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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174 votes
7 answers
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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 ...
Mark's user avatar
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108 votes
27 answers
40k views

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 ...
90 votes
3 answers
13k views

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 ...
Joel David Hamkins's user avatar
87 votes
8 answers
15k views

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 ...
KConrad's user avatar
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85 votes
9 answers
16k views

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 ...
Michael Greinecker's user avatar
72 votes
4 answers
22k views

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 ...
Anweshi's user avatar
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67 votes
2 answers
14k views

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 ...
Damek Davis's user avatar
65 votes
9 answers
13k views

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 ...
ThiKu's user avatar
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63 votes
6 answers
12k views

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 ...
user57888's user avatar
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63 votes
19 answers
95k 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 ...
61 votes
7 answers
16k views

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 ...
Anton Geraschenko's user avatar
61 votes
5 answers
7k views

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. ...
Tom Leinster's user avatar
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51 votes
4 answers
6k views

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 ...
Fabrice Pautot's user avatar
50 votes
4 answers
22k views

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 ...
Bugs Bunny's user avatar
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48 votes
7 answers
12k 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 ...
Tom E's user avatar
  • 481
47 votes
3 answers
12k views

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 ...
Alex M.'s user avatar
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43 votes
5 answers
6k views

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 ...
an12's user avatar
  • 1,272
42 votes
0 answers
791 views

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)=\...
Taras Banakh's user avatar
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40 votes
5 answers
11k 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{...
Sune Jakobsen's user avatar
40 votes
5 answers
10k views

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 ...
Kenny Easwaran's user avatar
40 votes
2 answers
3k views

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" ...
Matthias Ludewig's user avatar
39 votes
4 answers
15k views

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 ...
Bill Johnson's user avatar
38 votes
4 answers
4k views

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 ...
Piotr Hajlasz's user avatar
38 votes
2 answers
3k views

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, ...
domotorp's user avatar
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37 votes
6 answers
5k views

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 ...
Gordon Craig's user avatar
  • 1,625
37 votes
5 answers
4k views

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 ...
Dmitri Pavlov's user avatar
36 votes
14 answers
5k views

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 ...
José Figueroa-O'Farrill's user avatar
33 votes
1 answer
2k views

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 \...
Siméon's user avatar
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33 votes
2 answers
2k views

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 ...
Taras Banakh's user avatar
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32 votes
3 answers
3k views

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 ...
Vladimir Sotirov's user avatar
32 votes
2 answers
1k views

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 $N$?
Null's user avatar
  • 321
32 votes
1 answer
4k views

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 $\mathcal M(\mathbb Z)$ ...
Valerio Capraro's user avatar
32 votes
1 answer
4k 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 $\...
Sergei Ivanov's user avatar
31 votes
3 answers
7k views

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 ...
bort's user avatar
  • 313
31 votes
4 answers
4k views

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 ...
Gene S. Kopp's user avatar
  • 2,190
29 votes
2 answers
2k views

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 ...
Mohammad Golshani's user avatar
29 votes
1 answer
4k views

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 ...
Andreas Thom's user avatar
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28 votes
7 answers
13k views

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 ...
Matthew Daws's user avatar
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27 votes
4 answers
11k views

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} /\...
Mark Bell's user avatar
  • 3,125
27 votes
4 answers
2k views

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 ...
Joel David Hamkins's user avatar
27 votes
3 answers
6k views

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 ...
CKD's user avatar
  • 373
27 votes
2 answers
1k views

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 ...
Piotr Hajlasz's user avatar
27 votes
1 answer
1k views

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 ...
Rami Luisto's user avatar
26 votes
2 answers
2k views

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 ...
Sergei Ivanov's user avatar
26 votes
3 answers
7k views

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 ...
Nate Eldredge's user avatar
26 votes
2 answers
2k views

Axiom of choice: ultrafilter vs. Vitali set

It is well known that from a free (non-principal) ultrafilter on $\omega$ one can define a non-measurable set of reals. The older example of a non-measurable set is the Vitali set, a set of ...
Stefan Geschke's user avatar
26 votes
3 answers
5k views

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 ...
Hadi's user avatar
  • 731
25 votes
6 answers
5k views

Proof of Krylov-Bogoliubov theorem

Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel ...
Quinn Culver's user avatar
25 votes
2 answers
2k views

Writing a function on $\mathbb{R}$ as a sum of two injections

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. It is well-known that, using transfinite recursion with a well-ordering of $\mathbb{R}$, one can construct two injective functions $g,h: \...
Burak's user avatar
  • 4,105
25 votes
2 answers
3k views

Most significant results in motivic integration theory?

I am thinking about reading a course on motivic integration. I have already read certain introductions to the subject; yet I am not sure that they mention all the significant parts of the theory. So, ...
Mikhail Bondarko's user avatar

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