Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
3,074 questions
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$\tau$-additive measures on a complete metric space are tight
Let $X$ be a complete metric space. Are all $\tau$-additive Borel measures on $X$ tight?
In Bogachev's "Measure Theory", vol. 2, in the proof of Theorem 8.9.4 (end of page 213) it says:
Note that ...
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measurability of the limit of functions parameterized by real numbers
Let $f_s: \mathbb R \to \mathbb R$ a be family of Borel measurable functions parameterized by $s\in \mathbb R$. Consider the limit function
$$ F(t)=\limsup_{s\to 0} f_s(t). $$ Is the function $F$ ...
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Is every probability measure a pushforward of Lebesgue measure?
If $m$ is a probability measure on a measurable space $(X, \Sigma)$, is there necessarily a measurable function $f : [0, 1] \to X$ such that $m(A) = \mu(f^{-1}(A))$ for all $A \in \Sigma$?
($\mu$ is ...
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Distribution of good diophantine approximations
Let $\langle x \rangle: \mathbb{R} \to (-1/2,1/2]$ be the periodic function with period $1$ which is $x$ for $x \in (-1/2,1/2]$. Is there some function $D(a,b)$ of real numbers $a<b$ such that, for ...
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Is there a canonical uniform probability measure on compact subsets of Banach spaces?
One can construct a finite measure on a compact metric space $(X,d)$ by the following procedure:
Fix a non-negative sequence $\{\epsilon_n\}$, $\epsilon_n \to 0$. Let $Y_{\epsilon_n}$ be the minimal ...
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Proof that the subspace of signed measures integrating d(x,e) is closed
Let $\mathcal{M}(S)$ be a space of finite signed measures on a metric space $S$ ($=\mathbb{R}^2$ in my case) equipped with the total variation norm. Let
$\mathcal{M}_1(S)=\{\mu \in \mathcal{M}(S):\...
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Ordered measurable spaces
Let $(X, \leq)$ be a partial order and $\Sigma_X$ a $\sigma$-algebra on $X$. Is the set $\{(x, y) \in X\times X \mid x \leq y\}$ measurable with respect to the product $\sigma$-algebra?
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Are there uniformly discrete paradoxical subsets in $\mathbb{R}^3$?
I think there aren't any discrete paradoxical subsets in $\mathbb{R}^2$ (any isometry that mapped a discrete subset into itself would have to either be a glide-reflection, a translation or a rotation ...
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Random sets and invariant extensions of Lebesgue measure
Given AC, is there a probability measure $\mu$ on $2^{[0,1]}$ and a translation-invariant extension $\lambda$ of Lebesgue measure on $[0,1]$ such that: for all permutations $\pi$ of $[0,1]$ and all ...
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Inverting the cumulative probability function to find roots of stochastic function
Given a function:
$$f[x]=a\, \Phi \left[-x+\sigma \sqrt{\tau}\right]-\left(b+c\, e^{-d \tau}\right)\Phi \left[-x\right]$$
where $\Phi$ is the cumulative density function of the standard normal ...
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Haar measurable sets and quotient maps
Let $G$ be a locally compact Hausdorff group with a Haar measure $\mu$, let $H$ be a closed normal subgroup of $G$, and let $q: G \to G/H$ be the quotient homomorphism. Let $\nu$ be a Haar measure ...
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Approximate a set of functions by step functions on one partition
Given a probability space $\Omega$ and a countable set $M$ of measurable functions $f\colon \Omega\to \mathbb{R}$, I am looking for conditions on $M$ such that the following holds: For all $\...
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Kolmogorov continuity theorem and Holder norm
The Kolmogorov Continuity theorem (see for example the Wikipedia page) lets us prove that a stochastic process $X_t$ (on some complete metric space $(S,d)$) is Holder continuous almost surely provided ...
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Two questions related to Dirichlet spaces and Sobolev spaces
I want to ask a question that arises from reading this paper.
Let $X$ be a locally compact space which is countable at infinity and let $\xi$ be a Radon measure on $X$. Suppose $V$ is a Hilbert ...
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Decay of positive definite function in $L^p$
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous positive-definite function with $f(0)=1$. Positive-definiteness of $f$ means
$$
\sum_{i=1}^{n}\sum_{j=1}^{n}f(x_i-x_j)y_i y_j \geq 0
$$
for all $...
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Counter-example to the completeness of the Wasserstein metric
$\newcommand{\P}{\mathcal{P}}$
Let $(E,d)$ be a complete metric space, let $\P(E)$ be the set of all probability measures on $(E,\mathcal{B}(E))$. Let $W_d$ be the $1$-Wasserstein (Kantorovich) ...
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Set Functions: Monotonicity and Sub-additivity are Independent?
While my question is not high-level, I've had lots of views and no answers on Stackexchange (https://math.stackexchange.com/q/2391423 ) so I'm hoping for better luck posting it here .....
I'm trying ...
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What does $\pi$ in the term $\pi$-system stand for?
In measure theory, what does the $\pi$ in $\pi$-system stand for? Also, what about the $\lambda$ in $\lambda$-system? I want to know why Dynkin chosen these names, and why these names make sense.
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Wiener Measure measure on functions?
I know that the Wiener measure for the Brownian motion $\{B_t\}_{t\ge 0}$ on the probability space $(\Omega, \mathscr{F},P)$ can be defined as $\mu=P\circ B^{-1}$ acting on the sigma-algebra generated ...
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Extreme Points of a set of distributions with moment and/or support constraint
Let $X$ be a random variable with the distribution $F$ (cdf).
What are the extreme points of the sets of the form:
\begin{align}
P_1&=\left\{ F: \int |x|^k dF\le c \right\},\\
P_2&=\left\{ F:...
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Possible subsets of reals that equal the set of continuity of a function
This should be an easy question, but I don't quite know how to approach it. It may be somewhat related to the concepts mentioned in the context of this past question, though it was motivated mainly by ...
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Are functions of bounded variation a.e. differentiable?
In general, it is well known that, on the real line, say on $[0,1]$, if a function $f$ is of (pointwise) bounded variation, meaning that
$$
\sum_{i=1}^n |f(x_i)-f(x_{i-1})| <+\infty
$$
for every ...
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Does every compact metric space have a canonical probability measure?
Edit: Shortly after this post it was rightly pointed out by @AntonPetrunin that the measure $\mu$ may not be unique. @R W then showed how one can construct a metric space where the limiting measure is ...
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Compactly supported functions and projections
Let $\Omega$ be an open subset of $\mathbb{R}^n$ and take a family of continuous compactly supported functions $f_n$ on $\Omega$ normalized to one (in the $L^2$ sense).
Then, these functions span a ...
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Free probability with unbounded random variables?
This is partially inspired by this question and this blog post.
When trying to express classical probability in the "free probability" setting one takes an algebra of random variables equipped with ...
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Measure of bounded fourth (and below) moment distributions?
Many results in probability theory/random matrix theory/etc require probability distributions with finite fourth moments; what is the measure of such probability distributions (in the space of ...
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What is $\mu$-approximablity in Loeb measure (conflicting statements in books)?
In Loeb measure, a set is Loeb measurable iff it is $\mu$-approximable, where $\mu$ (roughly speaking) is a finitely additive hypervalued measure over internal sets.
But I found the definitions of $\...
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Discrete support for Radon measures
I have a sequence of Radon measures (on some set $X$, compact subspace of $\mathbb{R}^d$ so nothing too fancy), say $\mu_n$, which are actually $L^1(X)$ functions. In the limit I want to prove that I ...
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Measure of the rate of convergence for filtration and conditional expectations
This question is cross-posted at MSE with a soon to expire bounty that hasn't generated much discussion.
Let $(\Omega, \mathcal{F},P)$ be a probability space and $(\mathcal{F}_n)_n$ a filtration that ...
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product of power sets
For a set $X$, let $\mathcal P(X)$ denote its power set and let $\mathcal P(X)\otimes\mathcal P(X)$ denote the product $\sigma$-algebra in $X^2$. When $|X|\leq\aleph_0$ then $\mathcal P(X)\otimes\...
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Countable products of total measures
Suppose $\kappa = \mathfrak{c} =2^{\aleph_0}$ is a real valued measurable cardinal with a witnessing measure $m:\mathcal{P}(\kappa) \to [0, 1]$ - So $m$ is a diffused (points have zero measure) $\...
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Convex support of an exponential family and its mean parameter space $\mathcal{M}$
This question comes up in studying mean parametrization of exponential families of distributions. (See Brown's 1986 book on the subject.)
Let $\nu$ be a (Borel) measure on $\mathbb R^d$. Let $p(\...
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Ultrafilter theorem and translation invariant measures
The usual Vitali construction of a non-Lebesgue measurable set generalizes to a proof that there are no (non-trivial) translation invariant measures on $\mathcal P\mathbb R$.
On the other hand, there ...
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A question on probability measure on the unit ball of Banach spaces
Let $X$ be a Banach space and let $(x^{*}_{n})_{n}$ be a sequence in $X^{*}$. Suppose that $\sum_{n}|\langle x^{*}_{n},x\rangle |\leq \|x\|$ for all $x\in X$.
Question: Is there a probability measure ...
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When is the set of points for which limit of $f_n $ exists measurable?
This is not a research level question, but I have not received any feedback on the other side, so I thought I'd try here. My apologies if the answer (counterexample) is a triviality: Suppose $(X,\...
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Let $\mathcal{M}(\Xi)$ set of all probability destributions on $\Xi$. Supremum over $\mathcal{M}(\Xi)$ is equal to sup over Dirac distributions
This doubt is born because I am reading an article in this link in pag 12 in order to use these ideas to prove another problem that raised me. My doubt is following:
Let $(\Xi,\mathcal{E})$ be a ...
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reference request : constructive measure theory
As the title said, I would like to know if constructive measure theory has been developed somewhere ?
I am more precisely interested in the (constructive) theory of completely continuous valuation on ...
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Terminology for this notion of "$\sigma$-algebra" in a topos
Let $\mathcal{E}$ be a Grothendieck topos. I want to define a sort of "$\sigma$-algebra" for it, and I'm asking about what it should be—or already is—called. I know from nlab that Cheng spaces are an ...
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Weak* convergence of measures on Boolean algebras
The Dieudonné-Grothendieck theorem asserts that given a compact Hausdorff space $K$ and a uniformly bounded family $\mathcal{K}\subset C(K)^*$ (the dual to the Banach space of continuous real-valued ...
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Measurability of a particular set generated by discrete probability measures
Suppose that $(S,\Sigma)$ is a measurable space with $S$ Polish and $\Sigma$ its Borel sigma algebra. Let $\mathcal{C}$ be the collection of discrete probability measures on $S$ having countably ...
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The (Sigma) Algebra of Convex Sets
This is a question-by-proxy for a colleague from computer science. I'm sure many people here are already aware that convex decomposition forms an important sub-field of both computational geometry and ...
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strong measurability question
Let $X$ be a separable Banach space and $\mathcal L$ the collection of bounded linear operators on $X$. The strong operator topology has the sub-basis $\{B_{x,y,\epsilon}\colon x,y\in X,\epsilon>0\}...
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Does it make sense to regard the graph of any function as being a "sort-of-null set"?
Following the nice answer to Do the Lebesgue-null sets cover "all the sets can naturally be regarded as sort-of-null sets"?, the particular situation that I am especially interested in (...
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Do the Lebesgue-null sets cover "all the sets can naturally be regarded as sort-of-null sets"?
Let $F$ be the set of bijective Borel-measurable functions $f \colon [0,1] \to [0,1]$ that preserve the Lebesgue measure.
Is it the case that for every non-Lebesgue-measurable set $A \subset [...
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Characterizing the sum $L^1 + L^\infty + L^{1,\infty} + L^{\infty, 1}$ of iterated Lebesgue spaces "by duality"
For the usual Lebesgue spaces $L^p (\mu)$ ($p \in [1,\infty]$) on a ($\sigma$-finite) measure space $(X,\mu)$, it is well-known that one has the characterization
$$
L^p (\mu) = \left\{f : X \to \Bbb{...
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Pointwise convergence implies uniform convergence?
Let $K$ be an integral kernel of a bounded operator $S:L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n) $ defined like
$$(Sf)(x)= \int_{\mathbb{R}^n}K(x,y)f(y)dy.$$
Assume that $K\in C^{\text{bounded}...
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Is there a characterization of the Hausdorff measures?
It is known that there is a unique measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$ such that the measure of the rectangle $\prod_i [a_i,b_i[$ is $\prod_i (b_i-a_i)$. This is the Lebesgue ...
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Stone-Weierstrass analogue for $L^p$
Let $A$ be a complex algebra of bounded measurable functions on the measure space $(X,\mu)$ (case of $[0,1]$ with Lebesgue measure is enough for me) closed under conjugation. Assume that $A$ separates ...
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1
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Tensor product of measure spaces
For a compact topological space $X$, denote by $\mathcal{M}(X)$ the Banach space of finite signed Borel (Radon) measures on $X$ with the total variation norm. This is canonically isometric to the dual ...
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2
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Topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$
Are there complete TVS topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$
This question is strongly linked to
is the space of all borel measures on $\...
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