Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
793
questions with no upvoted or accepted answers
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Cutting a piece of cake that $n$ people value as exactly $w$
Stromquist and Woodall (1985) study the problem of Sets on which several measures agree. There are $n$ non-atomic value measures on the unit circle, and a parameter $w\in(0,1)$. The goal is to find a ...
7
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455
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 \...
7
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853
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Uniformization/measurable selection theorems
Let $X,Y$ be measurable spaces and $F\subseteq X\times Y$. We say that $f:X\to Y$ is a uniformization map for $F$ if $(x,f(x))\in F$ for each $x\in \pi_X(F)$ where $\pi_X$ is the left projection map. ...
7
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1k
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Prokhorov's theorem for finite signed measures?
Prokhorov theorem provides a useful characterization of relatively compact sets w.r.t. narrow topology (topology induced by narrow convergence) in the space of probability measure.
Notation used ...
7
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372
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Why has Sacks' "Measure-theoretic uniformity" not been more influential?
In the 1969 paper "Measure-theoretic uniformity in recursion theory
and set theory," Trans. Amer. Math. Soc. 142 1969 381–420, Sacks gave
a measure-theoretic approach to several results previously ...
7
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205
views
Simultaneous Strong Law of Large Number classes?
Say that $C$ is a SSLLN class of subsets of some topological space $V$ provided that for every sequence of i.i.d.r.v.s $X_1,X_2,...$ with values in $V$, we almost surely have: For every $A$ in $C$, $\...
7
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611
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"Liftings" of L^\infty functions
This is motivated by this question: Is there an inclusion of $L_\infty(G)$ into $C_0(G)^{**}$? and Bill Johnson's comments there.
Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon ...
7
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725
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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 ...
7
votes
1
answer
225
views
Hausdorff dimension and sigma finiteness
If a function $ f : \mathbf{R} \to \mathbf{R} $ is $\mathscr{C}^{0,\alpha}$ for every $ 0 < \alpha < 1 $ then its graph has Hausdorff dimension $1$.
I would like to see an example of such a ...
6
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262
views
Is there a uniform version of Lebesgue's differentiation theorem?
Let $\mu$ be a finite measure on $\mathbb R$ and $f,g : \mathbb R \to \mathbb R_{\geq 0}$ two measurable maps such that $\int_{x\in\mathbb R} f(x)\ \mu(dx) \leq 1$ and that $g(x) \leq 1$ for all $x$. ...
6
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263
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Existence of a limit of alpha-difference quotient for Hölder functions
Let $f:\mathbb{R}\to \mathbb{R}^d,d\geq 1,$ be an Hölder function with exponent $\alpha\in (0,1)$, meaning that
\begin{equation}
\sup_{x, y \in \mathbb R, \,x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}<...
6
votes
0
answers
116
views
Weak-type inequality for the partial Fourier sum operator
I'm studying harmonic analysis by myself. One of the online notes gives the following claim as a remark:
For any $N \in \mathbb{Z}^{+}$, let's use $S_{N}$ to denote the partial ($N$ terms) Fourier sum ...
6
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answers
890
views
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
$$
W_p(\mu, \nu)^p = \inf_{\gamma \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^p\; d\...
6
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257
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Preiss' theorem on Riemannian manifolds
This may be a silly question as it seems one of the several Euclidean results in that carry over to Riemannian manifolds by passing in coordinates, but I suspect that the issue is subtler, and I am ...
6
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335
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Radon-Nikodym derivatives with parameters?
Suppose that $(A,\Sigma_A)$ and $(X,\Sigma_X)$ are measurable spaces, and that
$$
\mu,\nu \: : \: A \times \Sigma_X \rightarrow [0,1]
$$
are Markov kernels, i.e. probability measures on $X$ ...
6
votes
0
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169
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The Fubini property of the $\sigma$-ideal generated by closed subsets of measure zero
Question. Can a Borel subset $B\subset\mathbb R\times \mathbb R$ be covered by countably many closed sets of measure zero in the plane if for every $(a,b)\in\mathbb R\times\mathbb R$ the horisontal ...
6
votes
0
answers
4k
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Interchange of supremum and integral
Let $f : X \to Y$, $X \subset R^n$, $Y$ Banach space, $g : X \times Y \to R \cup \{ \infty \}$, $L^n$ the n-dimensional Lebesgue measure.
Are there some results under which the following interchange ...
6
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127
views
How big may the maximum set of entire function be?
Let us consider an entire function of several complex variables $f(z_1,\dots,z_n)$ and its modulus maximum
$$M(r,f):=\max \{ |f(z_1,\dots,z_n)|: |z_1|\le r,\dots,|z_n| \le r \} $$ with $r\ge 0$. How ...
6
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0
answers
385
views
Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance
Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...
6
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729
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What is the Banach dual of the Bochner space $L^\infty(\Omega;X)$?
Suppose $\Omega$ is a $\sigma$-finite measure space (I'm happy to take $\Omega = \mathbb{N}$) and let $X$ be a Banach space. It's pretty well known that the Banach dual of $L^\infty(\Omega)$ can be ...
6
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186
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Measure-minimizing simplex with fixed inradius
Let $\Delta^n$ be an $n$-simplex in $\mathbb{R}^n$. Let $V$ be the volume and $r$ the inradius (radius of the inscribed sphere) of $\Delta^n$. There is a well-known result that
$$
V \geq \frac{n^{n/2}(...
6
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0
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357
views
Approximating a measurable function from a second-countable, locally compact Hausdorff group to a separable Banach space
Let $ G $ be a second-countable, locally compact Hausdorff group and $ B $ a separable Banach space.
We say that a function $ f: G \to B $ is Bochner-measurable if and only if it is the everywhere ...
6
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0
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959
views
What relates to measure spaces as topological spaces relate to metric spaces ?
Has there been study of a generalization of measure spaces along the following or similar lines ?
Given a measure space $(X, \Sigma, \mu)$, define for $U\in\Sigma$ a $\mu$-ball of radius $r$ of $U$ ...
6
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297
views
Generating stationary, ergodic random fields on a homogeneous space
Consider a homogeneous space $M$, which for the sake of concreteness, let's take to be $M = \mathbb R^d$. Fix some space $A$, and consider the space of functions $X = C(M,A)$, along with its Borel $\...
6
votes
0
answers
8k
views
Dual space of continuous functions
Let $C_b(\Omega,V )=$ { $ f:\Omega\rightarrow V $ } is the Banach space of all bounded continuous functions in Banach space $V$ with a norm $\|\cdot\|$ defined as $\|f\|_\infty=\sup _{x\in\Omega}\|f(x)...
5
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182
views
When does the Fourier transform of a measure decay?
Let $\mu$ be a Borel measure on $\Bbb R^d$.
It is well known that $\mu= |f|dx$ with $f\in L^1(\Bbb R^d)$ then its Fourier transform satisfies
$$\widehat{\mu}(\xi)\to0,\qquad \xi\to\infty.$$
However if ...
5
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answers
179
views
Applications of Baire's theorem on functions of first class
I found the following theorem on page 32 of John Oxtoby's Measure and Category.
Theorem 7.3. If $f$ can be represented as the limit of an everywhere convergent sequence of continuous functions, then $...
5
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answers
150
views
Is there a natural finitely additive measure for which Vitali sets have measure zero?
Vitali sets are nonmeasurable and in particular are not null sets. But all Vitali sets are in some sense small, as described below. Let $V$ be any Vitali set and let $k \in \mathbb{N}$. For each $i \...
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218
views
Concentration inequalities for random measures
For random variables $X_1,\dots,X_n$ with common mean $\mathbb{E}[X_i]=\mu$ and common bounds $a\leq X_i\leq b$, we have the very useful Hoeffding's inequality:
$$\mathbb{P}\left(\left|\mu -\frac1n\...
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answers
121
views
Criteria for tightness of Gaussian measures on Banach spaces
In Bogachev's book "Gaussian Measures" (Example 3.8.13) sufficient conditions for the (uniform) tightness of a sequence of centered Borel Gaussian probability measures on a separable Hilbert ...
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267
views
Completeness of the space $L^p$ and the Axiom of Countable Choice
I am thinking about the proof that the usual $L^p$ spaces are complete.
So, let $(X,\mathcal{F},\mu)$ be a measure space and let
$p\in[1,+\infty)$.
Important: by a measure I mean a nonnegative $\sigma$...
5
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answers
144
views
Bochner–Minlos Theorem for locally convex spaces and their duals
Let $(X,\tau)$ be a locally convex space and $(X^{*},\tau_{s})$ be its topological dual space equipped with the strong topology. Denote by $S(X,X^{*})$ the collection of operators from $X$ to $X^{*}$ ...
5
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answers
133
views
Are open subsets of a $\sigma$-compact LCH space $\mathcal{K}$-analytic?
I'm reading Guedj and Zeriahi's Degenerate Complex Monge-Ampère Equations Chapter 4 which talks about capacities. Specifically Corollary 4.13 claims that when $X$ is a locally compact Hausdorff $\...
5
votes
0
answers
221
views
Does there exist a “fat” Thomae’s function?
Definitions and some motivation:
Thomae’s function, also known as the raindrop function has several curious properties. One of which is the following - both its discontinuity set and continuity set ...
5
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0
answers
285
views
Feynman path integral and Wilsonian renormalization
Everything below is to be viewed in the Euclidean setting with $d$ dimensions and all measures are understood to be Borel measures.
The usual problem of Quantum Field Theory is to make sense of ...
5
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130
views
Relationship between continuous vector fields and divergence measure fields in dimension $\ge 2$
Let $\Omega \subset \mathbb R^d$ with $d \geq 2$ (I am mostly interested in the case when $\Omega$ is the unit ball). A vector field in $L^p(\Omega,\mathbb R^d)$ is called a divergence measure field ...
5
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339
views
Computing the infinite dimensional Lebesgue measure of "cubes"
There is no Lebesgue measure in infinite dimensions—this slogan is familiar to every student interested in analysis. One possible, precise statement of this result may be as follows: if $X$ is an ...
5
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0
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148
views
Continuity of the Green function with respect to the measure
Let $G$ be a finitely generated group and let $\mu$ be a finite measure on $G$. Define the Green function as
$$G(\mu)=\sum_{n\geq 0}\mu^{*n}(e),$$
where $\mu^{*n}$ is the $n$th convolution power of $\...
5
votes
0
answers
183
views
Invariant measure on coset space and integrable functions
Let $G$ be a locally compact abelian group, and $H$ a closed subgroup. Let $C_c(G)$ be the space of continuous, compactly supported complex valued functions on $G$. General theory of Haar measure ...
5
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answers
236
views
Polish transversals
A subset of $X$ an indecomposable continuum $Y$ is called a composant transversal if $X$ has exactly one point from each composant of $Y$.
So a continuum has a composant transversal precisely when ...
5
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0
answers
452
views
Partitioning $\mathbb{R}^n$ into closed sets
Let $n$ be a positive integer. It is well-known that $\mathbb{R}^n$ cannot be non-trivially partitioned into open sets, since it is connected.
Let $\frak P$ be a partition of $\mathbb{R}^n$ into ...
5
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0
answers
303
views
Points of continuity of Kullback-Leibler divergence with respect to weak convergence
I know that the Kullback-Leibler
$D(\mu||\nu) := - \int_K\log\big(\frac{d \nu}{d \mu}\big) \, d\mu,$
over probability measures on a compact $K$ subset of $\mathbb{R}^d$, is only weakly lower ...
5
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answers
186
views
Heuristic and graphic representation of BV functions and their singularities
This question is about some heuristics and graphs of BV functions.
In 1-dimensional setting, two key examples of $BV$ functions $u: \mathbb R \to \mathbb R$ are
the Heaviside function, whose ...
5
votes
0
answers
617
views
Cadlag and adapted (usual conditions assumed) imply progressively measurable (related to Protter's Stochastic Calculus theorem 6)
Hi maybe someone on here can help me. I have been stuck on showing this fact for several months. I asked this question in the stack exchange and it has floated around for a while but to no avail.
...
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 ...
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 ...
5
votes
0
answers
382
views
Partition of the unit interval into uncountably many sets of full outer measure
Is it possible to construct an uncountable partition $(A_\delta)_{\delta\in[0,1]}$ of the unit interval $[0,1]$ such that $\mu (A_\delta)=1$ for each $\delta\in[0,1]$? ($\mu$ stands for the outer ...
5
votes
0
answers
293
views
Notation for calculus with measures?
One of the strengths of ordinary multivariable calculus is that you can use notation where functions are expressed pointwise (e.g. $\int_a^b x^2 \, \mathrm{d}x$ rather than merely $\int_a^b f$), and ...
5
votes
0
answers
182
views
measure of an image under an argmax function
I am trying to find any techniques to analyze the measure of an image of a set under an argmax function.
For example, let $\Omega\subset\mathbb{R}^n$ be compact and $\phi:\Omega\to\mathbb{R}$ be ...
5
votes
0
answers
255
views
Equidistribution of spheres in $\mathbb{R^2}/\mathbb{Z^2}$
Let $\mathbb{H^2}$ be the hyperbolic upper half place, and let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$ acting on $\mathbb{H^2}$. A proof of the equidistribution of spheres on $\mathbb{H^2/\Gamma}$ ...