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Questions tagged [measure-theory]

Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.

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2answers
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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 ...
33
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4answers
3k views

Reference for the Gelfand-Neumark theorem for commutative von Neumann algebras

The Gelfand-Neumark theorem for commutative von Neumann algebras states that the following three categories are equivalent: (1) The opposite category of the category of commutative von Neumann ...
3
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2answers
654 views

Borel vs measure for all Borel measures

Let X be locally compact and Hausdorff, and let $f:X\rightarrow\mathbb R$ be a function. Suppose that for all finite regular (positive) Borel measures $\mu$, we know that $f$ is $\mu$-measurable. ...
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2answers
571 views

Percolation Model and Complex Probabilities

Let $d>0$ be an integer and consider the first neighbors independent bond percolation model in $\mathbb Z^d$, where each edge is open with probability $p\in[0,1]$. I would like to know, if can we ...
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2answers
223 views

disjointlize an arbitrary sequence in a ring?

In a ring R (nonempty class of sets closed under difference and finite union), any sequence (here means a function on natural numbers $\mathbb N$) {$E_i$} in R can be disjointlized to a disjoint ...
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3answers
2k views

terminology about ring/algebra in abstract algebra and measure theory

Both in abstract algebra and measure theory is there term ring/algebra, but their definition are different and we can not deduce one from the other, the only requirement in definition they share is ...
9
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2answers
690 views

Partition of R into midpoint convex sets

We say that a subset $X$ of $\mathbb{R}$ is midpoint convex if for any two points $a,b\in X$ the midpoint $\frac{a+b}{2}$ also lies in $X$. My question is: is it possible to partition $\mathbb{R}$ ...
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7answers
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Regular borel measures on metric spaces

When teaching Measure Theory last year, I convinced myself that a finite measure defined on the Borel subsets of a (compact; separable complete?) metric space was automatically regular. I used the ...
9
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2answers
4k views

approximate a probability distribution by moment matching

Suppose we want to approximate a real-valued random variable $X$ by a discrete random variable $Z$ with finitely many atoms. Suppose all moments of $X$ is finite. We want to match the moments of $X$ ...
2
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1answer
1k views

If a Borelian set has positive measure, does it contain a non empty open set (minus a measure null set)?

Let A be a borelian set with postivie measure. I was asking myself if it is possible to find an open set $B\subseteq A$ such that $B$ is an open set minus a set of null measure...
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1answer
290 views

If $f_k \not\to 0$ a.e., does there exist a subsequence, a set of positive measure, and $c > 0$, on which $\liminf |f_{k_j}| > c$?

Here you are another question in basic measure theory... Let $f_k$ be a measurable sequence of functions on $(X,M,\mu)$ measure space. Suppose that $f_k$ does not go to 0 a.e.. Can I then find a set $...
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1answer
813 views

A question about sigma-ring

This question comes from the 4th line of the proof of Theorem E of Halmos' "Measure Theory", in page 25, which says that C is a sigma-ring. Because this website does not allow new users to link images,...
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1answer
1k views

Lebesgue measure of a set

Let m the n-dimensional Lebesgue measure on $R^n$.By definition of product measure, on each borelian set E $m(E)=\inf \left(\sum_{j=1}^\infty m(R_j),\:\: E\subseteq \bigcup R_j , \:\:R_j \text{ ...
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3answers
4k views

Is the sum of 2 Lebesgue measurable sets measurable?

Is the sum of two measurable set measurable? I think it is not...
30
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1answer
3k views

Is every smooth function Lebesgue-Lebesgue measurable?

This is motivated by pure curiosity (triggered by this question). A map $f:\mathbb R^n\to\mathbb R^m$ is said to be Lebesgue-Lebesgue measurable if the pre-image of any Lebesgue-measurable subset of $\...
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5answers
2k views

Existence of probability measure defined on all subsets

Let S be an uncountable set. Does there exist a probability measure which is defined on all subsets of S, with P({x}) = 0 for any element x of S ? If I remove the condition P({x}) = 0, then I can ...
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2answers
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How to show that x-y is Lebesgue-Lebesgue measurable

Which is the cleanest way to show that the difference, $d:R^n\times R^n\rightarrow R^n$, $d(x,y)= x-y$, is Lebesgue-Lebesgue measurable? (i.e. foreach A lebesgue measurable in $R^n$, $d^{-1}(A)$ is ...
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2answers
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Defined Almost Everywhere [closed]

How can one prove that the convolution of $f \in L^1$ and $g \in L^p$ is defined almost everywhere? Here $f$ and $g$ are measurable functions in $R^n$. In general what techniques are there for ...
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2answers
6k views

lebesgue measure and countable sets [closed]

the lebesgue integral $ \int_{[0,1]} 1_{\mathbb{Q}} dm = 0 $ . and if we integrate the complement $ \int_{[0,1]} 1_{\mathbb{Q}^C} dm =1 $ which is the same as $\int_{[0,1]} dm $ to me this is still ...
8
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1answer
361 views

Estimating flat norm distance from a planar disc

Let $D\subset\mathbb R^2\subset\mathbb R^n$ be a unit planar disc in $\mathbb R^n$. Let $S$ be an orientable two-dimensional surface in $\mathbb R^n$ such that $\partial S=\partial D$. Of course, we ...
15
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4answers
8k views

Difference between measures and distributions

On the one hand, Wikipedia suggests that every distribution defines a Radon measure: http://en.wikipedia.org/wiki/Distribution_(mathematics)#Functions_as_distributions On the other hand, Terry Tao ...
2
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1answer
243 views

Help determining the asymptotic behavior of an integral involving rational functions.

Let $\phi:\mathbb{P}^1\to\mathbb{P}^1$ be a rational function of degree $d\geq2$. How can one prove, using the normalized spherical measure, that $$\int_{\mathbb{P}^1(\mathbb{C})}|(\phi^n)'(z)|\ d\mu (...
10
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2answers
642 views

measure theory for regular cardinals

Measure theory is somewhat focused on the cardinal $\aleph_0$: First of all we have the usual $\sigma$-additivity, Polish (separable!) spaces such as $\mathbb{R}^n$, countable sequences and limits, ...
3
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2answers
418 views

Dimension of the measurable space $\mathbb{R}^n$

Consider $\mathbb{R}^n$ as measurable space with the Borel algebra. If $\mathbb{R}^n$ and $\mathbb{R}^m$ are isomorphic (in the category of measurable spaces, i.e. there are measurable maps in both ...
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3answers
1k views

Disintegrations are measurable measures - when are they continuous?

This is a sequel to another question I have asked. The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; ...
8
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4answers
2k views

When does a probability measure take all values in the unit interval?

Let $\mathbb{P}$ be a probability measure on some probability space $(\Omega,\mathcal{A})$. Are there conditions on the $\sigma$-algebra $\mathcal{A}$ such that for every real number $c\in [0,1]$ we ...
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3answers
1k views

When can a function be recovered from a distribution?

What properties does a distribution (in the generalized function sense) has to have in order to be a function. That is, when is $T(\varphi) = \int f \varphi$ for some $f$?
4
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1answer
536 views

What is the tensor product of $L^p(\bf R)$ with $L^q(\bf R)$?

I'm wondering: What is the tensor product of $L^p({\bf R})$ with $L^q({\bf R})$? (For p=q=2, the answer clearly should be $L^2({\bf R}^2)$; for other values of $p$ and $q$, it is not at all obvious ...
16
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8answers
4k views

Haar measure on a quotient, References for.

I remember reading Weil's "Basic Number Theory" and giving up after a while. Now I find myself thinking of it(thanks to some comments by Ben Linowitz). Right from the very beginning, Weil uses the ...
5
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0answers
470 views

Conditional probabilities in Banach spaces

This is the infinite-dimensional sequel to my question, Conditional probabilities are measurable functions - when are they continuous?. Let $\Omega = \Omega_1 \times \Omega_2$ be a probability space ...
3
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1answer
538 views

Non-existence of integral with respect to Poisson Random Measure

Let $\xi$ be a Poisson Random Measure of intensity $\mu$ (informally $\mathbb E\xi = \mu$). (For $f \ge 0$, say) when does $\xi f = \infty?$ Kallenberg (Foundations of Modern Probabilility) claims ...
15
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5answers
2k views

Conditional probabilities are measurable functions - when are they continuous?

Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting. Take $\mathcal F$ to be the Borel $\...
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2answers
933 views

Example for an integral, rectifiable varifold with unbounded first variation

I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation. Recapitulation for every $m$-rectifiable varifold $\mu$ exists a $m$-rectifiable ...
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2answers
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Non Lebesgue measurable subsets with “large” outer measure

It is well known that for any set A in R^d there exists a measurable set E such that E contains A and m*(A)=m*(E). Is it possible to go the other direction? In other words, is it true that for any ...
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5answers
10k views

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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5answers
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Why are abelian groups amenable?

A (discrete) group is amenable if it admits a finitely additive probability measure (on all its subsets), invariant under left translation. It is a basic fact that every abelian group is amenable. ...
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2answers
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Measure between the counting measure and the Lebegue measure

There are subsets of the real line that has infinite counting measure, but Lebegue measure 0, so the Lebegue measure is used for measuring larger sets than the counting measure. My question is: Is ...
38
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18answers
60k views

Suggestions for a good Measure Theory book

I have taken analysis and have looked at different measures, but I am currently looking at realizing a certain problem in a different light and feel that I need a better background in various measures ...
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7answers
8k views

What's the use of a complete measure?

A complete measure space is one in which any subset of a measure-zero set is measurable. For what reasons would I want a complete measure space? The only reason I can think of is in the context of ...
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1answer
865 views

Why is this generality in Vitali's Lemma useful?

In Vitali's Lemma it uses outer measure rather than measure. What are some results that depend on it this theorem applying to sets with only outer measure rather than measurable sets? Vitali's Lemma: ...
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14answers
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What are interesting families of subsets of a given set?

Motivation The usual starting point of both Topology and Measure Theory is the definition of a family of subsets of a set $S$. Indeed, one defines a topology on $S$ to be a family of subsets ...
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1answer
540 views

What is the “continuity” in “absolute continuity”, in general?

The wikipedia article on absolute continuity gives a delta-epsilon definition for a measure $\mu$ defined on the Borel $\sigma$-algebra on the real line, with respect to the Lebesgue measure $\lambda$:...
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2answers
5k views

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 ...
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2answers
933 views

Definable collections of non measurable sets of reals

Is there a definable (in Zermelo Fraenkel set theory with choice) collection of non measurable sets of reals of size continuum? More verbosely: Is there a class A = {x: \phi(x)} such that ZFC proves "...
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1answer
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Why is 3 a bad constant in the Vitali covering lemma?

Hi, recently I had to do with the Hardy-Littlewood maximal function and we used there the Vitali covering lemma with constant 5. Then, given an advice, I proved it with constant k>3. But I cannot ...
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5answers
8k views

Do sets with positive Lebesgue measure have same cardinality as R?

I have been thinking about which kind of wild non-measurable functions you can define. This led me to the question: Is it possible to prove in ZFC, that if a (Edit: measurabel) set $A\subset \mathbb{...
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1answer
1k views

measurable sets not depending on even coordinates

Let $A\subset\{0,1\}^\omega$ be a measurable set (w.r.t. the usual borel sigma algebra) which does not depend on any even coordinate (that is, if $x\in A$ and $x$ and $y$ agree except on a finite ...
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2answers
587 views

What's the standard name for sets of a given size with maximal probability (or a given probability and minimal size)?

The definition I'm going to give isn't quite the concept I really want, but it's a good approximation. I don't want to make the definition too technical and specific because if there's a standard name ...
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4answers
2k views

measure spaces as presheaves?

I recently had the idea that maybe measure spaces could be viewed as sheaves, since they attach things, specifically real numbers, to sets... But at least as far as I can tell, it doesn't quite work -...
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1answer
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Lebesgue measure of boundary of Caccioppoli set

Can anything be said about the measure of the topological boundary of a Cacciopoli set in $R^n$? Of course, the reduced boundary has finite (n-1)-dimensional Hausdorff measure, but this does not say ...