Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
2,911
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On vanishing signed measure
It is quite easy to show that if $\mu$ is positive finite Borel measure on, say $[0,1]$, and for all $n \in \mathbb{N}$
$$\int_{[0,1]} e^{-nx}\mu(dx)=0$$
holds true, then $\mu=0$. Does this still ...
- 53
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Universally measurable map coincides a.e. with a Borel map
Let $X$ be a standard Borel space: that is, a topological space equivalent to a Borel subset of $\Bbb R$. It is known that for any probability measure $p$ on $X$ and any universally measurable set $A\...
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Is the Hausdorff metric on sub-$\sigma$-fields separable?
Let $(X,\mu,\mathcal{F})$ be a probability space. The paper Equiconvergence of Martingales by Edward Boylan introduced a pseudometric on sub-$\sigma$-fields (sub-$\sigma$-algebras) of $\mathcal{F}$ ...
- 6,257
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Volume of a shape whose boundary consists of portions of spheres symmetrically placed about the origin in $d\gg 1$ dimensions
We are given a convex shape $S$ in the $d$-dimensional Euclidean space, whose boundary is formed by portions of $2d$ different spheres, one portion per sphere. The radius of each sphere is the same, $...
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Infimum of Fourier transform of singular measure
Let $\mu$ be a finite non negative singular measure on $\mathbf{R}^d$. I would like to know if there exists some result on the infimum of the absolute value of its Fourier Transform $$\hat{\mu}(t)=\...
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355
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How to maximize the number of people that mingle with each other?
Some motivation:
A company has a certain number of workers. On each given day from Monday to Friday, a fixed number of people are to work. The company wants to foster teamwork, bonding and camaraderie ...
- 4,832
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318
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Spherical average of $\frac{1}{x}$
Let $X_1,...,X_n$ be points on $\mathbb S^1.$
We then define the expectation value $E(X)=\frac{1}{n}\sum_{i=1}^n X_i.$
Let $\frac{dS(X_1)}{2\pi}$ be the normalized surface measure of $\mathbb S^1,$ i....
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When is the Radon-Nikodym derivative locally essentially bounded
Let $\mu\lll\nu$ be $\sigma$-finite Borel measures, which are not finite, on a topological space $X$. Under what conditions is $0<\operatorname{ess-supp}(\frac{d\mu}{d\nu}I_K)<\infty$ for every ...
- 4,969
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Hausdorff dimension of the graph of a BV function
Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function.
Is the Hausdorff dimension of the graph of $u$ equal to $N$? How can we prove it?
Update.
In an answer to this post, it ...
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Conditions for existence of dominating $\sigma$-finite measure for all conditional distributions
Suppose $X$ and $Y$ are two real-valued random variables with a specified joint probability distribution $P_{X,Y}.$ I wish to determine if there is a $\sigma$-finite measure $\mu$ on the real line ...
- 1,269
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Question abouth Prokhorov metric
Let $X$ and $Y$ be two random variables with first order moments, i.e. $E[|X|]$, $E[|Y|]<+\infty$. Assume further that
$$E\left[|X-Y|\right]<\varepsilon.$$
Set $Law(X)=\mu$ and $Law(Y)=\nu$, ...
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A variant of Freiling's Axiom of Symmetry and a weak form of the Continuum Hypothesis in models where all sets of reals are Lebesgue measurable
Consider the following variant of Freiling's Axiom of Symmetry, $\mathsf{AS}$, which will be denoted $A_{< 2^{\aleph_0}}$:
given any function $f$ from $\mathbb{R}$ into the families of of subsets ...
- 6,024
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On a.e. approximate differentiability of certain continuous real functions
I have the following question:
If $f:[0,1]\to \mathbb{R}$ is a bounded continuous function of $\sigma$-finite variation in sense 1, then is it true that $f$ is approximately differentiable a.e. on $[...
- 1,861
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Coupling of non-probability/sub-probability measures
A coupling of two probability measures $P,\tilde P$ on a Borel space $X$ is any probability measure on $X^2$ whose one-dimensional marginals are $P$ and $\tilde P$. In particular, for any such ...
- 1,605
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Stationary, ergodic measures from the structuralist point of view
Stationary, ergodic measures are a class of objects very familiar to probabilists. In a sense, these are the weakest generalization of the classic case of independent, identically distributed random ...
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Intuitive meaning of Giry monad's $\sigma$-algebra
The Giry monad $G : \textbf{Meas} \to \textbf{Meas}$ maps a measurable space $(X, \mathcal{F})$ to its set of probability measures. The $\sigma$-algebra of $G(X, \mathcal{F})$ is the smallest algebra ...
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Unifying two definitions of $L^\infty$
Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon measure on $X$.
Definitions:
A subset $E\subseteq X$ is called locally Borel if $F \cap E$ is Borel for every Borel set $F\subseteq X$ ...
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A measurable set that acts as a speedometer
Definitions and some motivation:
Say a car is driving on a straight road. All we know is where it starts, and how much time it spends in certain stretches of the road. With just this much information, ...
- 4,832
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Do 1-additive maps admit tensor products?
Let $\mathcal{F}$ be a set algebra (or a Boolean algebra). Following Kalton, let me call a function $f\colon \mathcal{F}\to \mathbb R$ $\delta$-additive ($\delta \geqslant 0$), whenever $f(\varnothing)...
- 11.3k
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What is this disintegration-like theorem?
This is cross-posted at MSE.
I'm looking for a reference for the following result. It seems like it must be known, or follow quickly from something known, but I have not been able to find it in any ...
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Alberti rank one theorem and a blow-up argument
In this paper, it is written that Alberti’s rank
says that the singular part $D^s u$ with respect to $\mathcal L^d$ of the distributional derivative $Du$ of a function $u \in BV_{loc}(\mathbb R^d; \...
user123457
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Base zero-dimensional spaces
Definition. A zero-dimensional topological space $X$ is called base zero-dimensional if for any base $\mathcal B$ of the topology that consists of closed-and-open sets in $X$, any open cover $\mathcal ...
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Conditions under which a linear functional on a space of measures must be integration of a function
Let $X$ be a measurable space, and let $M(X)$ be the vector space of finite signed measures on $X$. Are there natural conditions on a linear functional $f:M(X)\rightarrow\mathbb{R}$ that are ...
- 2,090
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700
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Hausdorff approximating measures and Borel sets
Suppose $ 1 \leq m \leq n $ are integers and for each $ 0 < \delta < \infty $ let $\mathscr{H}^{m}_{\delta} $ be the size $ \delta $ approximating measure of the $ m $ dimensional Hausdorff ...
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Approximation of monotone Sobolev functions
Let $f\in W_{loc}^{1,2}(\mathbb R^2)$ be a continuous monotone (real valued) function (monotone in the sense that the maximum and minimum of $f$ in a precompact open set are attained at the boundary)....
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Error estimate in the spectral theorem of compact operators on a Hilbert space
Given a compact self-adjoint operator $K$ mapping $L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ as $f \rightarrow \int K(x,y) f(y) d\mu(y)$, let us define its eigenvalues $\lambda_i$ and eigen-...
- 2,146
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Operator-valued measurable functions
Let $H$ be a non separable Hilbert space and $\Omega$ be a measurable space.
Naturally, we say that $f:\Omega\to B(H)$ is $w$-measurable if $f^{-1}(O)$ is measurable for any open set $O$ in the weak ...
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Lower semi-continuity of the Hellinger-Fisher-Rao distance
I am currently working on unbalanced optimal transport, where the Hellinger (or sometimes Fisher-Rao) distance
$$
H^2(\rho,\mu)=\int_{\Omega}\left|\sqrt{\frac{d\rho}{d\lambda}}-\sqrt{\frac{d\mu}{d\...
- 5,163
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Density of Gaussian measures on Banach spaces
I am trying to get my head around this question and was reading (1) which states the same a little bit more general:
Let $X$ be a separable Banach space and $X^*$ the dual space. The mean
value $...
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If $f : [-a,a] \rightarrow \mathbb{IR}$ is Scott continuous, why are $f^-$ and $f^+$ measurable?
In "A domain theoretic account of Picard's theorem" (http://www.doc.ic.ac.uk/~dirk/Publications/icalp2004.pdf), the authors assert the following.
Let $\mathbb{IR}$ be the interval domain $\lbrace [a^-...
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Existence of doubling non-Polish metric measure spaces
Let $(X,d,\mu)$ be a metric measure space (i.e. $(X,d)$ is a metric space and $\mu$ is a Borel measure on $X$). Let's say that $X$ is doubling if there exists a constant $C \geq 1$ such that $0 < \...
user14166
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If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?
Let $ (X,\Sigma,\mu) $ be a $ \sigma $-finite measure space and $ B $ a Banach space. A function $ f: X \to B $ is said to be strongly $ \mu $-measurable iff it is the almost-everywhere pointwise ...
- 1,406
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dual space of the subspace of the space of probability measures [closed]
I have a question which maybe so naive but I want to know the result about it.
Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Then by some materiau such as ...
- 1,837
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Lifting a semicommutative von Neumann algebra into an algebra of operator-valued functions
Given a complete probability space $\Omega$ and a von Neumann algebra $\mathcal M$, it is well-known that the tensor product von Neumann algebra $\mathrm L^\infty(\Omega)\overline\otimes\mathcal M$, ...
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Is the topology of weak+Hausdorff convergence Polish?
Let $X$ be a compact metric space, $P_X$ the set of Borel probability measures on $X$, and $K_X$ the set of non-empty closed subsets of $X$. I will define the "topology of weak+Hausdorff ...
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Dense subcategory of measurable spaces
Recall the notion of a dense subcategory $\mathcal{D}$ of a category $\mathcal{C}$. It means that the restricted Yoneda functor $\mathcal{C} \to \mathrm{Hom}(\mathcal{D}^{op},\mathbf{Set})$, $A \...
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How much time does a function spend above or below its average value around a point?
Given a locally integrable function $f: \mathbb R \to \mathbb R$, define $
K: \mathbb R \times \mathbb R+ \to \mathbb R$ by
$$
K(x, r) :=
\begin{cases}
1, & \text{if }f(x) > \dfrac{1}{2r}\...
- 2,039
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Abstract transverse measure theory
After reading Noncommutative Geometry book (see here) I came across the notion of the so called abstract transverse measure theory which is a generalization of standard measure theory well adapted to ...
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sequence of graphs converge in the sense of varifold to multiplicity 2 plane
Say in $R^3$, is there a sequence of smooth graphs $f_n$ over some plane P, such that the graphs as submanifolds in $R^3$ converge in the sense of varifold (as Radon measures on $R^3 \times Gr(2,3)$ ) ...
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483
views
Is the range of a (nonnegative or signed) measure a closed set?
Halmos showed that the range of a non-negative, finite measure is a closed subset of real numbers.
Is this true for non-negative, even infinite measures?
Is this true for signed measures? If so, can ...
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Cameron Martin space
I have seen two definitions of Cameron Martin space of a Gaussian measure $\nu$ on a Banach space (say $W$) and am unable to establish their equivalence. Any help would be appreciated.
1) It is the ...
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Is there a survey of recent work relating to the Hausdorff dimension of sets defined through some restriction of digits?
I am familiar with the work of Helmut Cajar, but his book is thirty years old and it's clear that there has been substantial progress since then. I have been spending a lot of time looking through ...
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Difference between Beta Process and Dirichlet process
I'm trying to understand the definition of a Beta process, as given in the paper:
www.ece.duke.edu/~lcarin/Paisley_BP-FA_ICML.pdf
The problem is that from the definition it follows that every ...
- 233
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346
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Integrating on orbits of algebraic groups
Suppose $G$ is a $\mathbb{Q}$-algebraic group (I am interested in the semisimple case) acting rationally on a vector space $V_\mathbb{Q}$. Let $x \in V_\mathbb{Q}$ be a non-zero rational vector. ...
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About weak integrals: Appendix of Folland's book "A course in abstract harmonic analysis"
Consider the following fragment from Folland's book "A course in abstract harmonic analysis":
All integrals are here to interpreted in the weak sense (see p285 in Folland's book). Why is ...
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What is the 'right' definition of zero measure subsets of Banach spaces?
Question. There are several ways of defining a notion of a 'zero measure' subset of a Banach space $X$. Which one is the 'right' or failing that, the preferred notion? [See below for a more precise ...
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349
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continuity entropy with respect gibbs measures
Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider Bernoulli measures on $X$ only.
Let $f:X\to \mathbb{R}$ be Holder continuous. The measure $\mu$ is a Gibbs measure with potential $f$ if there ...
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418
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Improving equi-integrability for a family $\mathcal F$ in $L^1(\Omega)$
Let $\mathcal F$ be a weakly compact subset of $L^1(\Omega)$. Dunford–Pettis theorem says that $\mathcal F$ is uniformly integrable. Also, by de la Vallée-Poussin theorem we can find an increasing ...
- 1,245
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203
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Measurable $\epsilon$-optimal selection with an analytically measurable stochastic kernel
Let $(X, \mathcal{X})$ and $(A, \mathcal{A})$ be standard Borel spaces, $D \subseteq X \times A$ be an analytic set, and $D_x := \{a \in A : (x, a) \in D\}$ denote the $x$-section of $D$ at $x \in X$.
...
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765
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Prohorov's theorem for random elements of Hilbert space: weak convergence
Let $(\Omega,\mathcal{F},P)$ be a probability space and let $(E,\mathcal{E})$ be a separable Hilbert space ($E$) with Borel $\sigma$-algebra $\mathcal{E}$. For concreteness let us set $E=L^{2}[a,b]$ ...
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