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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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\...
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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 $...
Titouan Vayer's user avatar
5 votes
0 answers
337 views

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 ...
truebaran's user avatar
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1 vote
1 answer
181 views

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 ...
user502940's user avatar
2 votes
2 answers
413 views

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, ...
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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-...
mw19930312's user avatar
1 vote
1 answer
181 views

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. ...
ljjpfx's user avatar
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645 views

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 ...
Mushu Nrek's user avatar
0 votes
1 answer
55 views

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....
Sofia's user avatar
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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 $f≠0$ (...
fsp-b's user avatar
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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 ...
Mushu Nrek's user avatar
4 votes
0 answers
144 views

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\...
pi66's user avatar
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3 votes
1 answer
216 views

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 ...
Vasily Ilin's user avatar
4 votes
2 answers
517 views

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(...
Pablo Lessa's user avatar
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2 votes
1 answer
156 views

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(...
0xbadf00d's user avatar
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4 votes
1 answer
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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 ...
Aryeh Kontorovich's user avatar
1 vote
2 answers
333 views

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 ...
Paul Cusson's user avatar
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3 votes
2 answers
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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 ...
Léo D's user avatar
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1 answer
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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 $\rho$, $\mathcal M_1(E)$ ...
0xbadf00d's user avatar
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1 vote
1 answer
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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 $\...
0xbadf00d's user avatar
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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 ...
Redeldio's user avatar
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1 answer
108 views

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: $$ \qquad\forall n\geq 1~:~ f_n(t)=g_n(t)~~ \text{ a.e.}\tag1 $$ $$ \...
Karim KHAN's user avatar
-4 votes
1 answer
62 views

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 $\mathcal{A}$...
Made's user avatar
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55 views

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 ...
Norbert Dąbrowski's user avatar
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1 answer
124 views

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). ...
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1 answer
343 views

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 : ...
Viv Bičak's user avatar
5 votes
2 answers
2k views

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 $\...
AMath91's user avatar
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8 votes
1 answer
191 views

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 ...
user159631's user avatar
2 votes
0 answers
58 views

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 ...
Claudio Rea's user avatar
2 votes
1 answer
112 views

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 (...
0xbadf00d's user avatar
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1 vote
0 answers
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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, ...
Cameron Zwarich's user avatar
5 votes
2 answers
525 views

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 ...
aduh's user avatar
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1 vote
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 &...
James Baxter's user avatar
  • 2,029
0 votes
1 answer
532 views

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)...
Jialong Deng's user avatar
  • 1,749
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 \, ...
lrnv's user avatar
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1 vote
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 $\...
aduh's user avatar
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0 votes
0 answers
46 views

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 ...
Nate's user avatar
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4 votes
1 answer
466 views

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 ...
Denis White's user avatar
-3 votes
2 answers
155 views

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, ...
ABIM's user avatar
  • 4,989
1 vote
0 answers
186 views

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 ...
wlad's user avatar
  • 4,792
2 votes
1 answer
158 views

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 ...
A beginner mathmatician's user avatar
0 votes
1 answer
366 views

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<\...
A beginner mathmatician's user avatar
-1 votes
1 answer
111 views

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 ...
Learning math's user avatar
9 votes
1 answer
741 views

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 ...
Dmitri Pavlov's user avatar
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 \...
Seyhmus Güngören's user avatar
0 votes
0 answers
307 views

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 ...
A beginner mathmatician's user avatar
1 vote
0 answers
147 views

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 ...
VS.'s user avatar
  • 1,816
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: $$ \...
Karim KHAN's user avatar
3 votes
1 answer
212 views

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 $(\...
aduh's user avatar
  • 839
5 votes
1 answer
138 views

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 $...
leo monsaingeon's user avatar

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