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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A conjecture: Given $nD$ finite measures that bisects

Given $nD$ finite measures $µ_1, . . ., µ_{nD}$ in $\mathbb{R}^n$ does there exists a set of at most D hyperplanes that bisect each of the measures? How far is this problem explained? That is ...
Ma Joad's user avatar
  • 1,611
2 votes
1 answer
66 views

Set with positive relative measure in all intervals [closed]

Let $A$ be a (Borel-)measurable subset of $[0,1]$.Let $\lambda$ denote Lebesgue measure. Is it possible that there exists a constant $c>0$ such that for all intervals $I \subset [0,1]$ we have $$ \...
Kurisuto Asutora's user avatar
7 votes
2 answers
1k views

Conditional Expectation for $\sigma$-finite measures

Someone knows of some definition or reference of how to define conditional expectation for a measure space with $\sigma$-finite measure. I think it should be as follows: Let $(X,\mathcal{B},\nu)$ ...
Rusbert's user avatar
  • 173
2 votes
1 answer
1k views

Understanding measure-preserving transformation [closed]

Given measure space $(S, \mathcal{S}, \mu)$, and measurable function $\phi: S \to S$. $\phi$ is measure-preserving if $\forall A \in \mathcal{S}, \mu(A) = \mu(\phi^{-1}(A))$. My confusion is that why ...
Jokerr's user avatar
  • 23
4 votes
1 answer
263 views

Zero-one law for an independence-like structure

I am a number theorist by profession, so apologies if the answer to this question is "trivially true" or "trivially false". Let $(\Omega, \mathcal{A}, P)$ be a (non-atomic) probability space. Let $(\...
Kurisuto Asutora's user avatar
4 votes
1 answer
1k views

Is the Wasserstein-1 metric translation invariant?

Define the Wasserstein-1 metric (or the Earth mover's distance) between two positive measures $\mu_1$, $\mu_2$ by $$ W(\mu_1, \mu_2) = \inf_{\gamma \in \Gamma (\mu_1, \mu_2)} \int \|x_1 - x_2\| \, \...
Jonas Adler's user avatar
1 vote
1 answer
259 views

Recover norm from integral

I am given the following expression where $f \in L^2(\mathbb{R}^2, \mathbb{R}^{2 \times 2})$ $$\int_{\mathbb{R}} \int_{\mathbb{R}} \langle g(x), f(x,y) h(y)\rangle dx dy.$$ The functions $g$ and $h$ ...
user avatar
0 votes
2 answers
464 views

Semifinite measure and spectral theorem

Let $H$ be a complex Hilbert space (not necessary separable). Spectral Theorem: Let $A_1$ and $A_2$ be two commuting normal operators, then there exists a measure space $(X,\mathcal{E},\mu)$, two ...
Student's user avatar
  • 1,154
4 votes
2 answers
171 views

Geometric mean of positive measures

Let me given with an obvious example. Let $\Omega\subset{\mathbb R}^n$ be an open domain. If $f,g\in L^1(\Omega)$ and $f,g\ge0$, then $\sqrt{fg}\,\in L^1(\Omega)$. Now let me replace the absolutely ...
Denis Serre's user avatar
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2 votes
0 answers
201 views

Bilinear form representation

It is known, that the following result holds: let $X,Y$ be compact sets in $\mathbb{R}^N$ and $\mathbb{R}^M$ respectively and let $B:C(X)\times C(Y)\to\mathbb{R}$ be a continuous bilinear form ($C(X)$ ...
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4 votes
0 answers
195 views

A kind of 0-1 law?

Suppose that P is a Borel subset of Baire $\times$ Baire, such that for every pair $x,x'$ of reals in the horizontal copy of Baire, if: $x,x'$ are $E_0$-equivalent (that is, $x(n)=y(n)$ for all but ...
Vladimir Kanovei's user avatar
2 votes
1 answer
282 views

Regarding characterisation of outer functions in a Hardy space

Please see the definition of Hardy spaces on the unit disc here. This is regarding outer functions on a Hardy space. I know that outer functions can have no zeroes in the open unit disc since it is ...
user510271's user avatar
7 votes
1 answer
560 views

Haar measure on $\mathrm{SL}_3(\mathbb{Z}) \backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}_3(\mathbb{R})$

The bi-invariant Haar measure on the quotient $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathrm{SL}_2(\mathbb{R}) / \mathrm{SO}_2(\mathbb{R})$ represents the moduli space of rank two real lattices modulo ...
user120980's user avatar
6 votes
2 answers
420 views

Measurable maximal independent set in infinite graph of bounded degree

We have a graph $G$. The vertices of $G$ are a measurable subset of $\mathbb{R}^n$ for some $n$. The degree of each vertex is bounded by some absolute finite constant $K$. Q1. Does $G$ have a ...
Brendan McKay's user avatar
7 votes
2 answers
403 views

Characterizations of infinite compact Abelian groups and probability spaces based on the forcing notion they give

Let $G$ be an infinite compact Abelian group with the collection $\mathcal{B}$ of Borel subsets of $G$, and $m$ the (unique) normalized Haar measure on $\mathcal{B}$. This gives a natural forcing ...
Mohammad Golshani's user avatar
2 votes
2 answers
197 views

If $\mu$ is a signed measure and $\mathcal S$ is a subfield, then the variations of $\mu$ and $\left.\mu\right|_{\mathcal S}$ coincide on $\mathcal S$

Let $\Omega$ be a set $\mathcal A$ be a $\sigma$-algebra on $\Omega$ $\mu:\mathcal A\to\mathbb R$ be $\sigma$-additive If $\mathcal S\subseteq2^\Omega$ with $\emptyset\in\mathcal S$, then $$\...
0xbadf00d's user avatar
  • 161
1 vote
0 answers
49 views

Extension of $\sigma$-additive vector measures on a ring and the relationship between the corresponding total variation functions

Let $\Omega$ be a set $\mathcal S\subseteq2^\Omega$ be a set with $\emptyset\in\mathcal S$ $\mathcal S_{\text{loc}}:=\left\{A\subseteq\Omega:A\cap S\in\mathcal S\text{ for all }S\in\mathcal S\right\}$...
0xbadf00d's user avatar
  • 161
1 vote
1 answer
328 views

Weak-convergence of probability measures implies the convergence of the measure of a continuity set

Let $\Omega$ be a Polish space and $\mathcal{B}(\Omega)$ be its Borel $\sigma$-algebra. Let $\{\mu_n\}$ be a sequence of probability measures on $\mathcal{B}(\Omega)$ such that $\mu_n$ weak-converges ...
sdj's user avatar
  • 13
6 votes
0 answers
169 views

The Fubini property of the $\sigma$-ideal generated by closed subsets of measure zero

Question. Can a Borel subset $B\subset\mathbb R\times \mathbb R$ be covered by countably many closed sets of measure zero in the plane if for every $(a,b)\in\mathbb R\times\mathbb R$ the horisontal ...
Taras Banakh's user avatar
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1 vote
1 answer
81 views

Function is almost everywhere 1 w.r.t. sequence of regular Borel probability measures

Let $\epsilon>0$ be given. Let $Y$ be a compact, Hausdorff space and let $U\subseteq Y$ be an open subset. Assume that $(\mu_n)_{n\in\mathbb{N}}$ is a sequence of regular Borel probability measures ...
Maria's user avatar
  • 11
0 votes
1 answer
635 views

Upper bound for KL divergence on compact space

Let $(\Omega, \mathcal{F}, \mu)$ be a probability space and let $Q$ be the uniform distribution on $(\Omega, \mathcal{F})$ such that $q = dQ / d\mu$ exists. Then the KL-divergence for some probability ...
bvn's user avatar
  • 3
2 votes
0 answers
163 views

An operator valued Egoroff's theorem

The following statements suggests $B(H)$-valued Egoroff's theorem when $H$ is a separable Hilbert space. Probably it will be hold even if a von Neumann algebra $M$ whose predual is separable is ...
ABB's user avatar
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3 votes
1 answer
356 views

Transformations of càdlàg functions

Denote by $D[0,1]$ the space of càdlàg functions on $[0,1]$. Take a Borel set $B$ in $\mathbb R$ such that $0\notin \overline{B}$ and consider the function $$(Tf)(t) = \sum_{s\leqslant t, f(s)-f(s-)\...
user512365's user avatar
4 votes
1 answer
162 views

Relative volume increase of $\delta$-fattening of a connected set

The following question was asked very recently at Relative volume increase of δ-fattening of a compact set: Is the following inequality true for all non-empty, compact sets $A \subseteq \mathbb{R}^n$ ...
Iosif Pinelis's user avatar
4 votes
1 answer
258 views

Surface/Volume-Ratio of an $\epsilon$-extension of a compact subset $S \subset \mathbb R^n$

For a non-empty, compact set $S \subset \mathbb{R}^n$, the $\epsilon$-extension of $S$, $S_\epsilon$, is defined to be the set $$ S_\epsilon = \cup_{a \in A} B_{\epsilon}(a), $$ where $B_\epsilon(a)$ ...
Lucas L. 's user avatar
6 votes
1 answer
441 views

Relative volume increase of $\delta$-fattening of a compact set

For a non-empty, compact set $A \subseteq \mathbb{R}^n$, the $\delta$-fattening of $A$, $A_\delta$, is defined to be the set $$ A_\delta = \cup_{a \in A} B_{\delta}(a), $$ where $B_\delta(a)$ denotes ...
Cenk Baykal's user avatar
-1 votes
1 answer
75 views

transformation of two measures on different space

Let $\{e_1,e_2,...,e_n\}=E $ be the standard bases of $\mathbb{R}^n$, and $U\subset\mathbb{R}^n$ be a linear space generated by $\{e_1,e_2,...,e_n\}$. Let $\Sigma_U$ be the smallest $\sigma-$ field ...
di sun's user avatar
  • 1
13 votes
1 answer
526 views

Entropy of composition

I asked this at math.stackexchange.com, but got no answers. Let $(X,B,\mu)$ be a probability space. Let $T,S:X→X$ be two measurable measure preserving maps that commute (i.e $TS=ST$). Let $A$ be a (...
Whiskey's user avatar
  • 133
2 votes
0 answers
112 views

Density of $C^0(\Bbb R^{n}\times (0,T))$ and $C^{\infty}_c(\Bbb R^{n}\times (0,T))$ in $L_{p,q}(\Bbb R^n_T)$

Let the space $L_{p,q}(\Bbb R^n_T)$ be defined as the set of all measurable $f:\Bbb R^{n}\times(0,T)\to\Bbb R$ such that $||f||_{p,q}<\infty$, where $$ ||f||_{p,q}:=\left(\int_0^T\left( \int_{\Bbb ...
BigbearZzz's user avatar
  • 1,245
2 votes
1 answer
247 views

Regarding outer functions again

Consider the Hardy space $H^p, 0<p\leq\infty$ (defined here). It is said that given any two outer functions $x_1$ and $x_2$ in $H^p$, there exists $a_1$ and $a_2$ in $H^\infty$ such that $a_1x_1=...
user510271's user avatar
2 votes
1 answer
150 views

Regarding representation of an outer function

Theorem 2.1 in the book ‘Theory of Hp spaces by Peter. L Duren states that : Any function $f$ analytic on the unit disc belongs to the Nevanlinna class iff it is of the form $\frac{g}{h}$ where $g$ ...
user510271's user avatar
3 votes
1 answer
413 views

Is the total variation of a vector measure $\mu$ a (classical) measure, even when $\mu$ is not of bounded variation?

Let $(\Omega,\mathcal A)$ be a measurable space $E$ be a $\mathbb R$-Banach space $\mu:\mathcal A\to E$ with $\mu(\emptyset)=0$ and $$\mu\left(\biguplus_{n\in\mathbb N}A_n\right)=\sum_{n\in\mathbb N}\...
0xbadf00d's user avatar
  • 161
4 votes
1 answer
173 views

Absolute continuity of measures - reference sought

For two measures $\mu, \nu$ on the same space say that $\mu$ is absolutely continuous with respect to $\nu$ ($\mu \ll \nu$) whenever $\nu(A)=0$ implies that $\mu(A)=0$ too. Let $(\Omega, \mathsf P$) ...
Tomasz Kania's user avatar
  • 11.3k
6 votes
0 answers
4k views

Interchange of supremum and integral

Let $f : X \to Y$, $X \subset R^n$, $Y$ Banach space, $g : X \times Y \to R \cup \{ \infty \}$, $L^n$ the n-dimensional Lebesgue measure. Are there some results under which the following interchange ...
yon's user avatar
  • 303
3 votes
1 answer
807 views

Approximation by simple functions on a product $\sigma$-algebra

Let $(\Omega_i,\mathcal A_i)$ be a measureable space $\mathcal M_i\subseteq2^{\Omega_i}$ be a $\pi$-system with $\Omega_i\in\mathcal M_i$ and $\sigma(\mathcal M_i)=\mathcal A_i$ $\mathcal E(M_1\times\...
0xbadf00d's user avatar
  • 161
4 votes
1 answer
233 views

Regarding outer functions

Please see the definition of Hardy spaces on the unit disc here. Let $0<p\leq\infty$. Let $f\in H^p$ with $\|f-1_e\|_p<1$ (Where $1_e$ Is the constant function one). Then is $f$ an outer ...
user510271's user avatar
2 votes
1 answer
121 views

Why do we define the Doléan measure of a continuous square-integrable martingale only on the predictable sets?

If $M$ is a continuous square-integrable martingale on a filtered probability space $(\Omega,\mathcal A,(\mathcal F_t)_{t\in[0,\:T]}\operatorname P)$ and $[M]$ denotes the quadratic variation of $M$, ...
0xbadf00d's user avatar
  • 161
0 votes
1 answer
111 views

a continuity question concerning metrics on probablility measures

For a metric space $M$, I'll write $Prob(M)$ for the Borel probability measures on $M$. I am interested in metrics on $Prob(M)$, such as the Kantorovich distance (or other metrics). If $f: M \...
Larry Moss's user avatar
2 votes
0 answers
102 views

Is this concrete set generically Haar-null?

This question is related to this problem on MO about generically Haar-null sets in locally compact Polish groups but is more concrete. First we recall the definition of a generically Haar-null set in ...
Taras Banakh's user avatar
  • 40.8k
6 votes
0 answers
127 views

How big may the maximum set of entire function be?

Let us consider an entire function of several complex variables $f(z_1,\dots,z_n)$ and its modulus maximum $$M(r,f):=\max \{ |f(z_1,\dots,z_n)|: |z_1|\le r,\dots,|z_n| \le r \} $$ with $r\ge 0$. How ...
user64494's user avatar
  • 3,309
2 votes
0 answers
56 views

Absolute continuity of DOS measure for Schrödinger operators

Kotani theory gives roughly that for ergodic operators there is a certain equivalence between absolutely continuous spectrum and an absolutely continuous density of states measure. I would like to ...
DDriggs's user avatar
  • 21
5 votes
0 answers
212 views

On generically Haar-null sets in the real line

First some definitions. For a Polish space $X$ by $P(X)$ we denote the space of all $\sigma$-additive Borel probability measures on $X$. The space $P(X)$ carries a Polish topology generated by the ...
Taras Banakh's user avatar
  • 40.8k
2 votes
1 answer
275 views

Density in Wasserstein space

I am wondering whether the following result is true: Let $\mathcal W_p(\mathbb R^d)$ be the Wasserstein space of order $p$ and let $\eta$ and $\gamma$ be two probability measures in $\mathcal W_p(\...
Ryan's user avatar
  • 325
7 votes
1 answer
241 views

Convergence in Lebesgue measure

It is well known that if $K_n$ are compact sets in $\mathbb{R}^n$ converging in Hausdorff distance to $K$ compact as well, then it does not follow that their Lebesgue measures converge (even if the ...
JeffreyStone's user avatar
0 votes
0 answers
108 views

Capacity and measure

Fix $p\in [1, 2)$ and denote the $p$-capacity of a compact set $K$ as $p$-$\text{cap}(K)$, i.e., \begin{equation} p\text{-cap}(K)\equiv\left\{\int_{\mathbb{R}^2}|D\varphi|^p\ \mathrm{d}x\ \Big|\ \...
Nirav's user avatar
  • 347
2 votes
0 answers
679 views

Recognizing "complete" $\sigma$-algebras

Disclaimer : I asked a very similar question on MSE two days ago. Let $E$ be a set, let $\mathcal{E}\subset\mathcal{F}$ be two $\sigma$-algebras on $E$, and let $\mathcal{X}\subset\mathcal{F}$ be a ...
Olivier Bégassat's user avatar
4 votes
0 answers
2k views

Does rate of convergence in probability come from a metric?

In general, when we talk about convergence of a sequence, we need a topological space. If we want to talk about a rate of convergence, we need to quantify how far away one element of the sequence is ...
Froomfondel's user avatar
12 votes
1 answer
993 views

Structure of the Cantor part of the derivative of a BV function

It is well known that an integrable function $u \colon \mathbb R^d \to \mathbb R$ is said to be of bounded variation iff the distributional gradient $Du$ is (representable by) a finite Radon measure, ...
Romeo's user avatar
  • 960
4 votes
0 answers
109 views

Reference to an explixit construction of a locale from a measurable space

In A sheaf theoretic approach to measure theory shows that measures on a measurable space are equivalent to measures on some locale whose open sets are the $\sigma$-ideals of the $\sigma$-algebra. The ...
Lolman's user avatar
  • 369
1 vote
0 answers
90 views

Measure of the boundary of the support of a certain function defined by an expectation

Suppose: $\mathcal{S} = \{ S \in \mathbb{R}^d \ | \ S_i > 0, \forall i = 1,...,d \} $ $R$ is a random vector (on some probability space, $\Omega$) such that, $R: \Omega \to \mathcal{S}$. $h : ...
d_797's user avatar
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