains $\Omega$ and is closed under complements and finite disjoint unions. Is it true that the monotone class generated by ...

8
votes

2
answers

527
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### In which category is a measure on a measurable space a morphism?

I'd like to be able to say that a measure $\mu$ on a measurable space $X$ "is" a morphism $R \to X$, where $R$ is some incarnation of the real numbers in an appropriate category.
In other ...

-1
votes

2
answers

200
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### Conditional expectation: commuting integration and supremum

Let $X$ and $A$ be compact Polish spaces endowed with Borel $\sigma$-algebras. Let $\mathcal{A} = X\times \mathcal{B}(A)$ be the $\sigma$-algebra consisting of cylinders whose projections on $A$ are ...

0
votes

0
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23
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### Convergence of approximate solution sequence to measure valued solution for incompressible Euler equation

I recently studied the measure valued solution of incompressible Euler equations.
In Majda and Bertozzi's book ‘Vorticity and Incompressible Flow’:
Theorem 12.10. Let $\{v^\epsilon\}$ be an ...

2
votes

0
answers

120
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### Banach space of vector measures

Let $S$ be a set and $\Sigma$ be a $\sigma$-algebra of subsets of $S$. Let $A$ be a Banach space over the field of complex numbers. A countably additive map $\mu:\Sigma\to A$ is called a vector ...

8
votes

1
answer

580
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### Measure without measurable sets

This question is a little on the softer and speculative side, so bear with me.
Usually a measurable space is $(\Omega, \Sigma)$, a set $\Omega$ and sigma algebra $\Sigma$ of subsets. A measurable ...

2
votes

1
answer

56
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### Product sigma-algebra: approximating elements arbitrary good using the generating sets

I am struggling to find a reference for the following statement, which I still believe to be true.
"Let $(\Omega_1, \mathcal{A}_1, \mu_1), (\Omega_2, \mathcal{A}_2, \mu_2)$ be finite measure ...

4
votes

2
answers

222
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### Product of locally Borel sets locally Borel

Let $X$ be a locally compact Hausdorff space with a fixed Radon measure (= Borel measure that is finite on compact subsets, inner regular on open subsets and outer regular on Borel sets) $\mu$ . A ...

2
votes

1
answer

274
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### Radon-Nikodym derivative in a compact Hausdorff space

Let $X$ be a compact Hausdorff space where $X$ have infinitely many points and the topology is non-discrete, $m$ be a regular probability measure defined on the Borel $\sigma$-algebra of $X$, and $g$ ...

2
votes

0
answers

44
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### Regularity on $\mathbb{T}^3$ of the "functional average" of a map $S : C^\infty(\mathbb{T}^3, \mathbb{R}) \to L^2(\mathbb{T}^3, \mathbb{R})$

For simplicity, let $C^\infty(\mathbb{T}^3, \mathbb{R})$ be the real Frechet space of periodic smooth functions on $\mathbb{R}^3$. Here, $\mathbb{T}^3$ is the $3$-dimensional torus.
For a fixed ...

1
vote

1
answer

50
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### How to characterize the Borel sets of product between finite and uncountable space?

Consider the product space $Z=X\times Y$, where $X$ is a finite set with discrete topology and $Y$ is an uncountable compact subset of $\mathbb{R}^n$ with the usual subspace topology. Denote with $\...

3
votes

1
answer

99
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### Singular distribution F such that convolution F and F is an absolutely continuous distribution?

F is a singular distribution function concentrated on the positive half-line. Is it possible that 2-fold convolution F*F is an absolutely continuous distribution function? Please, give me an example.

1
vote

1
answer

196
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### Total variation distance

Let $\mathcal{X}$ be the input or feature space, let $\mathcal{B}$ be Borel $\sigma$-algebra on $\mathcal{X}$ and $P(\mathcal{X})$ denotes the set of all probability measures on $(\mathcal{X},\mathcal{...

1
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0
answers

84
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### Chain complexes indexed over measurable subsets of $\mathbb{R}$: Towards a measurable notion of Euler Characteristic

I have for a while tried to generalize the notion of a chain complex in a way to obtain a "continuous" or at least "measurable" notion of Euler Characteristic.
I have come up with ...

1
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0
answers

81
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### Extreme points of a two-dimensional convex body in terms of its surface area measure

Let $K \subset \mathbb{R}^2$ be a nonempty compact convex set.
For any $t \in S^1$, define the unit vector $u_t = (\cos t, \sin t)$ making an angle of $t$, and let $l_K(t)$ be the tangent line of $K$ ...

1
vote

0
answers

90
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### Proving more stronger fomula for discrepancy of a sequence [closed]

I am reading famous book about uniform distribution of sequences by Kuipers and Niederreiter and have questions about solving below exercise from that book. Before going to main exercise I will write ...

3
votes

0
answers

117
views

### Naïve definition of a measure on a fractal

This question was previously posted on MSE.
Let $K\subset \mathbb R^2$ be a compact fractal of Hausdorff dimension $1<d<2$. I want to define a natural measure on $K$.
One option would be to use ...

2
votes

0
answers

94
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### Extreme confusion with the Gaussian measure on $\mathcal{S}'(\mathbb{R}^n)$ supported on $C^\infty(\mathbb{R}^n)$ and the issue of Borel sets

Let
\begin{equation}
C_a(x,y):=\frac{1}{(4\pi)^{n/2}} \int_a^\infty \frac{dk}{k^{n/2}}e^{-km^2-\lvert x-y \rvert ^2/(4k)}
\end{equation}
be a covariance operator with a cutoff $a>0$. Here, $m>0$ ...

5
votes

0
answers

157
views

### When does the Fourier transform of a measure decay?

Let $\mu$ be a Borel measure on $\Bbb R^d$.
It is well known that $\mu= |f|dx$ with $f\in L^1(\Bbb R^d)$ then its Fourier transform satisfies
$$\widehat{\mu}(\xi)\to0,\qquad \xi\to\infty.$$
However if ...

-2
votes

1
answer

186
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### Why surreal numbers cannot be extended further in this way using measure approach?

Basically, a lebesgue measure of dimension $n$ of a set of the same dimension $n$ is $n$-volume, $\lambda_n(S)$.
If the dimension of a set is greater than the dimension of the measure, the measure is ...

1
vote

1
answer

68
views

### When is the probability measure on the "direct product" via the Kolmogorov extension theorem supported on the "direct sum"?

Let me restrict to the case of Hilbert spaces, which seem simplest.
Let $\{H_n\}$ be a sequence of (possibly infinite dimensional) Hilbert spaces and $\{ \mu_n \}$ be a sequence of Borel probability ...

0
votes

1
answer

180
views

### Riemann-Liouville integral of $f$ is zero implies $f =0$ a.e

The Riemann-Liouville integral is defined by
$$
I^\alpha f(x)=\frac{1}{\Gamma(\alpha)} \int_a^x f(t)(x-t)^{\alpha-1} d t
$$
where $\Gamma$ is the gamma function and $a$ is an arbitrary but fixed base ...

2
votes

0
answers

155
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### Shift invariance of the Lebesgue measure

I am trying to write a brief introduction to the Lebesgue integration in $\mathbb{R}^m$ from the general viewpoint. The students do not specialize in this field. So I formulate a theorem without proof....

1
vote

0
answers

47
views

### Domain where the fractional Laplacian operator is a closed operator

Consider the fractional Laplacian defined by
$$(-\Delta)^s u(x) = P.V. \int_{\mathbb{R}^N} \frac{u(x) - u(y)}{|x - y|^{N + 2s}}dy, \ s \in (0,1).$$
Also consider that
$$D((-\Delta)^s) = \{u \in H^s(\...

1
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0
answers

60
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### Does a Borel transform uniquely determine a Borel measure?

It is a known fact that Borel measures are uniquely determined by their Fourier transforms. This is the motivation for the following question.
I came across the concept of a Borel transform of a Borel ...

4
votes

0
answers

376
views

### Fixing the duality $L^\infty(X)= L^1(X)^*$ for Radon measure spaces

Consider the following fragment from Folland's book "A course in abstract harmonic analysis":
Let me denote the Borel subsets of $X$ by $\mathscr{B}(X)$. Folland claims that if $\mu$ is a ...

0
votes

1
answer

45
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### If $ \mathcal{H}^k(B_1(0)\cap S)\leq A\omega_k $ when $ \mathcal{H}^k(B_r(x)\cap S)\leq A\omega_kr^k $ for all $ 0<r<\delta $, $ x\in\mathbb{R}^n $?

Let $ S\subset\mathbb{R}^n $ is of finite $ k $-dimensional Hausdorff and $ 0<\delta<1 $ is a constant. If for any $ x\in\mathbb{R}^n $ and $ r>0 $, we hae
$$
\mathcal{H}^k(S\cap B_r(x))\leq ...

6
votes

1
answer

211
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### Integration along fibres of continuous map on compact Hausdorff spaces

Let $p:Z\to X$ be a continuous surjective map between compact Hausdorff spaces.
Does there exist a family $m=(m_x)_{x\in X}$ of Radon probability measures on $Z$, such that
the support of $m_x$ is ...

0
votes

1
answer

63
views

### Sufficient conditions for L1 convergence of exponentials

Suppose $(X,B,m)$ is a finite measure space and $(f_n)$ is a
sequence of functions converging almost surely and in $L^2(X,m)$.
Moreover, I know that for every $n$, $\int e^{f_n(x)} m(dx)<\infty$. ...

-1
votes

1
answer

110
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### Random variable as an integral of an indicator function

This answer says that if $X$ is a random variable and $X_+ = \mathrm{max}(0, X)$, then $X_+ = \int_0^\infty I_{\{X > x\}}\mathrm{d}x$. I'd like to know how to derive this starting with $A \in \...

2
votes

1
answer

127
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### A different way to try to define a measure on the unit-circumference circle

Let C denote the Lie group (ℝ/ℤ, +), and let ℤn denote the subgroup of C generated by [1/n].
Let (X,n) be an ordered pair where X ⊂ C is an arbitrary subset, and n ∊ ℤ+, such that that the set of ...

2
votes

0
answers

49
views

### Double quotient integral formula on $\Gamma \backslash G /K$

Let $G=\text{SL}(n,\mathbb R)$, $\Gamma=\text{SL}(n,\mathbb Z)$ and $K=\text{SO}(n,\mathbb R)$. Consider the double coset space $X= \Gamma \backslash G /K$ and its fundamental domain $\mathcal F\...

2
votes

1
answer

67
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### Total sets for $L^p$ for every $1\leq p < \infty$

Consider $L^p[ 0,1]$ for $1\leq p < \infty$ or, if you prefer, $L^p(\mu)$ where $\mu$ is a finite Borel measure with compact support. Let $(\phi)_{i\in I}$ be a subset of measurable functions that ...

0
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0
answers

43
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### Different measurability of Hilbert-space valued random variable

My question is motivated by this link.
Let $(\Omega,\mathcal{F})$ and $(Y,\mathcal{B})$ be measurable spaces, a measurable map $T:\Omega\to Y$ is called a $Y$-valued random variable.
Now let $H$ be a ...

5
votes

0
answers

124
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### Applications of Baire's theorem on functions of first class

I found the following theorem on page 32 of John Oxtoby's Measure and Category.
Theorem 7.3. If $f$ can be represented as the limit of an everywhere convergent sequence of continuous functions, then $...

2
votes

1
answer

85
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### Injectivity of two sided Laplace transform

Let $\mu,\nu$ be finite Borel measures on $\mathbb R$.
Assume that there is an open interval $(a,b)$ on which the Laplace transforms exist and coincide:
$$
\int_{-\infty}^\infty e^{-tx}\,d\mu(x) = \...

1
vote

1
answer

286
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### Takesaki lemma: existence Gelfand-Pettis integral

Consider the following fragment from Takesaki's second volume of "Theory of operator algebras" (lemma 2.4, chapter VI "Left Hilbert algebras").
In another post, it was explained ...

0
votes

0
answers

30
views

### Name for a regularity property of $\sigma$-ideals

Let $X$ be a topological space and let $\mathcal{B}$ be its Borel $\sigma$-algebra. Suppose $\mathcal{N} \subset \mathcal{P}(X)$ is a $\sigma$-ideal, i.e. $\emptyset \in \mathcal{N}$ and it is closed ...

2
votes

1
answer

140
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### Question on density of certain set of matrices

Let $B$ be an invertible real matrix and let $Q=\{A \text{ real}\mid AB^{T} \text{ is symmetric}\}$. Is the subset $S=\{ A \in Q\mid A+A(B^{-1}A)^{2} \text{ is symmetric}\}$ of measure zero in $Q$? I ...

1
vote

0
answers

50
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### Regularity of $\sigma$-finite measure pushforwarded by completion

Let $(X, d)$ be a metric space. Let $\mu$ be a $\sigma$-finite measure defined on borel subsets of $X$. Let $i \colon X \to \hat{X}$ be an isometry on image, where $\hat{X}$ is a complete metric space ...

5
votes

2
answers

314
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### Integrating on orbits of algebraic groups

Suppose $G$ is a $\mathbb{Q}$-algebraic group (I am interested in the semisimple case) acting rationally on a vector space $V_\mathbb{Q}$. Let $x \in V_\mathbb{Q}$ be a non-zero rational vector. ...

3
votes

0
answers

73
views

### Question on an integral inequality

I am reading van de Vaart and Weller, Weak Convergence and Empirical Processes With Applications to Statistics. And I am stuck in the proof of Theorem 2.6.7 on page 141.
For simplicity I restae the ...

1
vote

0
answers

108
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### Sets of Hausdorff measures zero

Let $H^g$ be the Hausdorff measure with respect to gauge function $g$.
I need to construct an example of a set E for which:
$H^g(E)=0$ for $g(r)=r^s$, $s>0$
and
$0<H^g(E)<+\infty$ for $g(r)=2^...

3
votes

1
answer

140
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### Closed graph correspondence which never contains the whole support

Let $I=[a,b]$ with $a<b\in\mathbb{R}$ and denote by $\mathcal{M}(I)$ the set of Borel probability measures on $I$ equipped with the topology induced by the weak convergence of measures.
Does there ...