Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
2,911
questions
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 \...
- 61
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 ...
- 40.9k
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 ...
- 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 ...
- 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 ...
- 40.9k
2
votes
1
answer
276
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(\...
- 325
7
votes
1
answer
244
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 ...
- 73
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|\ \...
- 347
2
votes
0
answers
682
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 ...
- 2,673
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 ...
- 63
12
votes
1
answer
1k
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, ...
- 960
4
votes
0
answers
110
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 ...
- 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 : ...
- 111
1
vote
0
answers
294
views
Approximation of Borel sets
Let $\nu$ be a finite Radon measure on $\mathbb{R}^2$ and denote the Lebesgue measure on $\mathbb{R}^2$ by $\mathcal{L}^2$. Assume that $\nu<<\mathcal{L}^2$.
We denote the boundary of $A\subset\...
- 347
1
vote
1
answer
169
views
Bochner integrability within a subspace
Let $(H,||\cdot||_H)$ be a Banach space and $K$ a (not necessarily closed) subspace. Suppose that $K$ is a Banach space under another norm $||\cdot||_K$, which satisfies
$$||x||_H\leq ||x||_K$$
for ...
- 1,871
5
votes
2
answers
661
views
Absolute continuity of measures on infinite binary sequences
Suppose $P$ and $Q$ are two probability measures on the space $\Omega = \{0,1\}^{\mathbb N}$ of infinite binary sequences equipped with the product $\sigma$-algebra generated by its cylinder sets, ...
- 51
4
votes
1
answer
610
views
Locally finite measures on a Polish space form a Polish space
I am looking for a reference where the following question is answered (hopefully affirmatively):
Let $S$ be a Polish space (maybe one needs to assume local compactness?). Is the space of locally ...
- 725
2
votes
1
answer
151
views
Linking error probability based on total variation
Consider probability measure $\mu_{XY}$ defined on $\mathbb{R}^d \times \{1,2,3\}$, and sub-probability measures $\mu_1$, $\mu_2$, and $\mu_3$ as $\mu_1(A):=P(X\in A, Y=0)$ and $\mu_2(A):=P(X\in A, Y=...
- 482
1
vote
0
answers
100
views
Paper by Diestel, Faires and Huff
I have been looking for a (long) while for the following paper:
J. Diestel, B. Faires, and R. Huff, Convergence and boundedness of measures on non-sigma complete algebras, preprint, 1976.
This ...
- 1,153
1
vote
1
answer
147
views
Conull subspace containing orbit of an (ergodically acting) group
I should probably start with a warning that this is my first post in this board and that I am sorry, if it is not up to standards. It would be great, if you could let me know how to improve the post.
...
- 13
1
vote
1
answer
181
views
Existence of a certain norm on space of measurable functions
Suppose $X$ is a measure space with measure $\mu$. Given a strictly increasing continuous (or sufficiently nice) function $\phi:[0, \infty)\to [0, \infty)$ with $\phi(0)=0$. Is it true that we can ...
- 13
6
votes
1
answer
351
views
Why are $\sigma$-algebras preferable to $\sigma$-rings?
The following is said without further explanation in Folland's Real Analysis:
Some authors prefer to take the domains of measures to be $\sigma$-rings rather
than $\sigma$-algebras. The reason is ...
user14319
8
votes
1
answer
450
views
A criterion for second countability
Let $(X,\tau)$ be a topological space.
Assume for any arbitrary topological base $\mathcal{E}$ of $\tau$ we have that: the Borel sigma algebras coming form $\mathcal{E}$ and $\tau$ are the same. ...
- 3,972
3
votes
1
answer
1k
views
Measurable functions in product space
I am reading a book by Billingsley (convergence of probability measures) and he makes a footnote on page 27 which I am struggling to understand. I'll explain the setup below.
Suppose $(X_n,Y_n)$ are ...
- 33
5
votes
0
answers
138
views
Disjoint covering number of an ideal
Let $\mathcal I$ be a $\sigma$-ideal with Borel base on an uncountable Polish space $X=\bigcup\mathcal I$.
Let $\mathrm{cov}(\mathcal I)$ (resp. $\mathrm{cov}_\sqcup(\mathcal I)$) be the smallest ...
- 40.9k
1
vote
1
answer
208
views
the "observable" space of a measure space [closed]
For a measure space $(X,\mathcal{A},\mu)$, the space of "observables" with respect to finite set $F$ which is endowed with counting measure on all of its subsets, is defined as follows:
$$obs (X, \mu,...
- 181
33
votes
2
answers
2k
views
Are the Sierpiński cardinal $\acute{\mathfrak n}$ and its measure modification $\acute{\mathfrak m}$ equal to some known small uncountable cardinals?
This question was motivated by an answer to this question of Dominic van der Zypen.
It relates to the following classical theorem of Sierpiński.
Theorem (Sierpiński, 1921). For any countable partition ...
- 40.9k
0
votes
0
answers
246
views
Expected value of parametrized Gibbs distribution w.r.t another probability distribution
Let $\mu$ be a compactly supported probability measure on a finite-dimensional euclidean space (for simplicity) $\mathbb E$, and suppose $\mu$ has density. For a random point $x \sim \mu$,
consider ...
- 6,726
1
vote
1
answer
70
views
Extension preserves the relation between two measures
Let $\rho_1$, $\rho_2$ be two measures(not necessarily nonnegative) on $(\Omega,\mathcal{F})$, where $\Omega$ is a set, and $\mathcal{F}$ is a $\sigma$-field in $\Omega$. Let $\mathcal{F}_0$ be a ...
- 113
1
vote
2
answers
260
views
Convergence of an iterated sequence
Let $K=[0,1]^2$ be a square and $p\in (0,1)$ be a fixed number. We define a map $F: K^2\to K^2$ as follows.
For $(x_1,y_1), (x_2,y_2)\in K$, it follows by a straightforward computation that there ...
- 345
0
votes
0
answers
81
views
Consequence of John and Nirenberg's lemma?
The lemma I'm referring to in the title is the following:
John and Nirenberg's lemma: Let $C_0 \subset \mathbf{R}^n$ a finite cube. Let $u \in L^1(C_0)$ and assume there exists a constant $k$ such ...
- 109
6
votes
1
answer
2k
views
About weak convergence of probability measure
Suppose $\mu_j$ is a sequence of measures on $\mathbb{R}$. By the definition of weak convergence of measures, $\mu_j$ weak converges to $\mu$ means that for any bounded continuous function $f$, there ...
- 71
16
votes
4
answers
6k
views
Good book for measure theory and functional analysis
I have taken advanced courses both in measure theory and also in functional analysis (Banach and Hilbert spaces, spectral theory of bounded and unbounded operators, etc.)
The connections between the ...
1
vote
0
answers
96
views
Random projection increases the distance?
Consider two absolutely continuous random variables $X: \Omega \mapsto \mathbb{R}^d$ and $Y: \Omega \mapsto \mathbb{R}^d$ for probability spaces $(\Omega, \mathcal{F},p_X)$ and $(\Omega, \mathcal{F},...
- 482
11
votes
1
answer
606
views
Translates of measure zero set
Suppose $X \subseteq \mathbb{R}$ has measure zero. Can we find an uncountable $A \subseteq \mathbb{R}$ such that $X + A = \bigcup \{a + x: a \in A, x \in X\}$ has measure zero?
Clearly, the answer is ...
- 111
3
votes
1
answer
376
views
measurable selection and values of optimization problem
In general, my problem can be formulated as follows: Let $X$ be a random variable with value in $\mathbb R^2$, and let $G:\mathbb R^2 \times \mathbb R\rightarrow \mathbb R$ be a function which is ...
- 325
0
votes
1
answer
261
views
Spectral Theorem, $AB = BA \implies B\Phi(f) = \Phi(f)B$
I'm studying the spectral theorem as appears in Reed and Simon's Functional Analysis.
Assume we have constructed the continuous functional calculus for a self adjoint bounded operator $A$ on a ...
- 181
3
votes
1
answer
388
views
Are induced transformations always measure-preserving on infinite measure spaces?
Let $T$ be a measure preserving transformation on a measure space $(X, \mathscr{F}, m)$ with infinite measure $m$. Let $A \in \mathscr{F}$ be such that $X = \cup_{k=0}^\infty T^{-k} A \pmod{m}$. Then ...
- 173
3
votes
1
answer
156
views
Definition of homogeneous or quasi-uniform or almost uniform measure
Let us call a measure $\Lambda$ homogeneous if there is an $\epsilon>0$ so that for all $r>0$ and $x,y$ in the support of $\Lambda$, we have
$$\Lambda(B(x,r))>\epsilon\Lambda(B(y,r))$$
...
- 141
0
votes
1
answer
302
views
How to characterize Radon Nikodym's derivative of a coupling with respect to any measure in the product space?
In my math essay of thesis I have defined the probability coupling as follows
$$\Pi(\mu,\nu)=\left\lbrace \pi \in \Omega \left\vert
\begin{matrix}
\pi(A\times\mathcal{Y})=\mu(A) \\
\pi(\mathcal{X} \...
user70004
4
votes
1
answer
309
views
$\tau$-additive measures on a complete metric space are tight
Let $X$ be a complete metric space. Are all $\tau$-additive Borel measures on $X$ tight?
In Bogachev's "Measure Theory", vol. 2, in the proof of Theorem 8.9.4 (end of page 213) it says:
Note that ...
- 2,129
1
vote
2
answers
112
views
measurability of the limit of functions parameterized by real numbers
Let $f_s: \mathbb R \to \mathbb R$ a be family of Borel measurable functions parameterized by $s\in \mathbb R$. Consider the limit function
$$ F(t)=\limsup_{s\to 0} f_s(t). $$ Is the function $F$ ...
- 833
8
votes
4
answers
2k
views
Is every probability measure a pushforward of Lebesgue measure?
If $m$ is a probability measure on a measurable space $(X, \Sigma)$, is there necessarily a measurable function $f : [0, 1] \to X$ such that $m(A) = \mu(f^{-1}(A))$ for all $A \in \Sigma$?
($\mu$ is ...
- 83
5
votes
1
answer
922
views
Has a discrete/quantum theory of probability based on the Cournot-Borel principle or something been developed?
In 1930, Émile Borel, the father of measure theory together with his student Lebesgue and a world-class expert in probability theory, published a short note Sur les probabilités universellement ...
- 1,016
3
votes
1
answer
223
views
Is there a canonical uniform probability measure on compact subsets of Banach spaces?
One can construct a finite measure on a compact metric space $(X,d)$ by the following procedure:
Fix a non-negative sequence $\{\epsilon_n\}$, $\epsilon_n \to 0$. Let $Y_{\epsilon_n}$ be the minimal ...
- 31
1
vote
1
answer
185
views
Proof that the subspace of signed measures integrating d(x,e) is closed
Let $\mathcal{M}(S)$ be a space of finite signed measures on a metric space $S$ ($=\mathbb{R}^2$ in my case) equipped with the total variation norm. Let
$\mathcal{M}_1(S)=\{\mu \in \mathcal{M}(S):\...
- 13
0
votes
2
answers
405
views
Ordered measurable spaces
Let $(X, \leq)$ be a partial order and $\Sigma_X$ a $\sigma$-algebra on $X$. Is the set $\{(x, y) \in X\times X \mid x \leq y\}$ measurable with respect to the product $\sigma$-algebra?
- 239
11
votes
2
answers
474
views
The "strong" measure number
Beyond measure zero we have yet another measure-y notion of smallness: strong measure zero. A set $S\subseteq\mathbb{R}$ is strong measure zero if, for any $f:\mathbb{N}\rightarrow\mathbb{R}_{>0}$, ...
- 19.3k
8
votes
2
answers
358
views
Extending the product measure on $2^\omega$
Consider the standard completed product measure $P$ on $\Omega=\{0,1\}^\omega$ corresponding to an i.i.d. sequence of fair coin-flips.
Given $n\in\omega$, let $\rho_n$ be the bijection of $\Omega$ ...
- 2,383
10
votes
1
answer
226
views
Distribution of good diophantine approximations
Let $\langle x \rangle: \mathbb{R} \to (-1/2,1/2]$ be the periodic function with period $1$ which is $x$ for $x \in (-1/2,1/2]$. Is there some function $D(a,b)$ of real numbers $a<b$ such that, for ...
- 151k