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
10
votes
2
answers
462
views
Graph metric approximating Euclidean metric
I've been reading Wolfram's recent articles about graph/mesh/grid structures as an analogy for physical space, and it seems to me that there will be a problem getting the notion of distance to work ...
- 119
18
votes
1
answer
510
views
Acting with a finite number of rotations on a set of positive measure can you fill almost the whole circle?
Let $E\subset S^1$ have positive Lebesgue measure. Do there exist finitely many rotations
$r_1, r_2, \dots ,r_n$ such that $r_1E\cup r_2E\cup \dots\cup r_nE$ has measure $2\pi$? Or is there a ...
- 441
0
votes
1
answer
54
views
Good upper-bound for $\mathbb E_A[e^{-t\|A\|_2}]$, for $t\ge0$ and random m by n matrix with iid entries with law $N(0,1)$
Let $A$ be a random $m$-by-$n$ matrix with iid $N(0,1)$ entries, $m$ and $n$ large with $n/m \longrightarrow \alpha \in (0, 1)$ . Let $\|A\|_2$ be the largest singular value of $A$ (i.e the spectral ...
- 6,726
6
votes
3
answers
549
views
Acting with all rational rotations on a subset of the circle having positive measure do you fill almost the whole circle?
Set $\Gamma$ for the group of the roots of the identity: $\Gamma=\{z\in \Bbb C | z^n=1$, for some $n\geq 0\}$ and for $E\subset S^1$ set
$\Gamma E=\{z\zeta, z\in \Gamma, \zeta\in E \}$
A trivial but ...
- 441
5
votes
1
answer
360
views
Inverse marginal property of a collection of $\sigma$-algebras
In my paper "On the inverse best approximation property of systems of subspaces of a Hilbert space"
I introduced the Inverse marginal property (IMP) for a collection of $\sigma$-algebras.
Let $(\...
- 925
2
votes
0
answers
114
views
Borel measurability
Suppose we have two locally compact Hausdorff spaces $X$ and $Y$. Let $i:X\to Y$ be a continuous injection. Under what condition the Borel $\sigma$-algebra of $X$ and $i(X)$ are isomorphic via the map ...
- 1,765
0
votes
1
answer
110
views
Can we say that : $ \exists f_{\infty}\in L_{\mathbb{R}}^{1} \text{ such that: } f_n\to f_\infty\text{ a.e and in } L_{\mathbb{R}}^{1} $ [closed]
Let $(E,\mathcal{A},\mu)$ be a finite measure space and $\{f_n\}\subset L_{\mathbb{R}}^{1}$ such that:
$$
\sum_{i=2}^{\infty}{\int_{E}{|f_n(t)-f_{n-1}(t)|d\mu(t)}}<+\infty
$$
Can we say that :
$$
\...
- 115
4
votes
0
answers
190
views
Classification of Euclidean-invariant measures?
Is there a classification of measures on $\mathbb R^n$ which are invariant under (Euclidean) isometries? Hausdorff measures of all kinds are examples -- could that be all of them? More precisely,
By ...
- 61.6k
0
votes
1
answer
104
views
$\sum_{n=1}^{\infty}{\frac{1}{n^{1+\epsilon}}\mathbb{E}\big((|X_n|\mathbb{1}_{|X_n|\leq n})^{1+\epsilon}\big)}<\infty,~~\forall\epsilon>0 $
Let $(E,\mathcal{A},\mathbb{P})$ be a probability space $\{X_n\}$ be a sequence of random variable, such that:
$$
(1)~.~~~\sup_n\mathbb E (|X_n|)<\infty\Rightarrow
$$
$$
(2)~.~~~\dfrac{M_j}{2}<...
- 115
3
votes
2
answers
336
views
Usefulness of the $\sigma(L^\infty,L^1)$ topology in the context of differential equations
In Brezis's Functional Analysis book through chapters 3-4, I've seen the $\sigma(L^\infty,L^1)$ topology on $L^\infty$ but did not see (so far) any application of it in differential equations. Is ...
- 597
4
votes
1
answer
496
views
Weak convergence in $L^1(X,\mu)$ space
I found an interesting property in some lecture notes on weak convergence: lemma 3.2 (2) on page 10: https://www.uio.no/studier/emner/matnat/math/MAT4380/v06/Weakconvergence.pdf . The idea is as ...
- 41
1
vote
1
answer
147
views
The existence of a copy of a random variable with conditional expectation constraint
Let there be two random variables 𝑋 and 𝑌 with a certain joint copula. Is it always true that there is another random variable 𝑍 independent from 𝑌 such as the vectors $(X,Y)$ and $(X,Z)$ have the ...
- 375
4
votes
1
answer
230
views
Is the intrinsic volume always positive for maximum dimension?
The intrinsic volume functions on $\mathbb{R}^d$ known from the Steiner formula and Hadwiger's Theorem can be extended to the domain of definable sets of an o-minimal structure $\text{Def}(\mathbb{R}^...
- 499
0
votes
1
answer
143
views
$ \{X_n\mathbb{1}_{X_n\in[-n,n]}\}$ is uniformly integrable
Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space.
Suppose $\{X_n\}$ is a sequence of random variables satisfying :
$$
\sup_{n}{\mathbb{E}(|X_n|)} <\infty
$$
Suppose that
$$
\dfrac{M_j}{...
- 57
1
vote
0
answers
63
views
Local time as a measurable map from Wiener space
Let $B$ be a Brownian motion on $[0,1]$. The local time of $B$, which I will denote by $L$, is defined as the process on $\mathbb R$ such that
$$\int_0^1 F(B_t)~dt=\int_\mathbb R F(x)L(x)~dx,\qquad\...
- 891
0
votes
1
answer
634
views
Convergence in $L^\infty([0,T];L^2(\Omega))$
I wonder whether this fact is true or not (if a counter-example exists, please just give a hint on how to construct it!):
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Consider a bounded ...
- 597
5
votes
1
answer
316
views
Function whose graph is a Borel relation
Suppose $f\colon\mathbb{R}^{\omega}\longrightarrow\mathbb{R}$ is a function such that
$$G(f):=\{(x,y)\in\mathbb{R}^{\omega}\times\mathbb{R}\mid f(x)=y\}$$
is a Borel set. Does it necessarily follow ...
- 1,612
-1
votes
1
answer
88
views
Show that: $ \int_{E}{g_\infty(t)d\mu(t)}\leq \lim_{n}{\int_{E}{g_n(t)d\mu(t)}} $ [closed]
Let $(E,\mathcal{A},\mu)$ be a finite measure space and $\{f_n\}$ be a sequence of $\mathcal{L}_{\mathbb{R}}^{1}$, such that $\{\min\bigl(0,f_n(\cdot)\bigr)\}$ (the negative parts of $f_n$) is ...
- 57
2
votes
1
answer
136
views
Duality form of $L^q$ norm, without assumption that $\int fg$ defined?
The following theorem is found, for example, in the Real Analysis books by Folland, by Yeh, and (in a slightly different form) by Royden.
Theorem. Let $(X,\mathcal{A},\mu)$ be a measure space.
Let ...
- 201
1
vote
0
answers
72
views
Maximal function corresponding to a family of functions
Let $(f_t)_{t\geq 0}$ be a family of measurable functions. Under which criterion we can say that $\sup_{t\geq 0}\lvert f_t\rvert$ defines a measurable function? I am working in the following ...
- 1,765
2
votes
0
answers
35
views
When does a measure-valued map admit a continuously parametrized density function?
Let $X$ and $Y$ be Polish spaces, let $\mathcal{P}(Y)$ be the space of Borel probability measures on $Y$ endowed with the smallest $\sigma$-algebra such that all functions of the form $\nu\mapsto\nu(A)...
- 12.6k
1
vote
1
answer
132
views
Lambda system generated by a non-atomic collection
Consider a probability space $(X,\Sigma,P)$.
Let say that a collection $\mathcal{B}\subseteq\Sigma$ is non-atomic if for every $E\in\mathcal{B}$ and $\alpha\in(0,P(E))$ there exists $F\in\mathcal{B}$ ...
- 11
4
votes
2
answers
454
views
Packing a Riemannian manifold with disjoints balls
Let $M$ be a smooth Riemannian manifold with Riemannian measure $\mu$. I don't suppose that $M$ is complete. Can we find a finite or countable disjoint collection of open (or closed) and relatively ...
- 111
0
votes
0
answers
163
views
Is the co-projection of the projection of a Borel set Lebesgue measurable?
Reading the classical book of Kechris "Classical descriptive set theory", I found the following facts
The projection of a Lebesgue measurable set need not be Lebesgue measurable, but the projection ...
- 21
1
vote
1
answer
97
views
Integral average near a point of dispersion
Let $\Omega\subset\subset\mathbb R^{n}$ be a bounded domain and let $E\subset \Omega$ be a Lebesgue measurable set. Let $f\in L^{1}(\Omega)$ and let $x\in \Omega$ be a point of dispersion of $E$, that ...
- 21
3
votes
1
answer
154
views
Is the inner/outer measure mapping continuous?
Let $\mathcal F$ be a field of subsets of a set $\Omega$. Equip the space $[0,1]^\mathcal F$ of functions from $\mathcal F$ into $[0,1]$ with the product topology. Then, the set $\Delta$ of finitely ...
- 839
2
votes
0
answers
60
views
Measurable extensions of probability measures
Let $X$ be a set, and let $\mathcal G \subset \mathcal F$ be $\sigma$-fields over $X$. Let $\Delta_\mathcal G$ (resp. $\Delta_\mathcal F$) be the set of probability measures on $\mathcal G$ (resp. $\...
- 839
0
votes
1
answer
112
views
Square integrable borel probability measures on Euclidean spaces are the law of random variables from an atomless polish space
Could someone provide me with a reference or proof for the following: Let $(\Omega, \mathcal{A}, P)$ be an atomless probability space, with $\Omega$ a Polish space. Given $f$ a random vector on $(\...
0
votes
1
answer
61
views
measures on groups without assuming a locally compact group topology
I'm interested in knowing whether there exists any kind of theory for measures on groups without assuming that it's the Haar measure for a locally compact group topology.
- 2,005
2
votes
0
answers
64
views
Measure of the convex hull of a ball and a point
I need to prove the following statement:
Let $B_s(z)$ be a ball centered at $z$ of radius $s$ s.t. $0\not\in B_s(z)$. Moreover let $K_s(z)$ the convex hull of $\{0\}\cup B_s(z)$.
Then
$$ \...
- 73
0
votes
0
answers
239
views
Constructing uncountably many independent random variables with same distribution from Brownian motion?
It is well known one cannot construct uncountable many independent random variables on $([0, 1], \mathcal{B}[0, 1], \lambda)$. ($\lambda$ Lebesgue measure.)
Also, one can clearly construct infinitely ...
user152718
0
votes
1
answer
146
views
separable support of Borel measure, with tau-additive measure and full support
I have a problem with Proposition 7.2.10 in Bogachev's Measure Theory Volume II book on page 77 (I have link to my drive with that book https://drive.google.com/file/d/...
- 127
3
votes
0
answers
93
views
For $\mathcal{L}^1$-a.e. $t\in R$, Hausdorff dimension of level sets of a locally Lipschitz function $f:R^n\to R$ is $n-1$?
Let $f:R^n\to R$ be a locally Lipchitz function. Denote $H^n$ the n-dimensional Hausdorff measure. We know that for any $H^n$-measurable subset $A\subset R^n$, for $\mathcal{L}^1$-a.e. $t\in R$, $A\...
1
vote
0
answers
53
views
Standard definition: vector-valued essential support
Let $f \in L^p(\mathbb{R}^n,\mathbb{R}^m)$. If $m=1$ then the essential support of $f$ is a mainstream definition; see here for example. However, when $m>1$ is the following definition used?
$$
\...
- 4,969
1
vote
0
answers
116
views
Reference request: harmonic analysis with non-Lebesgue reference measure
The Lebesgue measure on $\mathbb{R}^d$
admits the following polar decomposition:
$$
L(dx) = r^{d-1} dr \lambda(dy),
$$
where 𝜆 is the uniform measure on the Euclidean unit sphere of $\mathbb{R}^d$ ...
user153086
9
votes
1
answer
1k
views
Dual space of continuous Banach-space-valued functions
Let $X$ be a Banach space and $K$ some compact Hausdorff space. I am interested in the dual space of the Banach space
$$C(K; X) = \lbrace f: K \to X, \ f \text{ is continuous}\rbrace, \qquad \lVert ...
- 371
1
vote
0
answers
44
views
Decomposition of the space of Radon measures with respect fractional harmonic capacity?
It is well know that there is a generalization of Lebesgue decomposition theorem in the following way:
Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely ...
- 361
2
votes
3
answers
1k
views
Continuity of Brownian motion constructed from Kolmogorov extension theorem?
I'm trying to construct Brownian motion using the Kolmogorov extension theorem.
I am happy with the construction of a process with the required FDDs as (the canonical process associated with) a ...
user152718
16
votes
2
answers
1k
views
How often two iid variables are close?
Is there a constant $c>0$ such that for $X,Y$ two iid variables supported by $[0,1]$,
$$
\liminf_\epsilon \epsilon^{-1}P(|X-Y|<\epsilon)\geqslant c
$$
I can prove the result if they have a ...
- 1,268
1
vote
1
answer
168
views
Convergence of local means implies converge ae?
Let $f,f_n \in L^1(\mathbb{R},\mathbb{R}_+)$ with $\int_{\mathbb{R}} f = \int_{\mathbb{R}} f_n = 1$, $(\sqrt{f_n})'$ bounded in $L^2$, $\nabla \sqrt{f}\in L^2$ and such that $$\int_{ p+[0,1/n]} f_n = \...
- 23
0
votes
1
answer
96
views
Law of a step function and its generalization to two dimensions on an appropriate spaces
Let's consider two discontinuous functions defined on $D$ and $D \times [0,T]$, respectively:
A step function: $u_1(x)=\begin{cases}
u_{L}, x<c_1, \\[2ex]
u_{R}, x>c_1,
\end{cases}$
A "...
- 647
2
votes
1
answer
320
views
On the hereditary Lindelof topological spaces
I received the following interesting point in (1). I could not find any reference or clear proof. Any suggestion?
Theorem. A topological space $X$ is hereditary Lindelof if and only if for any ...
- 3,972
2
votes
0
answers
235
views
Reference for Borel $\sigma$-algebra of topology of convergence in probability
I'm pretty sure I can prove the "Theorem" given further below (without very much difficulty), but it seems way too basic not to have been noticed before.
So I'm wondering if there are any papers/...
- 678
9
votes
1
answer
683
views
Does every measurable subset of $\mathbb R$ of non zero Lebesgue measure contain arbitrarily long arithmetic progressions?
A subset $E$ of $\mathbb R$ is said to contain arbitrarily long arithmetic progressions, if for every natural $n$, there exists $a, d \in R, d$ nonzero, such that $a + kd$ is in $E$ for all natural $k ...
- 2,039
14
votes
1
answer
630
views
Does there exist a non-zero signed finite borel measure which is zero on all balls?
Let $(X,d)$ be a compact separable metric space. Let $\mu$ be a Borel, regular, finite, signed measure on $X$ such that for all $x\in X$, for all $r>0$, $\mu(B(x,r))=0$, where $B$ denotes the (...
- 335
2
votes
1
answer
206
views
The measure of ideals generated by random reals
We assume that for every real $x$, $L[x]$ only contains countably many reals.
Given a set $X$ of reals, then $L$-ideal generated by $X$ is the smallest set $I$ of reals so that
For any reals $x\in ...
- 4,201
2
votes
0
answers
137
views
What is the motivation to define measure valued solutions to a PDE model?
Consider the model
$$\partial _{t}\mu + \partial_{x}(b(t,\mu)\mu)+c(t,\mu) \mu=0,~~~\mu \in \mathcal{M}^{+}(\mathbb{R}^{+}), t \in [0,T], x \in \mathbb{R}^{+} $$
$$ \mu(0)=\mu_{0} $$
where $ \mu (t)$...
- 133
1
vote
0
answers
166
views
A question about Stroock's notes on the Weyl lemma
On p.4 of these notes, D. Stroock gives a quick and efficient construction of the Markov transition functions of a certain diffusion. The idea of his construction (on page 4) is to 'freeze' the ...
- 1,451
2
votes
0
answers
42
views
Commonly used metrics to compare two Young measures
Let $\Omega\subset \mathbb{R}^n$ be a bounded open set, $K\subset \mathbb{R}^d$ be a compact set, and $M_1(K)$ be the set of probability measures on $K$. Then a Young measure is defined as a $\textrm{...
- 483
1
vote
1
answer
81
views
Disintegration associative
Is the disintegration of two borelian probabilities measures is associative ? It means if $\mu = \mu_{y}^{1} \oplus h_{\#}^{1}\mu$ and $ h_{\#}^{1}\mu = \mu_{y}^{2} \oplus h_{\#}^{2} h_{\#}^{1}\mu$. ...
- 169