Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
3,071 questions
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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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Conditional law as a random measure and convergence of random measures
I'm looking for a reference book or article for the following two facts. In both statements, a Polish space $E$ and an ambient probability space $(\Omega, {\cal A}, \Pr)$ are given, and I consider ...
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probability measures with entropy equal to nonnegative number
Is it true that for a given nonnegative number, there exists a measure-theoretical entropy value (supremum of entropies of all partitions under a measure-preserving transformation) that equals this ...
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An almost orthogonality principle for L^p
I recently asked this question on Math StackExchange and someone suggested that it would probably be more suited for Math Overflow. Since it still has not been answered, here it goes:
If two ...
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Cutting a subset in many pieces with controlled perimeter
Let me premise that I am in no way an expert in the subject of this question.
Let's say we have a measure space $X$ with measure $m$ and a reasonable notion of perimeter for (nice enough) subsets of $...
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Measurability of subspace of set of all functions
Set $X=\mathbb{R}^n$ and let $X^{I}$, the space of maps from the (bounded or unbounded) interval $I$ to $X$, be endowed with the locally convex topology of pointwise convergence.
Is it true that the ...
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When is the support of a Radon measure separable?
Let $X$ be a topological space, equipped with its Borel $\sigma$-algebra $\mathcal B(X)$, and let $\mathbb P$ be a Radon probability measure on $(X, \mathcal B(X))$. Recall that the support of the ...
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Importance sampling of finite path of stochastic difference equation
Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...
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Weak convergence of the image of a weakly $L^1$ converging sequence
This is a follow-up on another question.
Can something be said about the image of a weakly converging sequence in $L^1$? More precisely
$u_k\ge 0$
$\|u_k\|_{L^1}=\int u_k=1$
$u_k$ converges to $u$ ...
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From universal measurability to measurability
Let $(\Omega,\Sigma)$ be a measurable space and $K$ be a compact
metrizable space endowed with its Borel $\sigma$-algebra
$\mathcal{B}(K)$. Let $A\subseteq\Omega\times K$ be universally
...
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When does a $W^*$-algebra have a standard Borel spectrum?
EDIT: André Henriques has commented below that the correct separability condition is not weak-* separability as I have written below, but separability of the predual.
This post came out a bit long, ...
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A question about "nice" functions
Let $f:\mathbb R \rightarrow \mathbb R$ be a function such that $\lambda(I)=\lambda(f(I))$ for each interval $I \subseteq \mathbb R$. ($\lambda$ is Lebesgue measure here.) Let us call such functions ''...
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Existence of dominating measure for weak*-compact set of measures
I have posted the following question also here a longer time ago, but due to no answers I thought it might fit better to MO.
Let $(\Omega,\mathcal F)$ be a measurable space and $\mathcal P$ a weak*-...
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Measurable multifunction
Let $f:[a,b]\times \mathbb{R}^{n} \times \mathbb{R}^{m}
\rightarrow \mathbb{R}^{n}$. Suppose $ f (.,x, u) $ is Lebesgue
measurable for each $(x,u)$. Suppose also that $ f $ is continuous
at $ (x, u) $ ...
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Variation on Fatou's lemma for Sobolev norms
Recall that Fatou's Lemma says that for every sequence $f_n$ of non-negative measurable functions
$$\int \liminf_{n\to \infty} f_n \ d\mu\leq \liminf_{n\to \infty} \int f_n\ d\mu \ .$$
If I am not ...
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How do these two Haar measures on SL(2,R) compare?
By using the Iwasawa decomposition, one obtains a (bi-invariant) Haar measure on $G:=\mathrm{SL}(2,\mathbb{R})$ which can be symbolically written as $\mathrm{d}x=\mathrm{d}a\,\mathrm{d}n\,\mathrm{d}k$,...
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How additive is Lebesgue measure in ZF+AD ?
What is known about the additivity of Lebesgue measure under the Axiom of Determinacy?
That is, for what cardinals $\kappa$ do we have
with $|I| = \kappa$, for all functions $f : I \to 2^\mathbb{R}$,...
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On dyadic cubes
I will copy the following definitions of Exercise 1.1.14 of the book "An introduction to measure theory" by Terence Tao.
Define a dyadic cube to be a half-open box of the form
\begin{align}
\left[\...
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extensions of lebesgue measure
The Hahn-Banach theorem implies that Lebesgue measure can be extended give a "measure" on all subsets of [0,1], but this measure is only guaranteed to be finitely additive. It might magically turn ...
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Nonseparable disintegration theory: references
I mean a theorem of the following kind. Let $A$ be a C*-algebra, and let $\pi: A\to B(H)$ be its representation. Then there exist a set $P$ with a positive measure $\mu$, a field of Hilbert spaces ...
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Preimage of zero measure sets
Denote by $|A|$ the measure of $A$ (Can be Lebesgue measure) under what conditions on a function $f:\mathbb{R}^m \to \mathbb{R}$ the preimage of a null set is zero. i.e.
$|A|=0 \Rightarrow |f^{-1}(A)|...
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On the proof of Modified Vitali Lemma.
I see the following theorem in Lihe Wang's A geometric approach to the Calderon--Zygmund estimates
(Modified Vitali) Let $0<\varepsilon<1$ and let $C\subset D\subset B_1$ be two measurable ...
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Is function from topological group to metric space Borel?
Let $G$ be a pseudometrizable compact abelian topological group, $X$ a compact
metric space and $f:X\rightarrow G$ a continuous bijective function.
Suppose there exists $g\in G$ such that if $d_{G}(...
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Regular Borel Measures equivalent definition
Please help me understand how the below definition is equivalent to the standard definition of regularity which says that a measure is regular if for which every measurable set can be approximated ...
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Existence of limit measure
Let $X$ be a separable metric space, $\mu_{n}$ a sequence of Borel probability measures
and $\mathcal{C}$ be a family of sets that is closed under finite unions and
interections, and that contains all ...
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Obtaining conditional probabilities as pushforwards of [0,1]
It is standard that every Borel probability measure on a polish space $X$ can be obtained as pushforward of the uniform measure $\lambda$ on $[0,1]$ along an almost-everywhere-defined Borel-measurable ...
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Product of two sigma fields
Let $\mathcal{F}$ and $\mathcal{G}$ be any two famillies of subset of a space $X$ (neither $\mathcal{F}$, nor $\mathcal{G}$ is a sigma-field).
$$\sigma( A\times B , A\in \mathcal{F}, B\in \mathcal{G})...
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Measurable projection theorem
Hi ;
i have this theorem from the book :Set-valued analysis
Let $(\Omega,\mathcal{A},\mu)$ be a
complete $\sigma$-finite measure space
, $X$ a complete separable metric
space and $G\in\...
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Random variables invariant under almost automorphisms.
Let $\Omega$ be a standard atomless probability space, we can assume $\Omega=(0,1)$ with Lebesgue measure. A bijection $f:\Omega/A_1\to\Omega/A_2$ is almost automorphism, if $P(A_1)=P(A_2)=0$, $f(A)$ ...
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Beck-Chevalley for measures?
A measurable set is a pair $(X,\Sigma)$ where $X$ is a set and $\Sigma$ is a $\sigma$-algebra on $X$. The elements $U\in\Sigma$ will be considered as subsets $U\subseteq X$. A morphism of measurable ...
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Is every bornological space measurable?
Every topological space is measurable, since we may canonically equip a topological space with its Borel $\sigma$-algebra. A bornological space is like a topological space, except the structure ...
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Change of variables formula for Riemann integration and Lebesgue Integration
I've put this question on math.SE for a while without getting any answers. I thought it must be a rather trivial question for MO so that I didn't put it here. But I do want to get some help anyway (...
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A set of positive measure with cardinality less than that of the continuum?
Is it consistent with ZFC that there is a subset of $[0,1]$ whose cardinality is less than that of the continuum but which has positive Lebesgue measure?
Obviously not given CH. And, given ZFC, ...
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Riesz representation theorem for vector-valued fields
Let $Q$ be a locally compact Hausdorff space, and let $V$ be a topological vector space. Consider the space $X = C_0(Q, V)$ of $V$-valued fields which vanish at infinity. Let $X^*$ denote the dual ...
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Product of Baire sigma-algebras
Suppose that $X$ is a Polish space and $\mathcal{E}$ is the $\sigma $-algebra of subsets of $X$ with the property of Baire. Consider the product $\sigma $-algebra $\mathcal{E}\otimes \mathcal{E}$ on $...
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Weak convergence in measure for negligible sets.
Let $X$ be a Polish space and $(P_n)$ a sequence of Borel probabilities which converges weakly in measure to a Borel probability $P$. By this i mean that for any $f\in C_b(X)$ which is continuous and ...
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Can random elements be defined in terms of a measure algebra?
Let $(\Omega,\Sigma,\mu)$ be a probability space, $(X,\mathcal{X})$ be a measurable space and $R(\Omega,X)$ be the set of equivalence classes of measurable functions from $\Omega$ to $X$ under almost ...
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Extension of measures from the ball sigma-algebra to the borel sigma-algebra
Let $X$ be a metric space, $\Sigma_{1}$ the borel sigma algebra and
$\Sigma_{2}$ the sigma algebra generated by balls (open and closed).
If $\mu$ is a probability measure on $\Sigma_{2}$ can it be ...
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Are all models of ZF + DC + "All set of reals are lebesgue measurable" also models of CH? [duplicate]
Possible Duplicate:
Lebesgue Measurability and Weak CH
I have studied a little set theory and I found that Solovay constructed a model of ZF+DC+"All set of reals are Lebesgue measurable" and I ...
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measure theory and continuum hypothesis
let's assume $\neg CH$, then there's a set $X$ such that $|\mathbb N|<|X|<|\mathbb R|$.
i'm wondering about the lebesgue measure of such set... is it even possible to measure it? would it be ...
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A terminal coalgebra of a certain functor on Mes
Let $\mathfrak C = \mathsf{Mes}$ be the category of meausurable spaces and measurable maps. For any object $X\in \mathfrak C_0$ we assign a measurable space $\mathcal P(X)$ whose elements $\mu$ are ...
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Does every commutative monoid admit a translation-invariant measure?
Let $T$ be a commutative monoid, written additively. The set $T$ is equipped with a canonical pre-order, defined by $s \le t$ when there exists $s' \in T$ so that $s + s' = t$. Consequently, $T$ may ...
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Extension of probability measure from a finite algebra to sigma-algebra with countable many generators
I apologize for probably trivial question, I am far from this field.
If $\mathcal A$ is a $\sigma$-algebra of subsets of $X$ (for example Borel sets of Cantor space $2^\omega$), can I extend to $\...
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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 ...
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Is a measurable homomorphism on a Lie group smooth?
Let $G$ be a Lie group, and let $\mathcal B(G)$ its Borel $\sigma$-algebra. Suppose that $f : G \to G$ is a Borel-measurable homomorphism. Is $f$ smooth?
Edit: My original question said "measurable ...
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Can one view the Independent Product in Probability categorially?
One can construct a category of probability spaces, but this category has no products. Now probability theory relies strongly on the ability to build independent products, the product measure. In a ...
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Is there a good concept of a measurable fibration?
In probability theory, there are many results which are valid in purely measurable settings, usually beginning with the assumption, "let $(\Omega, \mathcal F, \mathbb P)$ be an abstract probability ...
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Proving "almost all matrices over C are diagonalizable".
This is an elementary question, but a little subtle so I hope it is suitable for MO.
Let $T$ be an $n \times n$ square matrix over $\mathbb{C}$.
The characteristic polynomial $T - \lambda I$ splits ...
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Coupling of vectors
Let $X = (X_1,X_2)$ and $\hat X = (\hat X_1,\hat X_2)$ be two random variables where $X_i,\hat X_i$ are taking values over the Polish space $E_i$ endowed with their Borel $\sigma$-algebras, where $i=1,...
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Apollonian gasket and the degree of convergence
Let $r_1,r_2\dots$ be the radii of Apollonian gasket.
I would like to know for which values $\alpha$ we have
$$\sum_{n=1}^\infty r_n^\alpha<\infty.$$
I know that if three circles $A$, $B$ and $C$ ...
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