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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Why do probabilists take random variables to be Borel (and not Lebesgue) measurable?
I've been studying a bit of probability theory lately and noticed that there seems to be a universal agreement that random variables should be defined as Borel measurable functions on the probability ...
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Why should one still teach Riemann integration?
In the introduction to chapter VIII of Dieudonné's Foundations of Modern Analysis (Volume 1 of his 13-volume Treatise on Analysis), he makes the following argument:
Finally, the reader will ...
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Is every sigma-algebra the Borel algebra of a topology?
This question arises from the excellent question posed on math.SE
by Salvo Tringali, namely, Correspondence
between Borel algebras and topology.
Since the question was not answered there after some ...
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Demystifying the Caratheodory approach to measurability
Nowadays, the usual way to extend a measure on an algebra of sets to a measure on a $\sigma$-algebra, the Caratheodory approach, is by using the outer measure $m^* $ and then taking the family of all ...
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Why is Lebesgue integration taught using positive and negative parts of functions?
Background: When I first took measure theory/integration, I was bothered by the idea that the integral of a real-valued function w.r.t. a measure was defined first for nonnegative functions and only ...
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Non-Borel sets without axiom of choice
This is a simple doubt of mine about the basics of measure theory, which should be easy for the logicians to answer. The example I know of non Borel sets would be a Hamel basis, which needs axiom of ...
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Is there a category structure one can place on measure spaces so that category-theoretic products exist?
The usual category of measure spaces consists of objects $(X, \mathcal{B}_X, \mu_X)$, where $X$ is a space, $\mathcal{B}_X$ is a $\sigma$-algebra on $X$, and $\mu_X$ is a measure on $X$, and measure ...
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Axiom of choice, Banach-Tarski and reality
The following is not a proper mathematical question but more of a metamathematical one. I hope it is nonetheless appropriate for this site.
One of the non-obvious consequences of the axiom of choice ...
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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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Why isn't integral defined as the area under the graph of function?
In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also ...
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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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Is there a measure zero set which isn't meagre?
A subset of ℝ is meagre if it is a countable union of nowhere dense subsets (a set is nowhere dense if every open interval contains an open subinterval that misses the set).
Any countable set ...
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Pullback measures
Why do all measure theory textbooks present the concept of push-forward measure, but never the concept of pull-back measure? Doesn't the latter exist?
It's true that the naive treatment of such a ...
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When is $L^2(X)$ separable?
I have never studied any measure theory, so apologise in advance, if my question is easy:
Let $X$ be a measure space. How can I decide whether $L^2(X)$ is separable?
In reality, I am interested in ...
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A historical mystery : Poincaré’s silence on Lebesgue integral and measure theory?
Lebesgue published his celebrated integral in 1901-1902. Poincaré passed away in 1912, at full mathematical power.
Of course, Lebesgue and Poincaré knew each other, they even met on several occasions ...
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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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Nonstandard analysis in probability theory
I am quite new at nonstandard analysis, and recently I became aware of its use in probability theory mainly through the following two books:
Nelson (1987). Radically Elementary Probability Theory
...
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A kaleidoscopic coloring of the plane
Problem. Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap D)=\...
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How decreasing can a bijection $f:\mathbb{N}\to\mathbb{N}$ be?
This is a follow-up to this question by
Dominic van der Zypen. For each bijection $f:\mathbb{N}\to\mathbb{N}$, let
$$\operatorname{rc}(f) := \liminf_{N\to\infty} \frac{\left|\left\{(m,n)\in\{1,\dots,N\...
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Measurability and Axiom of choice
In some situations, you need to show Lebesgue-measurability of some function on $\mathbb{R}^n$ and the verification is kind of lengthy and annoying, and even more so because measurability is "obvious" ...
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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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Product of Borel sigma algebras
If $X$ and $Y$ are separable metric spaces, then the Borel $\sigma$-algebra $B(X \times Y)$ of the product is the $\sigma$-algebra generated by $B(X)\times B(Y)$. I am embarrassed to admit that I ...
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Applications of Rademacher's Theorem
Rademacher's Theorem (that every Lipschitz function on $\mathbb{R}^{n}$ is almost everywhere differentiable) is a remarkable result on the structure of the space of Lipschitz functions, but I was ...
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Is there a natural measures on the space of measurable functions?
Given a set Ω and a σ-algebra F of subsets, is there some natural way to assign something like a "uniform" measure on the space of all measurable functions on this space? (I suppose first ...
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Is there a finite family of functions such that the max of any two functions can be dominated by a third?
Is it true that for every $t$ there is an $n$ and there exists a finite function
family, $\cal F$, whose members are from $[n] \to \mathbb N$ (taking all different
values) and for any $f_1, \ldots, ...
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Existence of a strange measure
The answer to this question must be known, but I do not know where to find it. It is related to the Ulam measures I believe.
Question. Is there a finitely additive measure defined on all subsets of ...
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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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Reference for the Gelfand duality theorem for commutative von Neumann algebras
The Gelfand duality 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 ...
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Are the Sierpiński cardinal $\acute{\mathfrak n}$ and its measure modification $\acute{\mathfrak m}$ equal to some known small uncountable cardinals?
This question was motivated by an answer to this question of Dominic van der Zypen.
It relates to the following classical theorem of Sierpiński.
Theorem (Sierpiński, 1921). For any countable partition ...
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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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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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Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?
One might say, "a random subset of $\mathbb{R}$ is not Lebesgue measurable" without really thinking about it. But if we unpack the standard definitions of all those terms (and work in ZFC), it's not ...
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What are the obstructions for a Henstock-Kurzweil integral in more than one dimension?
I have recently come across the book The Kurzweil-Henstock Integral and its Differentials by Solomon Leader, in which, if I understand correctly, the HK integration process is modified in a way that ...
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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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Do invariant measures maximize the integral?
Update: The negative answer to the following question has been provided by Matthew Daws, who won, but also rejected, the bounty of 100 euro that I set over the question.
Let $\mathcal M(\mathbb Z)$ ...
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Counterexamples to differentiation under integral sign?
I'm exploring differentiation under the integral sign (I want to be much faster and more assured in doing this common task). So one thing I'm interested in is good counterexamples, where both ...
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Is there a probability theory developed in intuitionistic logic?
Since Boole it is known that probability theory is closely related to logic.
According to the axioms of Kolmogorov, probability theory is formulated with a (normalized)
probability measure $\mbox{...
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On the probability of the truth of the continuum hypothesis
First note that there exists a natural measure $\mu$ on $P(\omega \times \omega)$, inherited from the Lebesgue measure on the reals (by identifying the reals with $P(\omega)$ and $\omega$ with $\omega ...
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Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle
Hillel Furstenberg conjectured that the only $2$-$3$-invariant probability measure on the circle without atoms is the Lebesgue measure. More precisely:
Question: (Furstenberg) Let $\mu$ be a ...
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Does constructing non-measurable sets require the axiom of choice?
The classic example of a non-measurable set is described by wikipedia. However, this particular construction is reliant on the axiom of choice; in order to choose representatives of $\mathbb{R} /\...
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Construction of nonmeasurable sets
I have a history question for which I've had trouble finding a good answer.
The common story about nonmeasurable sets is that Vitali showed that one existed using the Axiom of Choice, and Lebesgue et ...
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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 ...
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A set that can be covered by arbitrarily small intervals
Let $X$ be a subset of the real line and $S=\{s_i\}$ an infinite sequence of positive numbers. Let me say that $X$ is $S$-small if there is a collection $\{I_i\}$ of intervals such that the length of ...
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Is arbitrary union of closed balls in $\mathbb{R}^n$ Lebesgue measurable?
Is an arbitrary union of non-trivial closed balls in the Euclidean space $\mathbb{R}^n$ Lebesgue measurable? If so, is it a Borel set?
@George
I still have two questions concerning your sketch of ...
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Weak and Strong Integration of vector-valued functions
This is probably an elementary question, but outside my area of expertise, and I was unable to find any suitable reference:
Suppose $f:X\to E$ is a continuous function from a compact spaces (endowed ...
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Rademacher theorem
If $f:\mathbb{R}^n\to\mathbb{R}^m$ is of class $C^1$ and $\operatorname{rank} Df(x_o)=k$, then clearly $\operatorname{rank} Df\geq k$ in a neighborhood of $x_o$. It is not particularly difficult to ...
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How can we not know the $s$-measure of the Sierpiński triangle?
I'm preparing a presentation that would enable high-school level students to grasp that the (self-similarity) dimension of an object needs not be an integer. The first example we look at is the ...
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Does every set of reals contain a measure-zero set of the same cardinality? Does it contain a meager set of the same cardinality?
This question arises from an issue in my post on Ashutosh's excellent question on Restrictions of the null/meager ideal.
Question 1. Does every set of reals contain a measure-zero subset
of the same ...
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About the definition of Borel and Radon measures
I am trying to understand the notion of Radon measure, but I am a little bit lost with the different conventions used in the litterature.
More precisely, I have a doubt about the very definition of ...
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Dual of bounded uniformly continuous functions
Let $(X,d)$ be a metric space, and let $C_u(X)$ be the Banach space of bounded uniformly continuous functions on $X$ (with the uniform norm). How can I characterize its dual space $C_u(X)^*$?
I ...
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