Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
2,897
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Remainder-balancedness of primes
Let $\mathbb{N}_+$ denote the set of positive integers. Consider the remainder function $\text{rem}:\mathbb{N}_+\times \mathbb{N}_+ \to \mathbb{N}\cup\{0\}$ defined by $$(n,d) \mapsto n - \Big(\Big\...
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1
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174
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Can we further restrict the space of test functions to $C_c^\infty (X)$ in weak convergence?
Let
$X := \mathbb R^n$,
$C_b(X)$ the space of all real-valued bounded continuous,
$C_c(X)$ the space of all real-valued continuous functions with compact supports, and
$C_c^\infty(X)$ the space of ...
6
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1
answer
270
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A characterisation of continuous real functions
Let $f: \mathbb R^n \to \mathbb R$ be a measurable function.
We say $f$ is precise if for every $x \in \mathbb R^n$ and every compact subset $K$ of $\mathbb R^n$ such that for $|K \cap B_\delta (x)|&...
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1
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90
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Sums of powers of measures of $p$-adic balls
Let $(a_n,k_n) \in \mathbb{Z}_p \times \mathbb{N}$ for $n \in \mathbb{N}$ and consider the sequence of closed $p$-adic balls $B(a_n,k_n) = a_n + p^{k_n}\mathbb{Z}_p$. I assume that the $(a_n,k_n)$ are ...
7
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2
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165
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Bisector of two points in a Riemannian manifold has measure $0$
Let $p,q\in M$, $p\neq q$, where $M$ is a Riemannian manifold. We will let the bisector of $p,q$ be $\mathcal{B}(p,q)=\{x\in M;d(p,x)=d(q,x)\}$. Does $\mathcal{B}(p,q)$ have measure $0$?
I was ...
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2
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152
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When can we extend a function on a $\lambda$-system to a probability measure?
Let $\Omega$ be a nonempty set and let $\mathcal{L}$ a $\lambda$-system on $\Omega$. That is,
(i) $\Omega \in \mathcal{L}$,
(ii) if $A, B \in \mathcal{L}$ and $A \subseteq B$, then $B \setminus A \in \...
2
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1
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179
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References on tilting distributions
I would be interested in any book, paper, or other reading material that gives a comprehensive treatment of tilted distributions using the following notion of "tilting" (or equivalent):
...
2
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3
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337
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More natural example of measurable but not progressive process
All examples of measurable but not progressive processes I have ever seen seemed to be based on the huge difference between $\mathcal{F}$ and $\mathcal{F}_\infty$. Here is what I mean.
Consider ...
2
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114
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A technical question concerning convolution product
Let $v\in L^p(\Bbb R^d)$, $1\leq p<\infty$ be nonzero function, i.e., $v\not\equiv 0$.
Define $$u(x)= |v|*\phi(x)= \int_{\Bbb R^d} |v(y)|\phi(x-y)d y$$ with $\phi(x)= ce^{-|x|^2}$ and $c>0$ so ...
6
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Can Theorem 1.40 in Rudin's Real and Complex Analysis be strengthened when the $\sigma$-algebra is Borel?
Let $(X, \mathcal F, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use the Bochner integral. Then we have Theorem 1.40 in Rudin's Real and Complex Analysis, i.e.,
...
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vote
1
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360
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Motivation for Ionescu-Tulcea extension theorem (as opposed to Kolmogorov's)
I recently asked a question on the differences between Ionescu-Tulcea and Kolmogorov extension theorems (ITET and KET for short). A lot of my confusion has been cleared there and what I understood ...
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Meyer's example of a separable process with no path regularity
This question is a cross-post from math.stackexchange.com. I am reposting it here since I didn't receive an answer there. The original post can be found by this link.
In the following excerpt from ...
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Does the set of square numbers adhere to Benford's law in every base?
Does the set of squares $S = \{n^2: n\in\omega\}$ adhere to Benford's law for the first digit in every base $b\geq 2$?
Precise formulation of what it means for a set $T\subseteq \omega$ to "...
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2
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Kolmogorov vs Ionescu-Tulcea extension theorem (again)
Disclaimer. This post is not a duplicate, I have carefully (best I could) read all posts on the subject both here and on math.se and my particular questions have not been asked there.
I've recently ...
3
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3
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364
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Progressive measurability intuition from Bichteler's *Stochastic integration with jumps* book
In the Stochastic Integration with Jumps Bichteler gives a very intuitive definition of progressive measurability I've never seen before:
Although I like this intuition very much, I cannot find a ...
4
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1
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520
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Optimal Transport: how is this transport map Borel measurable?
I'm reading Theorem 1.17. and its proof at page 14 of Santambrogio's Optimal transport for applied mathematicians. The content is not hard but a little bit long (because of related detail). Please ...
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1
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Let $c: X \times Y \to \overline{\mathbb R}$ be $\gamma$-measurable. Is $c_x:Y \to \overline{\mathbb R}, y \mapsto c(x, y)$ $\nu$-measurable?
Let $(X, \mathcal X, \mu)$ and $(Y, \mathcal Y, \nu)$ be $\sigma$-finite measure spaces. Let $\overline{\mathbb R} := \mathbb R \cup \{\pm \infty\}$.
$f:X \to \overline{\mathbb R}$ is called $\mu$-...
4
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285
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If a derivative is defined everywhere and $\pm1$ almost everywhere, is it constant?
Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that the set $A:=\{x\in\mathbb{R};f'(x)\not\in\{1,-1\}\}$ has measure $0$. Does this imply that $f'$ is constant?
Context: I was ...
4
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1
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339
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Does Szemerédi's theorem hold for sets with positive upper Banach density?
We say that a set of natural numbers $A\subseteq \omega$ has positive upper density if $$\lim\sup_{n\to\infty}\frac{|A\cap n|}{n+1} > 0.$$
Szeméredi's theorem states that every $A\subseteq \omega$ ...
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1
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200
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Upper density versus upper Banach density on $\omega$
For $A\subseteq\omega$ we define the upper density by $$d_u(A) = \lim\sup_{n\to\infty}\frac{|A\cap n|}{n+1}.$$ For $y\in \omega$ we set $A - y:= \{|a\setminus y|:a\in A\}.$ Note that $|a\setminus y|$ ...
2
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112
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Measure on the places of $\bar{\mathbb Q}$
Consider the set $S$ of all places of $\mathbb Q$ (i.e. the set of all absolute values up to equivalence). Then we can consider $S$ as a measure space with the counting measure $\mu$. Therefore $\mu(\{...
8
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2
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920
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Is there a measure theory for proper classes?
This question is naive, but I didn't get an answer at MSE: Is it straightforward to extend measure theory to proper classes?
Of course when one tries to define measures on "large sets" ...
2
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1
answer
314
views
Textbook definition for "path measure" or "probability measure over paths"
I need a formal definition for the path measure for stochastic differential equations.
Which textbook or paper should I consult?
3
votes
1
answer
204
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A level set of non-constant real analytic function
Assume that $u: B\to [0,1]$, where $B$ is an open ball in $\mathbb{R}^n$, is a nonconstant real analytic function and let $t\in[0,1]$ and let $\mu(t)=\operatorname{Volume}(\{x\in B: u(x)>t\})$. Why ...
3
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2
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235
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Is there something like a "self-avoiding Markov chain" on a continuous space?
If stumbled accross self-avoiding walks. They seem to be deterministically generated, but from a quick google search term seem to be randomly generated variants.
However, as far as I can see they are ...
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Measure preserving maps of pseudo-Lebesgue measure in infinite-dimensional vector space
Let $I =([0,1),\mathcal{B},\lambda)$ stand for the unit interval with a Lebesgue measure constrained on it. This is just a uniform probability distribution, an infinite power $I^{\infty}$ is a well ...
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1
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88
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Does an isometric automorphism of $L_p (X,\mu, E)$ preserve pointwise convergence?
Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use the Bochner integral. Let $p \in [1, \infty)$ and $q \in (1, \infty]$ such that $p^{-1}+q^{-...
1
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0
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39
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Does the constrained Wasserstein barycenter admit a blue noise property?
Let $(E,d)$ be a metric space and $\nu$ be a probability measure on $\mathcal B(E)$. In this paper, it is mentioned that sampling from $\mu$ can be described as choosing $n\in\mathbb N$, $x_1,\ldots,...
2
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0
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A version of Portmanteau theorem where $(\mu_n)_{n\in \mathbb N}$ is replaced by a net $(\mu_d)_{d\in D}$
Let $(E, d)$ be a metric space, $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$, and $\mathcal P(E)$ the space of all Borel probability measures on $E$. For $f \in \...
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0
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129
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Approximate range of Radon-Nikodym derivative in a dynamical system
Suppose $(X, G, \Omega, \mu)$ is a dynamical system where $(X, \Omega, \mu)$ is a Borel measure space and $G$ is acting on $X$ such that each group action $x\mapsto g\cdot x$ defines a measurable ...
4
votes
1
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264
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Supremum of infimum of measure of members of a free ultrafilter
For a set $A\subseteq \omega$ we let the upper density of $A$ be defined as $d^+(A) := \lim\sup_{n\to\infty}\frac{|A\cap(n+1)|}{n+1}$. Let $\text{FrU}(\omega)$ be the collection of free ultrafilters ...
3
votes
1
answer
187
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Property of sets of positive Lebesgue measure in $\mathbb{R}^2$
Let $P\subset \mathbb{R}^2$ be a set of positive Lebesgue measure. Is it always true that a suitable rotation and translation of $P$ always contains a set of the form $\{re^{i\theta}:r\in E, \theta\...
4
votes
2
answers
192
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Is the conditional expectation of a Caratheodory function a Caratheodory function?
Let $(Y, \Sigma,\mu)$ be measure space and $X$ a Polish space endowed with its Borel $\sigma$-algebra. Suppose that $f:Y\times X\to \mathbb R$ is a Carathéodory function (i.e. continuous in $x\in X$ ...
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0
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156
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Wiener Integral and its distribution
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space.
Let $(W(t))_{x \in \mathbb{R}^d}$ be a Gaussian random field.
Then, we can define Wiener integral $\int_{\mathbb{R}^d} f(\xi) \, dW(\xi)$...
2
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0
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125
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What are examples of infinite-dimensional Banach spaces that are also measure spaces?
I am interested in examples of infinite-dimensional vector spaces that
are Banach spaces or even Hilbert spaces
are measure spaces
Instead of the full vector space, subsets with measure structure ...
2
votes
0
answers
82
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A variant of disintegration theorem where the assumptions on $f$ and $g$ are exchanged
I have recently read about about disintegration theorem, i.e.,
Disintegration theorem Let
$X$ be a Polish space, $\mathcal X$ its Borel $\sigma$-algebra, and $\mu$ a Borel probability measure on $X$...
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1
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48
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Symmetric and nearly additive bounded functions
Let $(y_n)_{n\ge 1}$ be a sequence with values in $(0,1)$ such that $\lim_n y_n=1$. Let also $f: [0,1]\to \mathbb{R}$ be a bounded function such that $f(0)=0$ and satisfies
$$
\forall n\ge 1, \forall ...
4
votes
1
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129
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On partial absolute continuity
$\newcommand\B{\mathscr B}\newcommand\A{\mathscr A}\newcommand\si{\sigma}$Let $I:=[0,1]$, and let $\B$ and $\B^2$ denote the Borel $\si$-algebras over $I$ and $I^2$, respectively. Let $\A$ stand for ...
1
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1
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82
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Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace - Part II
This is a follow-up to this previous question, but under stronger assumptions.
Let $(X, \mu)$ be a (say, $\sigma$-finite) measure space, let $g \in L^2$ (say, over the real
scalar field). Let $\tilde ...
1
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2
answers
139
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On conditions for the existence of $h\in L^1$ such that $h>0$ a.e
I've been reading the chapter of Uniform Integrability on Probability Theory by Achim Klenke which has the following proposition.
If $\mu$ is $\sigma$-finite, then there is a function $h \in L^{1}(\mu)...
7
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2
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379
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Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace
Let $(X, \mu)$ be your favourite measure space (finite or $\sigma$-finite if you like), let $g \in L^2$ (say, the scalar field of $L^2$ is $\mathbb{R}$, though this probably doesn't matter). Let $\...
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If $H \in L_{q} (\mu, X^*)$ such that $\int \langle H, f \rangle \mathrm d \mu = 0$ for all $f \in L_{p}(\mu, X)$, then $H=0$ $\mu$-a.e
I'm reading Theorem 1 at page 98 of Vector Measures by Joseph Diestel, John Jerry Uhl. Here we use the Bochner integral.
Theorem 1 Let $(\Omega, \Sigma, \mu)$ be a $\sigma$-finite measure space, $1 \...
2
votes
0
answers
133
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On limits of positive linear functionals
I am looking for pointers to the literature on questions of the following kind. ($Y$ and $\Omega$ might be open subsets of some Euclidean space, but I am interested in the kind of conditions that need ...
1
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0
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61
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Prescribed class of measurable sets
Let $X\neq\emptyset$ and let $\mu:P(X)\to[0,\infty]$ be an outer measure. Recall that, a set $A\subseteq X$ is $\mu$-measurable if
$$
\mu(B)=\mu(A\cap B)+\mu(B\setminus A), \text{ for all }B\subseteq ...
1
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0
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148
views
Generalizations of Fubini-Tonelli's theoem
Fubini-Tonelli's theorem: If $(X, A, \mu)$ and $(Y, B, \nu)$ are $\sigma$-finite measure spaces and $f: X\times Y \to [0,\infty]$ is a measurable function, then
$$ \int _{X}\left(\int _{Y}f(x,y)\,{\...
1
vote
1
answer
146
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Show that a certain convergence of measures is equivalent to a certain convergence of integrals
Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of a measures. We know, by the Portmanteau Theorem, that:
$$\int f d \mu_n \to \int f d\mu, \quad \forall \, f \in C_b \hbox{(class of continuous and ...
2
votes
1
answer
539
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A question about the proof of the Levy-Khintchine representation Theorem
I'm studying Infinitely Divisible random variables using this Lecture Notes. And I have a question that is driving me crazy.
In the proof of the "only if" part of the Levy-Khintchine ...
0
votes
1
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244
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Does $L_{p}(\mu, X)^*=L_{q} (\mu, X^*)$ hold for $\sigma$-finite measure spaces?
I'm reading Theorem 1 at page 98 of Vector Measures by Joseph Diestel, John Jerry Uhl.
THEOREM 1. Let $(\Omega, \Sigma, \mu)$ be a finite measure space, $1 \leq p<\infty$, and $X$ be a Banach ...
2
votes
1
answer
161
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Joint irreducibility and aperiodicity of two independent Markov chains
Let $(X_i)_i, (Y_i)_i$ be two independent Markov chains on Polish state spaces $\mathcal{X}, \mathcal{Y}$ and with kernels $P, Q$. Suppose that they are both $\psi$-irreducible and aperiodic and have ...
1
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1
answer
227
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Is $L_p(X, \mu, E)$ uniformly convex for $p \in (1, \infty)$ if $E$ is a uniformly convex Banach space?
Let $(X, \Sigma, \mu)$ be a $\sigma$-finite complete measure space, $(E, |\cdot|)$ a Banach space, and $p \in (1, \infty)$. Let $L_p := L_p(X, \mu, E)$ be the Bochner space of all $\mu$-integrable ...