Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
2,898
questions
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123
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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 $\...
1
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0
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112
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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 ...
1
vote
1
answer
140
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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 $\...
0
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0
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78
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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 ...
5
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1
answer
189
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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 \...
2
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0
answers
45
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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 ...
4
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1
answer
664
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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 &...
3
votes
1
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114
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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 ...
1
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1
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80
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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
$$
I_T:= \int_0^T (M^\text{*}_s - M_s) \, \text{d}...
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0
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91
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Convex combination of positive mean-ergodic operators
Let $T_1,T_2:L^1([0,1],\mathrm{d}x)\to L^1([0,1],\mathrm{d}x)$ be positive mean-ergodic operators such that:
For every $h:[0,1]\to \mathbb{R}_+$ we have that
$$\int_0^1 T_1 h(x)\mathrm{d}x = \int_0^1 ...
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votes
1
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87
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(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 ...
2
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2
answers
785
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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 ...
3
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131
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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)}{...
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0
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76
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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 $\mathcal{B}_\mathbb{R}$.
...
6
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2
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264
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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 ...
5
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1
answer
233
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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 ...
3
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137
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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. ...
1
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1
answer
93
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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 ...
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0
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98
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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 ...
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1
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58
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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 ...
7
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1
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357
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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
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3
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680
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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
67
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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
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0
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72
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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
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0
answers
136
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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?
...
9
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1
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717
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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
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2
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376
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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 ...
10
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2
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695
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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 ...
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2
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328
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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
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0
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29
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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
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0
answers
154
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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
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664
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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
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1
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116
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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
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2
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245
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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
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1
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350
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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
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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 ...
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1
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71
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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
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1
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114
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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
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1
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285
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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
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93
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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
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92
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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
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0
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93
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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 ...
4
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0
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147
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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
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0
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125
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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
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 ...
-3
votes
1
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234
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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
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 ...
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 ...
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....
1
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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(\...