Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
856 questions with no upvoted or accepted answers
43
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A kaleidoscopic coloring of the plane
Problem. Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap D)=\...
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votes
0
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1k
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Example of a quasi-Bernoulli measure which is not Gibbs?
Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider measures on $X$ only.
A measure $\mu$ is quasi-Bernoulli if there is a constant $C\ge 1$ such that for any finite sequences $i,j$,
$$
C^{-1} \...
- 4,660
23
votes
0
answers
939
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A question about small sets of reals
In ZFC, does there exist an uncountable set of reals $A$ such that for every closed measure zero set of reals $B$, we have that $ A + B = \{a+b : a \in A, b \in B\} \neq \mathbb{R}$?
This question is ...
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0
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520
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Is the exponent of $2$ in the Pythagorean theorem the "same $2$" as $[\mathbb{C} : \mathbb{R}]$?
I posted this question in Math StackExchange a couple years ago; due to the recent surge in interest, and following the feedback of several users, I've decided to cross-post it here. I apologize for ...
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0
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1k
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A question about sigma algebras and rectangles
Let P be the statement: Every subset of plane belongs to the sigma algebra generated by $\{A \times B : A, B \subseteq \mathbb{R}\}$.
Let Q be the statement: Every continuum-sized family of subsets ...
- 9,631
18
votes
1
answer
2k
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Function of two sets intersection
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). Does ...
- 1,209
16
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0
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247
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Gap two Sierpinski set?
Is it consistent to have a set of reals $X$ of size $\aleph_3$ such that for every $Y \subseteq X$, $Y$ has measure zero iff $|Y| \leq \aleph_1$?
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15
votes
0
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3k
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Weak$^*$ convergence of measures vs. convergence of supports
Let $X$ be a compact metric space and let $\mathcal M(X)$ denote the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\text{supp} \mu$ for the support of $\mu$. It is easy to ...
- 1,658
15
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510
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Lebesgue density 1/2 (or bounded away from 0 and 1)
From the work of Preiss, we know that in infinite-dimensional spaces, one has violations of the Lebesgue density theorem. In particular, he has constructed examples of probability spaces where a set ...
- 6,541
15
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0
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1k
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A Kakeya-like problem: must a union of annuli fill the plane?
Let $S$ be a subset of $\mathbb{R}^2$ with the following property. For all $x \in \mathbb{R}^2$ and $\varepsilon \gt 0$, there exists a nontrivial interval $[a,b] \subseteq [1-\varepsilon,1]$, such ...
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13
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421
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A meager subgroup of the real line, which cannot be covered by countably many closed subsets of measure zero?
Is there a ZFC-example of a subgroup $H$ of the real line $\mathbb R$ such $H$ is meager, has zero Lebesgue measure, but cannot be covered by countably many closed subsets of measure zero in $\mathbb ...
- 41.8k
12
votes
0
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435
views
Uniform closure of subspaces of Baire class 1
Describe a uniformly closed linear subspace $A \subset C([0,1])$ such that the space $B_1(A)$ is not uniformly complete. Here $B_1(A)$ is the set of all bounded functions $f$ which are pointwise ...
- 1,704
11
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0
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488
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Are there 100 points that are part of every half-density part of the plane?
Is there a configuration $P$ that consists of 100 points of the plane such that every $X\subset\mathbb R^2$ whose density is half contains an isometric copy of $P$?
I am deliberately being vague ...
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11
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0
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263
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Which results in probabilistic group theory generalize from finite groups to compact Hausdorff groups (and which don't)?
Let $G$ be a finite group. It has been shown that:
If the probability that two randomly selected elements of $G$ generate an abelian group is greater than $5/8$, $G$ is abelian.
If the probability ...
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11
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0
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381
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Concerning Luzin-(N)-property
Definition: a function $f:\mathbb{R}\to \mathbb{R}$ has Luzin-(N)-Property if $f$ maps any null set to a null set.
By https://www.encyclopediaofmath.org/index.php/Luzin-N-property, it is known that ...
- 4,201
11
votes
0
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215
views
Shift invariant measurable selection theorem
Let $(X,\mathcal{F})$ be some measure space and endow $\mathbb{R}^\mathbb{Z}$ with the product topology and borel $\sigma$-field. Let $F$ be a point to set mapping $X^\mathbb{Z}\rightarrow \mathcal{P}(...
- 479
11
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0
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799
views
Errata for the Treatise of Analysis of Dieudonné
I was looking again at the beautiful and quite complete work of Dieudonné, his Treatise of Analysis, to refresh my memory about some aspects of classical analysis. I especially love this Treatise for ...
- 1,128
10
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0
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272
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What is the smallest $\sigma$-algebra of reals that is closed under addition of sets?
What is the smallest $\sigma$-algebra $\Sigma\subseteq\mathcal P(\Bbb R)$ containing the open sets and such that if $A,B\in\Sigma$, then $$A+B=\{a+b\mid a\in A,b\in B\}\in\Sigma?$$
I know that neither ...
- 3,011
10
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0
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172
views
Maximizing an integral w.r.t. a measure on the unit sphere
I would like to know if the answer to the following question is known.
Let $d \ge 3$. What is the value of
$$
\theta(d) := \max_{\mu} \int_{S^{d-1}} \int_{S^{d-1}} \cdots \int_{S^{d-1}} |x_1 \...
- 980
10
votes
0
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338
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Simultaneous strong law of large number classes?
Say that $C$ is a SSLLN class of subsets of some Polish space $V$ provided that for every sequence of Borel i.i.d.r.v.s $X_1,X_2,...$ with values in $V$, we almost surely have: For every $A$ in $C$, $\...
10
votes
0
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744
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Is the set of real-valued lower semi-continuous functions measurable in epigraph topology (= topology of Gamma convergence)?
Let LSC = LSC([0,1]) be the set of non-negative, lower semi-continuous functions on the unit interval which take values in $\mathbb{R}_+ \cup \{\infty\}$. We use epigraph topology on LSC, i.e. a ...
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0
answers
3k
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Quotients of Measurable Spaces?
Let $(\Omega,\Sigma)$ be a measurable space and $\Pi$ be a partition of $\Omega$. There is a projection $\pi:\Omega\to\Pi$ that maps each $\omega\in\Omega$ to the unique partition cell in $\Pi$ ...
9
votes
0
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256
views
Is the inverse of a measurably parametrised family of bijections between standard Borel spaces measurably parametrised?
It is known that a measurable bijection $f \colon [0,1] \to [0,1]$ has a measurable inverse. (Here, all measurability is simply with respect to the Borel $\sigma$-algebra of $[0,1]$.)
Now fix an ...
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9
votes
0
answers
217
views
Point-free topology, but with $\sigma$-algebras instead of spaces
I have a question about $\sigma$-algebras in relation to point-free topology. The question was inspired by a comment on a similar question I had:
If abstract $\sigma$-algebras (i.e. certain boolean ...
user30211
9
votes
0
answers
247
views
Covering inequality for sets of intervals
Let $I$ and $J$ be finite sets of open intervals $(a,b)\subset\mathbb R$. For a finite set of points $P\subset \mathbb R$ we denote those subsets of intervals from $I$ and $J$ containing some point ...
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0
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473
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A subset of plane that meets every line at length one
Is there a subset of plane whose intersection with every line has length one?
It is easy to construct such a set under the continuum hypothesis. Also, no such set is Lebesgue measurable - See ...
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9
votes
0
answers
624
views
Two questions about universally measurable sets
I have two questions about universally measurable sets:
(1) Is there a universally measurable set of reals which does not have the Baire property?
(2) Is there a universally measurable set of reals ...
- 9,631
9
votes
0
answers
223
views
Cramer's theorem in Hilbert spaces
I am interested under what conditions Cramer's theorem applies in random variables taking values in Hilbert spaces. Following these lecture notes, but using a Hilbert space:
Let $X_1,X_2,\cdots$, be ...
- 619
9
votes
0
answers
193
views
A small planar set containing a large family of curves
A beautiful construction by Besicovitch and Rado [1] produces an astounding example of a compact connected plane set of measure zero containing circles of all radii $r\in(0,1]$.
A corollary to a ...
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8
votes
0
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422
views
Non-affine smooth transformation of Gaussian is Gaussian
Suppose $Z\sim N(0,1)$ (standard Gaussian) and $f: \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f(Z)\sim N(0,1)$. My question is whether there exists any such $f$ other than $f(x)...
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0
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1k
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Wasserstein distance and Monge-Kantorovich-Rubinstein duality
The definition of Wasserstein $p$-distance between two measures $\mu$ and $\nu$ on a Polish space $X$ is given by
$$
W_p(\mu, \nu)^p = \inf_{\gamma \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^p\; d\...
- 1,407
8
votes
0
answers
339
views
The Cauchy Transform, and the convergence of the Fourier-Stieltjes transforms of a sequence of measures
Let $C\left(\mathbb{R}/\mathbb{Z}\right)$ denote the Banach space of continuous, $1$-periodic complex-valued functions on the unit interval, let $M\left(\mathbb{R}/\mathbb{Z}\right)$ denote its dual, ...
- 1,284
8
votes
0
answers
1k
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Can every set be measurable?
The Solovay model shows that ZF (plus an inaccessible) is consistent with every subset of $\mathbb{R}$ being measurable. How far can we go in that direction? We can always have non-measurable sets ...
- 6,870
8
votes
0
answers
463
views
When is the sigma-algebra generated by closed convex sets the same as the Borel sigma-algebra
For which topological vector spaces $E$ do we have the equality between the sigma algebra generated by the closed convex subsets of $E$ and the Borel sigma algebra of $E$ ?
More precisely, do we have ...
- 506
8
votes
0
answers
544
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A Banach-Tarski game
This is partially inspired by the question https://math.stackexchange.com/questions/1383397/cutting-a-banach-tarski-cake, which I find intriguing if unclearly written.
A paradoxical family of subsets ...
- 21.2k
8
votes
0
answers
469
views
A question about finitely additive extensions of Lebesgue measure
Suppose $m:P([0, 1]) \to [0, 1]$ is a finitely additive measure extending the Lebesgue measure. Must there exist some $X \subseteq [0, 1]$ such that $m(X \cap I) = |I|/2$ for every sub interval $I \...
- 9,631
8
votes
0
answers
212
views
Qualitative weakenings of probabilistic independence
In probability theory, independence of random variables is characterised by
$$(1)~~X~\text{independent}~Y \; \iff \; P_{(X,Y)} = P_X \otimes P_Y \enspace ,$$
where $P_{(X,Y)}$ is the joint probability ...
- 866
8
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0
answers
729
views
Density of countably additive measure in the set of all finitely additive measures.
Let $S$ be a countable discrete set, the following two results are quite easy to prove:
Every countably additive probability measure $\mu$ on $S$ commutes (in Fubini's sense) with every finitely ...
- 6,034
8
votes
0
answers
751
views
The log kernel and Bochner Theorem
I was wondering if it possible to find a measure $\eta$ on $\mathbb{R}$ such that
$$
L(x):=\log\frac{1}{|x|}=\int e^{itx}\;d\eta(t)
$$
for every $x\in [0,1/2]$.
On a structural ground, this ...
- 2,135
8
votes
0
answers
1k
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G-delta of measure 0 containig the rationals.
It is well known that $\mathbb{Q}$ is not a $G_\delta$. In fact no countable dense subset of $\mathbb{R}$ is a $G_\delta$. We order the rationals in a sequence: $\mathbb{Q}=\lbrace r_k\rbrace$, and ...
- 1,981
8
votes
1
answer
207
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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 ...
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7
votes
0
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162
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Relation between the additive Haar measure on $(K,+)$ and the multiplicative Haar measure on $K^{*}$ for a global field $K$
The following question comes from my studying of Alain Connes's paper Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function. In it, on p. 11, Connes notes that if $K$ is ...
- 1,805
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0
answers
169
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Relationship between measure theory and quantification
I was advised that this question might be better suited for mathoverflow, so I am reposting it here (original post).
In a 1978 paper published by David P. Ellerman and Gian-Carlo Rota, the duo discuss ...
- 201
7
votes
0
answers
205
views
Divergence as infinitesimal volume change on a Finsler manifold
Let $M$ be a smooth manifold and $Z$ a smooth vector field on it.
It generates a family of diffeomorphisms $\phi_t:M\to M$ by demanding that $\phi_0=\operatorname{id}$ and $\partial_t\phi_t(x)=Z(\...
- 8,086
7
votes
0
answers
282
views
Composition of couplings as a pullback construction
A metric measure space $(X,d,\mu)$ consists of a metric space $(X,d)$ together with a Borel measure $\mu$. A coupling between metric measure spaces $(X_1,d_1,\mu_1)$ and $(X_2,d_2,\mu_2)$ consists of ...
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7
votes
0
answers
177
views
Does this ideal in $B(L_1)$ have a (bounded) right approximate identity?
I will take a roundabout way to defining this ideal, because (a) this route is how my collaborators and I came to it (b) this alternative definition, rather than the standard one, may suggest a ...
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7
votes
0
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420
views
A discontinuous construction
Suppose we have an uncountable family of functions $f_r: [0, 1] \to R$ indexed by $r \in [0, 1]$ such that for each $r$, there exists a unique $x$ in $[0, 1]$ such that $f_{r}$ is positive on $x$ and $...
- 2,069
7
votes
0
answers
101
views
Volume of a neighborhood of singular matrices
Suppose we take the set of all $n\times n$ real matrices with entries in $[0,1]$ in Euclidean space. Let $N_\epsilon$ be the $\epsilon$ neighborhood of the set of all singular matrices in this space, ...
- 71
7
votes
0
answers
218
views
What (if any) was known about null sets before Lebesgue?
The notion af a null set, i. e., a set of Lebesgue measure zero, does not require a full blown construction of Lebesgue measure:
A set is $E\subset \mathbb{R}$ is called a null-set if it can be ...
- 8,992
7
votes
0
answers
264
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When is Radon-Nikodym derivative induced by a proper map of manifolds bounded?
Let $X,Y$, be compact complex manifolds, and let $f:X\to Y$ be a smooth, proper (i.e. for each $y\in Y$, $f^{-1}(y)$ is a compact set) and surjective map. Choose metrics on $X,Y$ and let $\mu_X, \mu_Y$...
