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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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 ...
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Measurability of optimal values and solutions
In Theorem 14.37 of Variational Analysis by Rockafellar and Wets, it shows that for any normal integrand $f:T\times\mathbb{R}^n\to\overline{\mathbb{R}}$, the function $p:T\to\overline{\mathbb{R}}$ ...
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Nonrandomized probability kernels
I've asked this question also on mathematics stackexchange, but despite nearly two dozen views, there isn't a single comment, nevermind an answer. Any help would be appreciated.
Update: See update 1 ...
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Elements of $L^p$ and nice representatives of equivalence classes
Considering $L^p$ $( 1 \leq p < \infty)$ as a normed vector space, each element of $L^p$ is actually an Equivalent class. Take $[f] \in L^p $ as an Equivalent class, What is the Nicest possible ...
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Whether $\varphi(E)$ is a Jordan measurable set?
Definition: A set $S \subset \mathbb {R^{n}}$ is Jordan measurable if it is bounded in $\mathbb {R^{n}}$ and its boundary is a set of Lebesgue measure zero.
The following conclusion has been ...
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Is there a Borel subset of $ \mathbb{R}^{2} $, with finite vertical cross-sections, whose projection onto the first component is non-Borel?
This question is related to another one that I asked two days ago.
Question. Does there exist a Borel subset $ M $ of $ \mathbb{R}^{2} $ with
the following two properties?
The ...
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Lipschitz kernel
We consider the following probability measure on $\mathbb{R}^2$:
$\mu = Leb\vert_{[0,1]} \times \delta_0$. Furthermore the following dilation, say $d$, is defined as $(x,0) \mapsto \frac{1}{2}(\delta_{...
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Measurability of a parametrized conditional expectation
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $\mathcal{G}\subset\mathcal{F}$ a Sub-$\sigma$-Algebra. Moreover, let $X:\Omega\rightarrow\mathbb{R}$ be a random variable and $F:\...
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How do we express measurable spaces using type theory?
A measurable space $(X,\mathcal X)$ consists of a set $X$ equipped with a $\sigma$-algebra of subsets $\mathcal X$. I would like to write computer programs involving measurable spaces, but to the best ...
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What is the mathematical characterization of sufficient statistics of a given $\sigma$-dominated probability model?
Given a probability model $\mathcal{P}=\{P_{\theta},\theta \in \Theta \}$ dominated by a $\sigma$-finite measure $\lambda$ (e.g. Lebesgue measure) on a locally compact space $\cal{X}$ along with $\...
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What are invariant measures of $E_m \times R_\alpha$ on $S^1 \times S^1$? Are they necessarily product measures?
For $m \in \mathbb{N}$, let $E_m \colon S^1 \to S^1$ be multiplication map $x \mapsto mx$. Also, let $R_\alpha$ be the map $x \mapsto x+\alpha$.
Now, consider $E_m \times R_\alpha \colon S^1 \times S^...
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Lebesgue Measurability and Weak CH
Let $LM$ denote "all subsets of $\Bbb{R}$ are Lebesgue measurable", and
$WCH$ (weak continuum hypothesis) denote "every uncountable subset of $\Bbb{R}$ can be be put into 1-1 correspondence with $\...
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Converse on the rectifiability of products of rectifiable sets
Let $1\leq k\leq m$ and $1\leq l\leq n$ fixed integers, $\mathscr{H}^k$ the $k$ dimensional Hausdorff measure and $E\subset \mathbb{R}^m$. We say that :
(1) $E$ is $k$ rectifiable if there exists $C\...
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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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Translates of null sets
Does there exist a null set of reals $N$ such that every null set is covered by countably many translations of $N$?
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Can the projection onto a compact set always be taken to be measurable?
This may be a very basic question.
Let $X$ be a complete metric space and let $T$ be a compact subset of $X$. Say that a function $\pi: X \to T$ is a projection if
$$
d(x, \pi(x)) = d(x, T) \quad \...
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Regularity of the reparametrization map between curves [closed]
I am looking for a reference for the following kind of results.
Let $\Gamma$ be the space of Lipschitz curves $\text{Lip}([0,1]; \mathbb R^d)$ equipped with the sup norm.
Let $B$ be a Borel subset of ...
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Are Bochner measurablity and Borel measurability compatible?
Let $(X,\mathfrak{M})$ be a measurable space and $E$ be a Banach space and $f:(X,\mathfrak{M})\rightarrow E$ be a function.
Question
Are Borel-measurable condition on $f$ and Bochner-measurable ...
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The Levy measure of the compound Poisson distribution
The compound Poisson distribution is defined as(see Levy processes and infinitely divisible distributions page: 18):
Let $c>0$ and $\sigma$ be a measure on $\mathbb{R}$ with $\sigma(\{0\})=0$, a ...
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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 ...
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What is known about the Gaussian measure of the unit ball in a Hilbert Space?
Let $X$ be an infinite dimensional separable Hilbert Space with norm $||\cdot||$ and let $\mu$ be a Gaussian measure on $X$ such that $\mu(X) = 1$. What do we know about $\mu(B(0,1))$, where $B(0,1)$ ...
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Conditions for a monotonic integral average
I am looking for conditions that ensure that an integral average of a function from $\mathbb R^n$ to $\mathbb R$ is a monotonic function of the averaging set.
To be more specific, let me start with ...
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what is the associated Borel set of a Borel measurable function on the extended real line?
This question comes from Theorem 19.B in page 81 of Halmos' "Measure Theory", as the image below shows.
In this theorem, we are given a function $\phi$ which is a Borel measurable function on the ...
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Von Neumann Algebras and Measure Theory
We know that a abelian von Neumann algebras $\mathcal{A}$ is isomorphic to $\mathscr{L}^{\infty}(X)$ for some special measure space $(X,\Sigma,\mu)$. For that reasone one can see an arbitrary von ...
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Measure space for trees and other algebraic datatypes
Given a measure space $\mathcal M$, I am wondering what kind of measure space $\mathcal T(\mathcal M)$ one could associate to the set of binary trees with elements from $\mathcal M$ at each node.
The ...
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Avoiding equal distances
Is the following consistent?
There exists $X \subseteq [0, 1]$, such that $X$ does not have measure zero and for every $Y \subseteq X$, if $Y$ does not have measure zero, then there are $y_1 < y_2 ...
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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$ ...
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Self convolutions of singular continuous measure
Let $\mu$ be a finite measure on $\mathbb{R}$. Define the measures $(\mu_n)_{n\geq 1}$ by $\mu_{n+1}=\mu\ast \mu_n$ and $\mu_1=\mu$
Is there a singular (with respect to the Lebesgue measure) ...
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For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?
I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...
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How quickly can the derivative of an everywhere differentiable function change sign?
Let $f : [a,b] \to \Bbb R$ be everywhere differentiable with $f'(a) = 1$ and $f'(b) =-1$.
By Darboux theorem, we know that $f'([a,b])$ is an interval containing $[-1,1]$. In particular, the set $\{x \...
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Event of positive probability occurs infinitely often in stationary ergodic sequence
Setup:
Suppose $X = \{X_n\}_{n\in\mathbb{Z}}$ is a stationary ergodic proces on the real line and let $A = \prod_{n\in\mathbb{Z}}A_n$ be a Borel measurable set such that
$$
P(X \in A) = P\left(X_n\in ...
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Non-isomorphic measurable spaces
Suppose that $X$ and $Y$ are measurable spaces with the property: there are measurable bijections $f:X \to Y$ and $g:Y \to X$. Is it possible to find non-isomorphic spaces $X,Y$ with this property? ...
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Invariant measureable function is constant
I'm wondering whether the action of $\mathbb{Q}/\mathbb{Z}$ on $S^1$ by multiplication is ergodic.
Note that this question is equivalent to each one of the following two statements
Is every ...
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Can a weaker version of the Hausdorff paradox be proved without AC?
The Hausdorff paradox is an incredibly counter-intuitive consequence of the axiom of choice; it is also important for demonstrating the non-existence, under AC, of a rotation-invariant measure on the ...
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author of a paradoxical decomposition of the interval
I am looking for the original author and the date of publication of the following result.
Theorem
There exist subsets $E_i\subset [0,1)$, $i\in {\bf Z}$, pairwise disjoints and real numbers $a_i$ ...
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Ergodicity and mixing of geodesic and horocyclic flows
I am reading a theorem about the ergodicity and mixing of geodesic and horocyclic flows on unit tangent bundle of a compact hyperbolic surface. I find that there are two ways (be listed below) to ...
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Diffuse measure space as a product of $[0;1]$ and another diffuse measure space
The title speaks of itself. How far is an arbitrary finite diffuse measure space from being almost isomorphic to a product of $[0;1]$ with another diffuse measure space? What would be reasonable ...
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How bad can the second derivative of a convex function be?
One can easily construct an example of a measurable function $f:(a,b)\to \mathbb{R}$ which satisfies the following property:
$$\label{p}\tag{P}
f\notin L^1(I),\ \mbox{for each interval}\ I\subset (a,...
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Is shifting $x \mapsto \theta_x\mu$ Borel measurable wrt total variation topology?
We work on a Polish group $G$ and consider finite signed measures. Let $\theta_x\mu(B) := \mu(x^{-1}B)$. Fix a $\mu$. It is clear that $x \mapsto \theta_x\mu$ is continuous when the codomain is given ...
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Dual of the space of all bounded functions, $B(X, \mathbb{R}).$
Let $X$ be a non compact separable metric space. Denote
by $B(X, \mathbb{R})$ the set of all bounded real functions
endowed with the sup norm, this is a Banach space. Denote by
$C_b(X,\mathbb{R})\...
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Dual space of $L^2(\mathbb{R},L^1(0,1))$?
I was wondering what the dual space of $L^2(\mathbb{R},L^1(0,1))$ is? (equipped with Lebesgue measures)
Formally, one would suspect that it is just $L^2(\mathbb{R},L^{\infty}(0,1))$. But this may be a ...
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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 ...
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Uniform martingale convergence of Radon-Nikodym derivatives of a convex set of probabilities
Cross posted at MSE here. I'm hoping someone here can help complete zhoraster's answer. Any hints or references are appreciated.
Let $(\Omega, \mathcal{F})$ be a measurable space equipped with a ...
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Prokhorov's theorem in non separable metric spaces
Recently, working in some calculations I needed to use the Prokhorov's theorem
about compactness for probability measures. However, a friend warned me that
I had not the hypotesis of separability ...
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An infinite-dimensional counterexample to a theorem of Lyapunov?
The following problem is related to work on topological dynamics, but I feel like the question is interesting on its own. I think the answer to the question below is likely to be well-known and I hope ...
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What is vague convergence and what does it accomplish?
For convenience, let's say that I have a locally compact Hausdorff space $X$ and am concerned with probability measures on its Borel $\sigma$-algebra $\mathcal{B}(X)$. Natural vector spaces to ...
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Integrating a function with respect to a mixture measure
This builds off on an old question about mixture measures: Generalized notions of mixture
Suppose $\mathcal{M}$ is a family of probability measures, and $Q$ is a probability measure over $\mathcal{M}$...
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Is "conditioning to a sub-$\sigma$-algebra" a measurable operation?
Let $\mathcal{B}$ be the Borel $\sigma$-algebra of $[0,1]$, and let $\mathcal{M}$ be the set of probability measures on $([0,1],\mathcal{B})$, equipped with the evaluation $\sigma$-algebra $\ \sigma(\...
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Convergence problem
Assume that $f_n:[0,1]\to [0,1]$ is a sequence of diffeomorphism that converges to a homeomorphism $f$ so that $$\int_0^1 \left[f'_n(x)+1/f'_n(x)\right]^2 dx<M.$$ Can we state that $f'_n$ converges ...
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Convergence of Radon Nikodym derivatives
I apologise in advance if my question is too basic.
Some notation:
$(X,\cal{X})$ denotes a measurable metric space
where $X$ is a metric space and
$\cal{X}$ is the associated Borel sigma algebra.
...
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