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
3
votes
1
answer
173
views
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 ...
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
$$
\...
7
votes
2
answers
1k
views
Conditional Expectation for $\sigma$-finite measures
Someone knows of some definition or reference of how to define conditional expectation for a measure space with $\sigma$-finite measure.
I think it should be as follows:
Let $(X,\mathcal{B},\nu)$ ...
2
votes
1
answer
1k
views
Understanding measure-preserving transformation [closed]
Given measure space $(S, \mathcal{S}, \mu)$, and measurable function $\phi: S \to S$. $\phi$ is measure-preserving if $\forall A \in \mathcal{S}, \mu(A) = \mu(\phi^{-1}(A))$. My confusion is that why ...
4
votes
1
answer
263
views
Zero-one law for an independence-like structure
I am a number theorist by profession, so apologies if the answer to this question is "trivially true" or "trivially false".
Let $(\Omega, \mathcal{A}, P)$ be a (non-atomic) probability space. Let $(\...
4
votes
1
answer
1k
views
Is the Wasserstein-1 metric translation invariant?
Define the Wasserstein-1 metric (or the Earth mover's distance) between two positive measures $\mu_1$, $\mu_2$ by
$$
W(\mu_1, \mu_2)
=
\inf_{\gamma \in \Gamma (\mu_1, \mu_2)} \int \|x_1 - x_2\| \, \...
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 $h$ ...
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 ...
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 ...
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 ($C(X)$ ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $$\...
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\}$...
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 ...
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 ...
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 ...
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 ...
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 ...
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-)\...
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 $A \subseteq \mathbb{R}^n$ ...
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 $B_\epsilon(a)$ ...
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 ...
-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 ...
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 (...
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
$$
||f||_{p,q}:=\left(\int_0^T\left( \int_{\Bbb ...
2
votes
1
answer
247
views
Regarding outer functions again
Consider the Hardy space $H^p, 0<p\leq\infty$ (defined here).
It is said that given any two outer functions $x_1$ and $x_2$ in $H^p$, there exists $a_1$ and $a_2$ in $H^\infty$ such that $a_1x_1=...
2
votes
1
answer
150
views
Regarding representation of an outer function
Theorem 2.1 in the book ‘Theory of Hp spaces by Peter. L Duren states that : Any function $f$ analytic on the unit disc belongs to the Nevanlinna class iff it is of the form $\frac{g}{h}$ where $g$ ...
3
votes
1
answer
413
views
Is the total variation of a vector measure $\mu$ a (classical) measure, even when $\mu$ is not of bounded variation?
Let
$(\Omega,\mathcal A)$ be a measurable space
$E$ be a $\mathbb R$-Banach space
$\mu:\mathcal A\to E$ with $\mu(\emptyset)=0$ and $$\mu\left(\biguplus_{n\in\mathbb N}A_n\right)=\sum_{n\in\mathbb N}\...
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 $(\Omega, \mathsf P$) ...
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 ...
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\...
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 ...
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 $M$, ...
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 \...
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 ...
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 ...
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 ...
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 ...
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(\...
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 ...
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|\ \...
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 ...
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 ...
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, ...
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 ...
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 : ...