Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
2,895
questions
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106
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Differentiation under the integral sign for a $L^1$-valued function (shape derivative)
Let
$d\in\mathbb N$;
$U\subseteq\mathbb R^d$ be open and $$\mathcal A:=\{\Omega\subseteq U:\Omega\text{ is bounded and open and }\partial\Omega\text{ is of class }C^{0,\:1}\};$$
$E:=\bigcup_{\Omega\...
0
votes
0
answers
84
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Regularity with respect to the Lebesgue measure through dimensions
Let us consider two probability measures $\mu \in \mathcal{P}(\mathbb{R}^{p})$ and $\nu \in \mathcal{P}(\mathbb{R}^{q})$ with $p,q \in \mathbb{N}^{*}$. We note $\#$ the push forward operator i.e for $...
5
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0
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337
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Computing the infinite dimensional Lebesgue measure of "cubes"
There is no Lebesgue measure in infinite dimensions—this slogan is familiar to every student interested in analysis. One possible, precise statement of this result may be as follows: if $X$ is an ...
1
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1
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181
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Friedland metric entropy
I was asking if it is possible to extend the definition of topological Friedland entropy for $\mathbb{Z}^d$ continuos actions to measure preserving actions.
The topologica Friedland entropy is ...
2
votes
2
answers
413
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Reference for the divergence theorem for embedded $C^1$-submanifolds of $\mathbb R^d$ with boundary
I'm aware of Gauss's theorem (aka the divergence theorem) for compact subsets $K$ of $\mathbb R^d$ with "$C^1$-boundary"$^1$.
I know that there are several generalizations of this theorem, ...
2
votes
1
answer
253
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Information theory for uncountably infinite-dimensional continuous random variable
I'm exploring the possibility to apply information theory on an uncountably infinite-dimensional scenario. I found the concept of generalized entropy for continuous random variables defined on finite-...
1
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1
answer
181
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Does Ahlfors–David regularity of a measure imply its Fourier asymptotic behavior?
Let $\mu$ be a Borel probability measure on $R^d$. If $\mu$ satisfies $\mu(B(x,r))\le Cr^\alpha$ for any $x\in R^d$ and $r>0$, then Strichartz (Fourier asymptotics of fractal measures, J. Funct. ...
4
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1
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645
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Integrate Radon-Nikodým derivatives against Lebesgue measure
I am struggling for quite some time, because of a problem involving Radon-Nikodým derivatives. I will try to describe the main features and perhaps somebody has an idea how to solve it.
I consider two ...
0
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1
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55
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Looking for a family of random variables such that only the second clause is fulfilled [closed]
Working with the epsilon-delta-criterium, a family $(X_i)_{i \in I}$ on $(\Omega,A,P)$ is uniformly integrable if
i) $sup_{i \in I} E(X_i) <\infty$
ii) $\forall \epsilon>0$ ex. $\delta>0$ s.t....
1
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0
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77
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Conditions for a function to vanish almost nowhere on its support?
Let $f:\mathbb{R}^d\rightarrow\mathbb{R}$ be a continuous function and $\mathrm{supp}(f) := \mathrm{cl}\{x\in\mathbb{R}^d\mid f(x)\neq 0\}$ its support.
Under which conditions is it true that $f≠0$ (...
-1
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1
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341
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Convergence of Radon-Nikodým derivative
Imagine we have a sequence of finite measures $\nu_n << \mu_n$ (on the torus $\mathbb{T}^2\subseteq \mathbb{R}^2$) converging weakly to some measures $\nu << \mu$. Do we automatically have ...
4
votes
0
answers
144
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Image of function contains identity elements
Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Let $f:U\...
3
votes
1
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216
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Measure theory on abstract Boolean ring
Since a σ-algebra in measure theory is indeed an algebra over $\mathbb{Z}_2$ with addition given by symmetric difference and multiplication given by intersection, does it mean we can put measure on ...
4
votes
2
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517
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Convergence of conditional measures for a convergent sequence of probabilities whose projection is constant
Setting
Suppose $\mu_n$ is a sequence of probability measures on $[0,1]\times [0,1]$ converging to a limit probability $\mu$ meaning that
$$ \lim_{n\to+\infty}\int f(x,y)d\mu_n(x,y) = \int f(x,y)d\mu(...
2
votes
1
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156
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Can we show that this transition semigroup preserves a certain Wasserstein space?
Let $E$ be a separable $\mathbb R$-Banach space, $v:E\to[1,\infty)$ be continuous, $$\rho(x,y):=\inf_{\substack{\gamma\:\in\:C^1([0,\:1],\:E)\\ \gamma(0)\:=\:x\\ \gamma(1)\:=\:y}}\int_0^1v\left(\gamma(...
4
votes
1
answer
229
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Measurable total order
Under what conditions on a metric space $X$, equipped with the Borel $\sigma$-algebra, does there exist a measurable total ordering of the elements of $X$?
By "measurable total ordering" we ...
1
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2
answers
333
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When is the inside of a Jordan curve open? [closed]
I'm working purely on intuition here. The Jordan curve theorem states that a Jordan curve separates the plane into a bounded component and an infinite component. For toy curves, it seems like this ...
3
votes
2
answers
2k
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Properties of the total variation norm on space of totally finite measure (from Bogachev)
Let $(X,d)$ be a metric space, $\mathcal{B}$ the Borel $\sigma$-algebra on $X$, and $\mathcal{M}(X)$ the space of totally finite measures on $\mathcal{B}$. Let $\|\mu\|_{TV}$ be the total variation ...
1
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1
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180
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If a Markov semigroup is eventually contractive, can we conclude that it admits a unique invariant measure?
Let $E$ be a separable $\mathbb R$-Banach space, $\rho$ be a complete separable metric on $E$, $\operatorname W_\rho$ denote the Wasserstein metric of order $1$ associated to $\rho$, $\mathcal M_1(E)$ ...
1
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1
answer
152
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If $(κ_t)$ is a semigroup with invariant measure $\mu$ and $ν$ is singular to $\mu$, then $νκ_t$ might not converge to $\mu$ in total variation norm
Let $E$ be a Polish space, $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal B(E))$, $\mu$ be a probability measure on $(E,\mathcal B(E))$ invariant with respect to $(\kappa_t)_{t\ge0}$ and $\...
4
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0
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251
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Continuity of the Lebesgue measure w.r.t the Hausdorff metric
I have a question linked to Interplay of Hausdorff metric and Lebesgue measure. Let us denote as $\mathcal K(\mathbb R^n)$ the space of compact subsets of $\mathbb R^n$ endowed with the Hausdorff ...
0
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1
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108
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Can we say that $f_\infty(t)=g_\infty(t) \text{ a.e}$?
Let $(E,\mathcal{A},\mu)$ be a finite measure space. Let $\{f_n\}$ and $\{g_n\}$ two uniformly integrable sequences such that:
$$
\qquad\forall n\geq 1~:~ f_n(t)=g_n(t)~~ \text{ a.e.}\tag1
$$
$$
\...
-4
votes
1
answer
62
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Can we say that $\{f_n\}\text{ is uniformly integrable over }E\setminus (\cap_p B_p)$? [closed]
Let $(E,\mathcal{A},\mu)$ be probability space and $\{f_n\}$ be sequence of functions such that
$$
\sup_n\int_{E}|f_n|d\mu<+\infty.
$$
Let $\{B_p\}$ be a sequence non-increasing in $\mathcal{A}$...
0
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0
answers
55
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Prove that infimum of a metric outer measure is a metric outer measure
Let $X$ be a compact measure space and $\mathcal{O}(X)=\{U\subset X, \textrm{U - open set}\}$
and let $\lambda$: $\mathcal{O}(X) \rightarrow \left[0,+\infty\right]$ be a metric outer measure.
Prove ...
0
votes
1
answer
124
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Integration against a measure that has an integral form
Suppose that $(X, \mathcal{X})$ is a measurable space and $(Y,\mathcal{Y}, \mu)$ is a measure space (in my particular application, they are Polish spaces endowed with their Borel $\sigma$-algebra). ...
0
votes
1
answer
343
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Conditions on continuity under Lebesgue measure
Let $h : X \times I \rightarrow \mathbb{R}$ be a continuous function, where $X$ is a compact set of $\mathbb{R}^k$, for some $k$.
Set $\hat{h}(x,t) = 1$ if $h(x,t) \neq 0$, $0$ otherwise.
Define $g : ...
5
votes
2
answers
2k
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Tight sequence of measures
This is probably a very easy question for experts in probability or measure theory.
I have a sequence of finite measures $\mu_{n}$ on a non-compact metric space $X$ such that $\mu_{n}$ converges to $\...
8
votes
1
answer
191
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Subspaces of $L_p([0,1])$ whose unit ball is compact for the topology of convergence in measure
Any information about the following questions would be welcome.
I wonder whether there are (well-known or easy) closed and infinite dimensional subspaces of $L_p([0,1])$ ($1<p<\infty$) whose ...
2
votes
0
answers
58
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Subharmonic functions with reasonably non absolute continuous laplacian
Does it exist a compact supported measure $\mu$ in the plane $\Bbb R^2$ with the following two properties?
1) Points have $\mu$-measure zero,
2) If $\Delta u=\mu$, then the polar set of $u$ has ...
2
votes
1
answer
112
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How is this bound for a Wasserstein contraction coefficient in this paper obtained?
I'm trying to understand the following conclusion from this paper (see below for the relevant paragraphs):
I'm not sure whether they really mean that it follows from the statements of Lemma 3.2 (...
1
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0
answers
57
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Outer-regular product of $\tau$-additive measures
Due to the deficiencies of the simple product measure defined on measurable rectangles, there have been many different constructions of product measures in more specialized circumstances.
Originally, ...
5
votes
2
answers
525
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Properties of measures that are not countably additive but have countably additive null ideals
This is a very naive question, maybe more of a reference request than anything else.
Let $(X, \mathcal X)$ be a measurable space. If $m$ is a real-valued function on $\mathcal X$, we say that $m$ has ...
1
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0
answers
64
views
Path connectedness of a certain subspace of measurable functions
Note: Functions that differ on a null set are not identified.
Consider the space of measurable functions $[0, 1] \to [0, 1]$ that are continuous exactly on a set of Lebesgue measure $r$ , $0 < r &...
0
votes
1
answer
532
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The properties of total variation metric
The questions was asked by me on Math StackExchange, but no answer appears, so I ask for help again.
Let $(X, d)$ be a complete (Hausdorff, separable, local compact and other nice properties you want)...
1
vote
1
answer
59
views
Can this be translated to a truncated multivariate moment problem?
Fix $\mathbf t \in \mathbb{R}_+^d$. I am looking for a r-atomic measure $\nu$ on $\mathbb{R}_{+}^{d}$ that solves the following 'moment' conditions.
$$\forall\, \mathbf i \in \mathbb{N}^{d} \, s.t \, ...
1
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0
answers
36
views
Does a total preorder on lotteries that preserves countable mixtures preserve arbitrary mixtures?
Let $X$ be a countable set. A lottery on $X$ is a function $\lambda: X \to [0,1]$ such that $\sum_x \lambda(x) = 1$. Let $\Delta X$ be the set of lotteries on $X$.
A total preorder $\preceq$ on $\...
0
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0
answers
46
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How to prove invariance implies quadrature about Haar measure?
I'm reading paper, which contains a lemma named as 'Invariance Implies Quadrature'. The lemma is stated as follows.
Lemma Let $f$ be a function from $O(d)$ to $\mathbb{R}$ and let $H$ be a finite ...
4
votes
1
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466
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Two definitions of $L^p$ spaces that are not always equivalent
There are two definitions of $L^p(S, \Sigma,\mu)$ in the literature. (Here $S$ is a set, $\Sigma$ is a $\sigma$-algebra of subsets of $S$ and $\mu$ is a positive measure.) The two definitions are ...
-3
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2
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155
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Getting almost certainty from uncountably many low-probability events
Let $(\Omega,\Sigma,\mathbb{P})$ be a complete probability space, $B\subseteq X$ be a non-empty Borel subset of a polish space $X$, $A$ be an uncountable indexing set, and $\{X_{\alpha,n}\}_{a \in A, ...
1
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0
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186
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Are measures better thought of as densities than differentials?
The standard notation for integrating with respect to a measure $\mu$ is:
$$\int f(x)\,d\mu(x).$$
But I've wondered if it could be better written as:
$$\int f(x)\mu(x)\,dx$$
where $\mu(x)$ is now ...
2
votes
1
answer
158
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Null preserving transformation
Suppose that $(\Omega,\mu)$ is a measure space. Let $\tau:\Omega\to\Omega$ is a measurable map such that $\mu\circ\tau^{-1}<<\mu$. Then $\tau$ s said to be null preserving. I want to prove the ...
0
votes
1
answer
366
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von Neumann ergodic theorem for $L_p$
Let $\tau:\Omega\to \Omega$ be a measure-preserving transformation with $\mu(\Omega)<\infty$. Define $T:L_p(\Omega)\to L_p(\Omega)$ as $Tf:=f\circ \tau$. I want to prove that for all $1\leq p<\...
-1
votes
1
answer
111
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On the concentration of Lipschitz functions near its expectation, where the vector has identical but not independent, components
Consider the random vector $X:=(X_1\dots X_1) \in \mathbb{R}^n, X_1 \sim \mathcal{N}(0,1).$ Notice the identical components, they're identically distributed but not independent.
Now, I was wondering ...
9
votes
1
answer
741
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Baire category theorem for uncountable unions
Any compact Hausdorff space $X$ is a Baire space:
if the set $X$ is a meager set (meaning a countable union of nowhere dense subsets,
also known as a set of first category),
then $X$ is empty.
I am ...
2
votes
0
answers
60
views
Is the set $\operatorname{Unif}(0,\frac{1}{n})$ for odd and even $n$ a 2-alternating capacity?
Let $\Omega$ be a complete metrizable space $\mathscr A$ its Borel $\sigma$-algebra and $\mathscr M$ the set of all probability measures on $\Omega.$ Every non-empty subset $\mathscr P \subset \...
0
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0
answers
307
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Definition of generic point
I am trying to read a paper named D.S. Ornstein, B. Weiss, Subsequence ergodic theorems for amenable groups, Israel J. Math. 79 (1) (1992) 113–127, doi:10.1007/BF02764805. In this paper the authors ...
1
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0
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147
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Is there difference in notion of measurability in classical versus constructive?
Are there notions of measure and examples of sets measurable in that measure in classical logic but not in constructive logic (I think there cannot be counterexamples in other direction)? Are there ...
2
votes
1
answer
70
views
$ \int_{E}^{*}{\psi (t) d\mu(t)}=\int_{E}{\phi (t) d\mu(t)} $
Let $(T, \mathcal{A}, \mu)$ be an arbitrary measure space.
The outer integral over $(T, \mathcal{A}, \mu)$ of a (possibly nonmeasurable) function $\psi: T\to (-\infty, +\infty]$ is defined by:
$$
\...
3
votes
1
answer
212
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A question about finitely additive integration
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space ($\mathbb P$ is countably additive). Let $\{p_\omega: \omega \in \Omega\}$ be a family of (countably additive) probability measures on $(\...
5
votes
1
answer
138
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completeness of $\mathcal M(\Omega)$ without any topological assumptions?
Let $(\Omega,\Sigma)$ be a measurable space (no reference measure is chosen!), and $V$ a finite-dimensional normed vector space.
Note carefully that I am not choosing any topology on $\Omega$, so the $...