Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
252 questions from the last 365 days
2
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2
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140
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An uncountable measurable subset of $\Bbb R$ containing no nonempty perfect set
$\newcommand\R{\Bbb R}$Assuming the axiom of choice, is there an uncountable Lebesgue-measurable subset $S$ of $\R$ that contains no nonempty perfect set?
Of course, such a set $S$, if it exists, ...
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3
votes
1
answer
68
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How irregular can the set of points of non-differentiability for an L1 function's primitive F get, before the FTC fails?
A Fundamental Theorem of Calculus for Lebesgue Integration, J. J. Koliha begins with the passage
Lebesgue proved a number of remarkable results on the relation between integration and differentiation....
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-3
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0
answers
65
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Exercise generalizing (related to) Hölder's inequality
I came across this exercise and feel absolutely stuck:
Let $p, q, r \in (1, \infty]$ be such that $1/p + 1/q = 1 + 1/r$. Suppose that $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ satisfies ...
- 1
12
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1
answer
393
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Is $X\times X$ homeomorphic to $X$ for a space of probability measures?
Let $\mathcal M_1(S)$ be the (compact, metrizable) space of probability Borel measures on the circle $S=\{z\in\mathbb C: |z|=1\}$ with its weak $*$ topology, so $\mu_n\to\mu$ if and only if
$$
\int_S ...
- 24.2k
6
votes
0
answers
105
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Dependence on Urysohn's Lemma in Cartan's Construction of Haar Measure
This question was posted by someone else on stackexchange three months ago, but no one has answered as of yet:
Cartan's 1940 paper, Sur la mesure de Haar, claims to provide a proof of the existence ...
- 85
3
votes
0
answers
61
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On the relative growth rates of occupancy times in ergodic theory
Let $(X, \mathcal{F}, \mu)$ be a general measure space, and let $T: X \to X$ be a measure-preserving transformation on $X$. Assume that $T$ is ergodic and satisfies the property that, for any set $A \...
- 105
0
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0
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42
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questions on stochastic kernels and pushforward operator
Let $f:X \rightarrow \Delta (Y)$ and $g:X \rightarrow \Delta (X)$ be two kernels. For any bounded measurable function $h_Y:Y \rightarrow \mathbb{R},$ define $F(h_Y):X \rightarrow \mathbb{R}$ such that ...
- 1
2
votes
0
answers
157
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About the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev
According to the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev we have the following result:
Let $(X,\tau)$ be a completely regular space and let $\Gamma$ be a
family of ...
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-3
votes
2
answers
195
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Which self homeomorphisms preserve measure on a torus, apart from affine? [closed]
Which self homeomorphisms preserve measure on a torus, apart from affines? Affine is the composition of rotation and automorphism. Measure is the Lebesgue measure.
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0
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1
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142
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Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine? [closed]
Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine? The affine is composition of rotation and continue automorphism.
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0
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0
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71
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Fourier decay implies what kind of regularity
We consider a function $f:\mathbb R^2 \to \mathbb C$ that is compactly supported and bounded. In addition, we know that
$$\lim_{\vert x\vert \to \infty} \vert x \vert^2 \vert \hat{f}(x)\vert =0,$$
...
3
votes
0
answers
93
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What axioms are needed to show that the range of a finitely additive diffuse measure on $\mathbb N$ is not closed?
The other day I learned of a small error in the book Theory of Charges: A Study of Finitely Additive Measures. Example 11.4.1 goes as follows.
Let $\mu_0$ be a finitely additive probability measure ...
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3
votes
1
answer
128
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Comparing two different principles of premeasure-to-measure extension
It is well-known that a premeasure $\mu_0$ (possibly taking infinite values) on a ring of subsets $\Omega_0$ of a set $X$ can be extended to a complete measure space $(X, \Omega_C, \mu_C)$ ($C$ for ...
- 133
0
votes
1
answer
66
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Does convergence in probability of iid samples imply convergence in measure of the sampled functions?
Let $g_i: [0, 1] \to \mathbb R$ be $L^1$ functions, equibounded in $L^1$ norm. Let $X_i$ a sequence of iid uniform random variables on $[0, 1]$. Suppose that
$$\frac{1}{n} \sum_{i = 1}^n g_i (X_i) \to ...
- 6,155
0
votes
0
answers
113
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Are measures singular with respect to all representing measures in $\mathbb{D}^n$ always concentrated on null-sets?
Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$.
We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if
$$\forall_{u \in A(\...
- 63
0
votes
0
answers
96
views
Derivative bounds for self convolution of the spherical measure in $R^d$
While reading this article on near $L^1$ estimates for the spherical lacunary maximal function, I came across the estimate
$$ |\partial^{\gamma} (\widetilde{\sigma} \ast \sigma)(x)| \lesssim |x|^{-(1 +...
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1
vote
1
answer
69
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Exhausting sequences contain a $\pi$ lift of a subset with a $(1-\delta)$ factor
Let $\pi : Y \to X$ be a measurable map between the $\sigma$-finite measure spaces $(Y, \mathcal{B}, \nu)$ and $(X, \mathcal{A}, \mu)$. Suppose there exists $c \in (0, \infty)$ such that for all $A \...
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0
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64
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A Problem using Limits of Sequences of Functions
Suppose $\{f_n\}$ is a sequence of nonnegative extended real-valued functions on $X$ and $\lim_{n\to\infty}f_n=f$. Take a simple function $0\leq\varphi\leq f$. If $X_{\infty}=\{x\in X: \varphi(x)=a>...
- 1
0
votes
1
answer
169
views
Existence of a "universal" measure-preserving transformation on the unit interval
Let $I = [0,1]$ be the unit interval equipped with the Lebesgue measure $\lambda$. Let $\mathcal{M}$ be the set of all Lebesgue measure-preserving transformations $T: I \to I$. We say a transformation ...
user545331
1
vote
2
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117
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If $f\in C([0,\infty))$, does $\delta>0$ and $g\in C^1((0,\delta))\cap C([0,\delta])$ s.t. $g\geq f$ on $[0,\delta]$ and $g(0)=f(0)$ exist?
The question is the following:
Suppose $f : [0,\infty) \rightarrow \mathbb{R}$ is a continuous function. Can I find $\delta \in (0,\infty)$ and a function $g : [0,\delta] \rightarrow \mathbb{R}$ such ...
- 309
0
votes
1
answer
78
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Intersection of sigma algebras generated by shifts
EDIT: Iosif's answer showed that my motivation for this question was mislead.
To keep this question interesting for a broader readership, let us forget about sequence spaces and tail algebras and ...
- 257
8
votes
1
answer
198
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Topological property of the space of probability measures
Suppose that $\mathbb{P}$ is the metric space of Borel probability measures on the interval $[0,1]$ equipped with the topology of $w^*$ convergence.
Consider also $\mathbb{P}_{ac}, \mathbb{P}_{s}$ the ...
4
votes
1
answer
130
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Restrict sigma algebra in measure-preserving system
Consider a measure space $(X,\mathcal{A},\mu)$ and a measure-preserving transformation $\phi \colon X\rightarrow X$, that is, $\phi$ is measurable and $\phi_*\mu = \mu$.
My intuition tells me that we ...
- 257
1
vote
0
answers
72
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Do almost all Gibbs' measures satisfy the weak-Poincare Inequality?
I am trying to interprete the discussion given in Section 3 of this paper,
https://core.ac.uk/download/pdf/82217936.pdf
Lets suppose we restrict to considering Gibbs's measures of the form $\sim e^{-...
- 617
0
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0
answers
32
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Hausdorff dimension: The dimension of boundary of a set [migrated]
I can't understand the following statement.
If (perhaps not closed) set $S$ has dimension $n$, then the boundary could have any dimension from $0$ to $n$. (Could someone give me an example?)
If S ...
- 129
1
vote
1
answer
182
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Metric currents on singular measures in $\mathbb R^d$
Unless I am misunderstanding a lot of works, it is my understanding that a finite and non negative measure $\mu=g\mathcal{H}^\alpha$, where $\mathcal{H}^\alpha$ is the $\alpha$-Haudorff measure, ...
- 391
7
votes
1
answer
211
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Existence of asymptotic sequence in ergodic measure-preserving transformations
Let $(X,\mathcal{F},\mu)$ be a measure space and let $T:X\to X$ be an ergodic measure-preserving transformation. We assume that $T$ satisfies the property that if $B \in \mathcal{F}$ and $T^{-1}B \...
- 113
2
votes
0
answers
69
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Link between Carathéodory's criterion and commutation in an orthomodular lattice?
In the theory of outer measures, Carathéodory's criterion constructs from an outer measure $\mu^*$ on $X$ a $\sigma$-algebra $\Sigma$ of subsets of $X$, on which the restriction of $\mu^*$ is a ...
- 193
4
votes
1
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256
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Approximating an $L^1$ function with Riemann sums
Note: Here all functions are genuine functions, i.e. pointwise defined measurable functions instead of defined only a.e.
Let $f: [0, 1] \to \mathbb R$ be an arbitrary $L^1$ function. Of course, $f$ is ...
- 6,155
11
votes
1
answer
500
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Uncountable families of measurable sets with pairwise positive intersections
Let $(X,\mathcal{B},\mu)$ be an arbitrary finitely additive probability measure space, let $a>0$ and let $(A_i)_{i\in I}$ be an uncountable family of subsets with measure $\geq a$.
Is there an ...
- 10.6k
1
vote
1
answer
102
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Every tight $\tau$-additive finite measure is Radon
According to the 7.2.2 Theorem of the book "Measure Theory" written by V.I. Bogachev, every tight $\tau$-additive finite measure is Radon. The proof says: "The restrictions of a $\tau$-...
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5
votes
1
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619
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Non-atomic probability measures on N
One can intuitively imagine picking a random natural number and ask to what extent the intuition can be axiomatized.
Using the axiom of choice, there is a total finitely additive (monotonic) averaging ...
- 7,545
0
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0
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38
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Are measures singular with respect to all representing measures in $\mathbb{D}^n$ always concentrated on null-sets? Will it also be a Henkin measure?
Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$.
We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if
$$\forall_{u \in A(\...
- 63
1
vote
0
answers
176
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If $f \in L^p(\Omega)$, then $f \in L^q(\Omega)$ for some $q < p + \epsilon$?
Loosely speaking, I would like to know whether membership in some Lebesgue space $L^p$ is stable under small perturbations of the exponent $p$.
Let $\Omega \subseteq \mathbb R^n$ be a bounded domain ...
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3
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0
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76
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Can we generalize the Kuratowski Extension Theorem to Souslin spaces?
The Kuratowski Extension Theorem says: Let $(X,\mathcal{A})$ be a measurable space, $Y$ be a polish space, $A\subseteq X$, and $f:A\to Y$ be a measurable map. Then there is a measurable function $F:X\...
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3
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1
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130
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Do sets of big returns contain sets of returns?
We say a subset $E$ of $\mathbb{N}$ is a set of returns if there is some measure preserving system $(X,\mathcal{B},\mu,T)$ and some $A\in\mathcal{B}$ with $\mu(A)>0$ such that $E=\{n\in\mathbb{N};\...
- 10.6k
3
votes
1
answer
227
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"Essential values" of a function at a point?
Recall that the essential range $\operatorname{ess.im} f$ of a measurable function $f \in L^\infty(\mathbb{R})$ is a compact set. Denote by $f_k$ the restriction of $f$ to the interval $[-1/k,1/k]$, ...
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3
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0
answers
129
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A Talagrand inequality for the supremum of partial sums over function classes under dependence. (Reference request)
As a consequence to the Talagrand concentration inequality, it is well known that for a measurable space $(S,\mathcal{S})$ and an i.i.d. sample $X_1,...,X_n$ of $S$-valued random variables, if $\...
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3
votes
0
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90
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Existence of symmetric total measures
Is it consistent that there is a total finitely additive measure $μ$ on $ℝ$ extending the Lebesgue measure such that for every Borel Lebesgue-measure-preserving bijection $f$ of $ℝ$, $∀α∈Ord \, ∀s∈Ord^...
- 7,545
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0
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78
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What properties does representing measure $\mu$ for $z\in\mathbb{D}^n$ has to satisfy so that $\nu=0$ is the only measure singular with respect to it?
Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$.
We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if
$$\forall_{u \in A(\...
- 63
1
vote
0
answers
51
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Questions about shear transformations
I am interested in the following shear transformation $T$, which is the linear transformation on $\mathbb{R}^n$ such that the $n$ by $n$ matrix representation is given by $T = I_n + ce_n e_1^{\perp}$ ...
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1
vote
0
answers
35
views
Estimation on function defined on diophantine approximation
Consider a convergent series $\sum_n a_n=1$ where $a_n\geq 0$. I am interested in a non-zero lower bound on the function
$$
S(t)=\sum_n a_n d(nt,\mathbb Z)^2
$$
where $d(nt,\mathbb Z)$ is the distance ...
- 1,352
5
votes
0
answers
160
views
Hartman uniform distribution of means
Background: for a discrete abelian group $G$, a character of $G$ is a homomorphism $\chi:G\to \mathbf S^1$, $\mathbf S^1$ being the circle group $\{z\in \mathbb C:|z|=1\}$ with ordinary multiplication....
- 929
5
votes
1
answer
188
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Girsanov's theorem for Gaussian measures as the Cameron-martin theorem with a random shift
Let $H \subset E$ be the Cameron-Martin space of a Gaussian measure $\mu$ on a separable Banach space $E$. The Cameron-Martin theorem states that for all $h \in E$ we have $h \in H$ if and only if $\...
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12
votes
2
answers
866
views
Sets that project to zero measure on all lines except one
It is a (difficult) exercise to show that there exists a measurable set $E \subset [0,1]^2$ (necessarily with zero 2-dimensional Lebesgue measure) such that the projection on every line passing ...
- 355
1
vote
1
answer
40
views
Envelopes of functions with respect to some convex cone $\mathcal{F}$
Let's say we are given a function $f:\mathbb R ^d\to \mathbb R$ continuous. Assume that $\mathcal F$ is a convex cone of continuous functions ($\mathbb R^d$ to $\mathbb R$) closed under maxima. I am ...
- 291
2
votes
0
answers
75
views
Have the open Questions 1 and 2 from Section 7 of the paper "Integrals with values in Banach spaces" been answered?
Some context: I had previously asked the post below on MSE, but someone suggested I ask it here and delete the original post.
In section 7 of the paper Integrals with values in Banach Spaces and ...
- 121
12
votes
2
answers
622
views
Countable set meeting uncountable family of positive measure sets
Suppose $\mu:\mathcal{P}([0, 1]) \to [0.1]$ is a probability measure and $\{A_i: i < \omega_1\}$ is a family of subsets of $[0, 1]$ such that $\mu(A_i) \geq 1/2$ for every $i < \omega_1$. Can we ...
- 129
1
vote
1
answer
103
views
Possible error in a paper about minimal sufficient statistic and minimal sufficient $\sigma$-algebra
According to Theorem 1 of [1], the stastistic $(X_1,\cdots,X_n)$ is minimal sufficient for the statistical model $X_1,\cdots,X_n\sim N(\theta,1)$ iid and $\theta\in\mathbb{R}$. This is false, as you ...
- 627
1
vote
1
answer
208
views
Function $g:\mathbb{R}\to \mathbb{R}^n$ such that $g(\sum_{i=1}^nx_i)=(x_1,\dotsc,x_n)$ a.e
Is there a measurable function $g:\mathbb{R}\to \mathbb{R}^n$ such that $g(\sum_{i=1}^nx_i)=(x_1,\dotsc,x_n)$ a.e.?
Due to the papers [1], [2], and [3] I'm obtaining a result that I think it's false. ...
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