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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Integration of vector function against vector measure

Let $X,Y,Z$ be Banach spaces and let $m\,:\,X\times Y\to Z$ be a bilinear map such that $\|m(x,y)\|\leq C \|x\|\|y\|$ for some fixed constant $C$. Moreover, let $\mu$ be a Borell vector measure on $\...
user72829's user avatar
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112 views

Mathematical justification for the use of an energy shell in the microcanonical ensemble

I would like to understand an identity used in the deduction of the explicit formula for the probability distribution of the microcanonical ensemble in statistical mechanics. Consider $\Lambda$ to be ...
MathMath's user avatar
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1 vote
1 answer
140 views

Convergence rate of a sequence of sets to a set-theoretic limit?

Suppose $n\in\mathbb{N}$ and set $A\subseteq\mathbb{R}^{n}$. If we define a sequence of sets $\left(F_r\right)_{r\in\mathbb{N}}$ with a set theoretic limit of $A$; how do we define the rate at which $\...
Arbuja's user avatar
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Purely non-atomic measure on the Gromov boundary of a finitely generated free group

In the set-up of my previous post, let $\theta$ be a purely non-atomic probability regular measure defined on the Borel $\sigma$-algebra of the metric space $(\partial F, d)$. We say $\theta$ admits a ...
Sanae Kochiya's user avatar
5 votes
1 answer
189 views

If every point is a Lebesgue point of $f$, does $f$ satisfy the intermediate value property?

Let $f: \mathbb R \to \mathbb R$ be a locally integrable measurable function. We say $f$ satisfies the intermediate value property if given any $a, b\in \mathbb R$ with $a < b$, whenever $u \in \...
Nate River's user avatar
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2 votes
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45 views

The world of non-weak*-topologies on $\mathcal{P}(X)$

Let $X$ be a metrizable space and consider $\mathcal{P}(X)$, the set of all probability measures on $X$. Typically, the weak*-topology is considered on $\mathcal{P}(X)$, which is a very natural ...
alhal's user avatar
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4 votes
1 answer
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Can a function that is continuous on a dense set be almost extended to a continuous function?

Note: All sets and functions defined below are assumed measurable. $\mu$ denotes the Lebesgue measure. Let $D$ be a dense subset of $[0, 1]$, and $f: D \to \mathbb R$ a function. Given $\varepsilon &...
Nate River's user avatar
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3 votes
1 answer
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Existence of a bounded measurable subset of $\text{SL}(d,\mathbb R)$ that is Borel isomorphic to $\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$?

$\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$ has two interesting properties: on one hand it is non-compact, but on the other hand it admits a unique $\text{SL}(d,\mathbb R)$-invariant finite ...
user506835's user avatar
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1 answer
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Integral of $M^\text{*} - M$ with respect to $M^\text{*}$ is zero for $M^\text{*}$ the running maximum of $M$ a continuous local martingale

Given $M$ a continuous local martingale, and $M^\text{*} = \sup_{0 \leq s \leq t} M_s$ its running maximum, we consider the finite variation integral $$ I_T:= \int_0^T (M^\text{*}_s - M_s) \, \text{d}...
George's user avatar
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Convex combination of positive mean-ergodic operators

Let $T_1,T_2:L^1([0,1],\mathrm{d}x)\to L^1([0,1],\mathrm{d}x)$ be positive mean-ergodic operators such that: For every $h:[0,1]\to \mathbb{R}_+$ we have that $$\int_0^1 T_1 h(x)\mathrm{d}x = \int_0^1 ...
Matheus Manzatto's user avatar
-1 votes
1 answer
87 views

(Rate of) Convergence in distribution and Laplace transform of random variables/stochastic processes

Let $X_t^n$ and $X_t$ be stochastic processes (with finite moments), and assume that for every $t>0$, $\lambda>0$ and bounded continuous function $\varphi$, \begin{equation} \int_0^te^{-\lambda ...
Wenguang Zhao's user avatar
2 votes
2 answers
785 views

Defining a measure of uniformity for measurable subsets of $[0,1]^2$ w.r.t dimension $\alpha\in[0,2]$

Let $(X,d)$ be a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,2]$ and $\text{dim}_{\text{H}}(A)$ is the Hausdorff ...
Arbuja's user avatar
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3 votes
1 answer
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If $f : [0,1] \to H$ has $t$-derivative with respect to the norm of $H$, and $H=L^2[0,1]$ itself, does the $t$-derivative exist in ordinary sense?

The question is as in the title. Let $H$ be a separable Hilbert space and $f : [0,1] \to H$ be a continuous mapping such that \begin{equation} f'(t):=\lim\limits_{\alpha \to 0} \frac{f(t+\alpha)-f(t)}{...
Isaac's user avatar
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Existence of a stronger notion of perfect measures

Let $\mathcal{X}$ be a measurable space with its $\sigma$-algebra $\mathcal{B}_\mathcal{X}$ and let $\mathbb{R}$ be the real numbers endowed with its Borel $\sigma$-algebra $\mathcal{B}_\mathbb{R}$. ...
Packo's user avatar
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6 votes
2 answers
264 views

Atoms for Markov kernels

Let $X$ and $Y$ be standard Borel measurable spaces. A Markov kernel $f : X \rightsquigarrow Y$ is a map $f(-|-) : \Sigma_Y \times X \to [0,1]$ such that: $f(-|x)$ is a probability measure on $Y$ for ...
Tobias Fritz's user avatar
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5 votes
1 answer
233 views

Intuitive meaning of Giry monad's $\sigma$-algebra

The Giry monad $G : \textbf{Meas} \to \textbf{Meas}$ maps a measurable space $(X, \mathcal{F})$ to its set of probability measures. The $\sigma$-algebra of $G(X, \mathcal{F})$ is the smallest algebra ...
A confused dove's user avatar
3 votes
0 answers
137 views

Eigenvalues of random matrices are measurable functions

I have read that if a random matrix is hermitian then its eigenvalues are continuous, hence also measurable. If the random matrix is not hermitian, the eigenvalues are not continuous in some cases. ...
Curtis74's user avatar
1 vote
1 answer
93 views

Asymptotic behavior of a Markov process on the set of $\{0,1\}$-polynomials

This question is cross-posted from https://math.stackexchange.com/questions/4711799/asymptotic-behavior-of-a-markov-process-on-the-set-of-0-1-polynomials I am trying to study the asymptotic behavior ...
Francesco Bilotta's user avatar
1 vote
0 answers
98 views

Concatenation of Markov processes and independence

In chapter 14 of Sharpe's General Theory of Markov Processes the concatenation of Markov processes $X^1$ and $X^2$ is described. I've posed the relevant part at the bottom of this post. It is rather ...
0xbadf00d's user avatar
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1 answer
58 views

The existence of a maximal “cross-sectional” filter on the Boolean algebra of measurable subsets of [0, 1] modulo almost everywhere equivalence

Let $\mathcal{B}([0, 1])$ be the Boolean algebra of measurable subsets of $[0, 1]$ modulo almost everywhere equivalence, i.e., two measurable sets which differ only by a Lebesgue null set are ...
David Gao's user avatar
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7 votes
1 answer
357 views

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 ...
Gro-Tsen's user avatar
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12 votes
3 answers
680 views

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 ...
David Gao's user avatar
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1 answer
67 views

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) ...
happyle's user avatar
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1 vote
0 answers
72 views

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 (...
Gregory V.'s user avatar
2 votes
0 answers
136 views

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? ...
Thomas Lehéricy's user avatar
9 votes
1 answer
717 views

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 $t$. ...
Nate River's user avatar
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2 votes
2 answers
376 views

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 ...
Moritz Schauer's user avatar
10 votes
2 answers
695 views

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 ...
Tim Campion's user avatar
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-1 votes
2 answers
328 views

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 ...
Vokram's user avatar
  • 109
0 votes
0 answers
29 views

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 ...
Nick's user avatar
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2 votes
0 answers
154 views

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 ...
user72829's user avatar
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8 votes
1 answer
664 views

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 ...
Amir Sagiv's user avatar
  • 3,544
2 votes
1 answer
116 views

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 ...
LoyoL's user avatar
  • 35
4 votes
2 answers
245 views

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 ...
Andromeda's user avatar
  • 189
2 votes
1 answer
350 views

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 $g$ ...
Sanae Kochiya's user avatar
2 votes
0 answers
55 views

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 ...
Isaac's user avatar
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1 vote
1 answer
71 views

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 $\...
cha0skampf's user avatar
3 votes
1 answer
114 views

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.
Silvestrov Dmitrii's user avatar
1 vote
1 answer
285 views

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{...
DRive's user avatar
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1 vote
0 answers
93 views

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 ...
The Thin Whistler's user avatar
1 vote
0 answers
92 views

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 $K$ ...
Jineon Baek's user avatar
1 vote
0 answers
93 views

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 ...
unit 1991's user avatar
  • 111
4 votes
0 answers
147 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 ...
Matheus Manzatto's user avatar
2 votes
0 answers
125 views

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, $m>0$ ...
Isaac's user avatar
  • 2,727
5 votes
0 answers
181 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 ...
Guy Fsone's user avatar
  • 1,033
-3 votes
1 answer
234 views

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 ...
Anixx's user avatar
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1 vote
1 answer
105 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 ...
Isaac's user avatar
  • 2,727
0 votes
1 answer
203 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 ...
Grandes Jorasses's user avatar
2 votes
0 answers
189 views

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....
Oleg Zubelewicz's user avatar
1 vote
0 answers
70 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(\...
José's user avatar
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