Questions tagged [measure-theory]
Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.
40,085
questions
243
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Is this really a categorical approach to integration?
Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A ...
- 45.7k
225
votes
7
answers
141k
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$L^p$ and $L^q$ space inclusion
Let $(X, \mathcal B, m)$ be a measure space. For $1 \leq p < q \leq \infty$, under what condition is it true that $L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m)$ and what is a counterexample ...
- 2,259
213
votes
5
answers
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The sum of an uncountable number of positive numbers
Claim: If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$
such that $\sum_{\alpha\in A}x_\alpha<\infty$, then $x_\alpha=0$ for all but at most countably many $\...
- 5,910
204
votes
4
answers
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Limit of $L^p$ norm
Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, $\displaystyle\lim_{p\to\infty}\|f\|...
- 3,354
145
votes
4
answers
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Is it possible for a function to be in $L^p$ for only one $p$?
I'm wondering if it's possible for a function to be an $L^p$ space for only one value of $p \in [1,\infty)$ (on either a bounded domain or an unbounded domain).
One can use interpolation to show that ...
- 8,633
137
votes
7
answers
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Construction of a Borel set with positive but not full measure in each interval
I was wondering how one can construct a Borel set that doesn't have full measure on any interval of the real line but does have positive measure everywhere.
To be precise, if $\mu$ denotes Lebesgue ...
- 8,633
122
votes
3
answers
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Does convergence in $L^p$ imply convergence almost everywhere?
If I know $\| f_n - f \|_{L^p(\mathbb{R})} \to 0$ as $n \to \infty$, do I know that $\lim_{n \to \infty}f_n(x) = f(x)$ for almost every $x$?
- 1,295
112
votes
9
answers
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What are Some Tricks to Remember Fatou's Lemma?
For a sequence of non-negative measurable functions $f_n$, Fatou's lemma is a statement about the inequality
$$\int \liminf_{n\rightarrow \infty} f_n \mathrm{d}\mu \leq \liminf_{n\rightarrow \infty}(...
- 7,358
111
votes
1
answer
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Lebesgue measure theory vs differential forms?
I am currently reading various differential geometry books. From what I understand differential forms allow us to generalize calculus to manifolds and thus perform integration on manifolds. I gather ...
- 10k
111
votes
8
answers
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If $S$ is an infinite $\sigma$ algebra on $X$ then $S$ is not countable
I am going over a tutorial in my real analysis course. There is
an proof in which I don't understand some parts of it.
The proof relates to the following proposition:
($S$ - infinite $\sigma$-algebra ...
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107
votes
2
answers
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If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$
Let $1\leq p < \infty$. Suppose that
$\{f_k, f\} \subset L^p$ (the domain here does not necessarily have to be finite),
$f_k \to f$ almost everywhere, and
$\|f_k\|_{L^p} \to \|f\|_{L^p}$.
Why is ...
- 8,633
95
votes
16
answers
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Reference book on measure theory
I post this question with some personal specifications. I hope it does not overlap with old posted questions.
Recently I strongly feel that I have to review the knowledge of measure theory for the ...
85
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7
answers
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Understanding Borel sets
I'm studying Probability theory, but I can't fully understand what are Borel sets. In my understanding, an example would be if we have a line segment [0, 1], then a Borel set on this interval is a set ...
- 1,221
77
votes
2
answers
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Integration of forms and integration on a measure space
In Terence Tao's PCM article: DIFFERENTIAL FORMS AND INTEGRATION, it is pointed out that there are three concepts of integration which appear in the subject (single-variable calculus):
the indefinite ...
user9464
77
votes
2
answers
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Cardinality of Borel sigma algebra
It seems it's well known that if a sigma algebra is generated by countably many sets, then the cardinality of it is either finite or $c$ (the cardinality of continuum). But it seems hard to prove it, ...
- 3,436
72
votes
5
answers
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Under what condition we can interchange order of a limit and a summation?
Suppose f(m,n) is a double sequence in $\mathbb R$. Under what condition do we have $\lim\limits_{n\to\infty}\sum\limits_{m=1}^\infty f(m,n)=\sum\limits_{m=1}^\infty \lim\limits_{n\to\infty} f(m,n)$? ...
- 839
71
votes
2
answers
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Integration with respect to counting measure.
I am having trouble computing integration w.r.t. counting measure. Let $(\mathbb{N},\scr{P}(\mathbb{N}),\mu)$ be a measure space where $\mu$ is counting measure. Let $f:\mathbb{N}\rightarrow{\mathbb{R}...
user54992
71
votes
4
answers
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How do people apply the Lebesgue integration theory?
This question has puzzled me for a long time. It may be too vague to ask here. I hope I can narrow down the question well so that one can offer some ideas.
In a lot of calculus textbooks, there is ...
user9464
70
votes
3
answers
13k
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Set of continuity points of a real function
I have a question about subsets $$
A \subseteq \mathbb R
$$
for which there exists a function $$f : \mathbb R \to \mathbb R$$ such that the set of continuity points of $f$ is $A$. Can I characterize ...
- 3,063
70
votes
2
answers
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Differences between the Borel measure and Lebesgue measure
I'm having difficult time in understanding the difference between the Borel measure and Lebesgue measure. Which are the exact differences? Can anyone explain this using an example?
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3
answers
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The $\sigma$-algebra of subsets of $X$ generated by a set $\mathcal{A}$ is the smallest sigma algebra including $\mathcal{A}$
I am struggling to understand why it should be that the $\sigma$-algebra of subsets of $X$ generated by $\mathcal{A}$ should be the smallest $\sigma$-algebra of subsets of $X$ including $\mathcal{A}$.
...
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66
votes
4
answers
8k
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Why is the Daniell integral not so popular?
The Riemann integral is the most common integral in use and is the first integral I was taught to use. After doing some more advanced analysis it becomes clear that the Riemann integral has some ...
- 2,231
66
votes
3
answers
9k
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Approximating a $\sigma$-algebra by a generating algebra
Theorem. Let $(X,\mathcal B,\mu)$ a finite measure space, where $\mu$ is a positive measure. Let $\mathcal A\subset \mathcal B$ an algebra generating $\cal B$.
Then for all $B\in\cal B$ and $\...
- 174k
65
votes
1
answer
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Formal definition of conditional probability
It would be extremely helpful if anyone gives me the formal definition of conditional probability and expectation in the following setting, given probability space
$ (\Omega, \mathscr{A}, \mu ) $ ...
- 4,187
63
votes
3
answers
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What is Haar Measure?
Is there any simple explanation for Haar Measure and its geometry?
how do we understand analogy Between lebesgue measure and Haar Measure?
How to show integration with respect to Haar Measure?
what do ...
- 1,157
63
votes
5
answers
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Difference between topology and sigma-algebra axioms.
One distinct difference between axioms of topology and sigma algebra is the asymmetry between union and intersection; meaning topology is closed under finite intersections sigma-algebra closed under ...
- 3,138
60
votes
10
answers
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Seeking a layman's guide to Measure Theory
I would like to teach myself measure theory. Unfortunately most of the books that I've come across are very difficult and are quick to get into Lemmas and proofs. Can someone please recommend a layman'...
59
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1
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Monotone Convergence Theorem for non-negative decreasing sequence of measurable functions
Let $(X,\mathcal{M},\mu)$ be a measure space and suppose $\{f_n\}$ are non-negative measurable functions decreasing pointwise to $f$. Suppose also that $\int f_1 \lt \infty$. Then $$\int_X f~d\mu = \...
- 1,855
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votes
5
answers
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Is there a change of variables formula for a measure theoretic integral that does not use the Lebesgue measure
Is there a generic change of variables formula for a measure theoretic integral that does not use the Lebesgue measure? Specifically, most references that I can find give a change of variables ...
- 581
58
votes
2
answers
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Lebesgue measurable but not Borel measurable
I'm trying to find a set which is Lebesgue measurable but not Borel measurable.
So I was thinking of taking a Lebesgue set of measure zero and intersecting it with something so that the result is not ...
- 14.7k
57
votes
4
answers
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Is every Lebesgue measurable function on $\mathbb{R}$ the pointwise limit of continuous functions?
I know that if $f$ is a Lebesgue measurable function on $[a,b]$ then there exists a continuous function $g$ such that $|f(x)-g(x)|< \epsilon$ for all $x\in [a,b]\setminus P$ where the measure of $P$...
- 7,137
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votes
4
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Is composition of measurable functions measurable?
We know that if $ f: E \to \mathbb{R} $ is a Lebesgue-measurable function and $ g: \mathbb{R} \to \mathbb{R} $ is a continuous function, then $ g \circ f $ is Lebesgue-measurable. Can one replace the ...
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56
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1
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Lebesgue measurable set that is not a Borel measurable set
exact duplicate of Lebesgue measurable but not Borel measurable
BUT! can you please translate Miguel's answer and expand it with a formal proof? I'm totally stuck...
In short: Is there a Lebesgue ...
- 2,085
56
votes
1
answer
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Are vague convergence and weak convergence of measures both weak* convergence?
For quite a long time, I have been confused about the definitions of weak convergence and vague convergence of measures among other modes of convergence that root from functional analysis, mainly due ...
- 47.7k
55
votes
7
answers
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False beliefs about Lebesgue measure on $\mathbb{R}$
I'm trying to develop intuition about Lebesgue measure on $\mathbb{R}$ and I'd like to build a list of false beliefs about it, for example: every set is measurable, every set of measure zero is ...
user365
54
votes
3
answers
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General Lebesgue Dominated Convergence Theorem
In Royden (4th edition), it says one can prove the General Lebesgue Dominated Convergence Theorem by simply replacing $g-f_n$ and $g+f_n$ with $g_n-f_n$ and $g_n+f_n$. I proceeded to do this, but I ...
- 6,534
54
votes
3
answers
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Preimage of generated $\sigma$-algebra
For some collection of sets $A$, let $\sigma(A)$ denote the $\sigma$-algebra generated by $A$.
Let $C$ be some collection of subsets of a set $Y$, and let $f$ be a function from some set $X$ to $Y$. ...
- 5,546
52
votes
3
answers
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Example where union of increasing sigma algebras is not a sigma algebra
If $\mathcal{F}_1 \subset \mathcal{F}_2 \subset \dotsb$ are sigma algebras, what is wrong with claiming that $\cup_i\mathcal{F}_i$ is a sigma algebra?
It seems closed under complement since for all $x$...
- 2,489
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5
answers
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Intuitive interpretation of limsup and liminf of sequences of sets?
What is an intuitive interpretation of the 'events'
$$\limsup A_n:=\bigcap_{n=0}^{\infty}\bigcup_{k=n}^{\infty}A_k$$
and
$$\liminf A_n:=\bigcup_{n=0}^{\infty}\bigcap_{k=n}^{\infty}A_k$$
when $A_n$ are ...
- 2,111
51
votes
3
answers
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On the equality case of the Hölder and Minkowski inequalities
I'm following the book Measure and Integral of Richard L. Wheeden and Antoni Zygmund. This is problem 4 of chapter 8.
Consider $E\subseteq \mathbb{R}^n$ a measurable set. In the following all the ...
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votes
8
answers
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Why do we restrict the definition of Lebesgue Integrability?
The function $f(x) = \sin(x)/x$ is Riemann Integrable from $0$ to $\infty$, but it is not Lebesgue Integrable on that same interval. (Note, it is not absolutely Riemann Integrable.)
Why is it we ...
- 2,924
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votes
3
answers
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Vitali-type set with given outer measure
Is it possible to construct a non-measurable set in $[0,1]$ of a given outer measure $x \in [0,1]$? This will probably require the axiom of choice. Does anyone have a suggestion?
Edit: I forgot to ...
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votes
3
answers
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What is the difference between outer measure and Lebesgue measure?
What is the difference between outer measure and Lebesgue measure?
We know that there are sets which are not Lebesgue measurable, whereas we know that outer measure is defined for any subset of $\...
- 611
49
votes
4
answers
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Infinite product of measurable spaces
Suppose there is a family (can be
infinite) of measurable spaces. What
are the usual ways to define a sigma
algebra on their Cartesian product?
There is one way in the context of
defining product ...
- 47.7k
48
votes
6
answers
7k
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What is the intuition behind Chebyshev's Inequality in Measure Theory
Chebyshev's Inequality Let $f$ be a nonnegative measurable function on $E .$ Then for any $\lambda>0$,
$$
m\{x \in E \mid f(x) \geq \lambda\} \leq \frac{1}{\lambda} \cdot \int_{E} f.
$$
What ...
- 4,503
48
votes
2
answers
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When exactly is the dual of $L^1$ isomorphic to $L^\infty$ via the natural map?
The dual space to the Banach space $L^1(\mu)$ for a sigma-finite measure $\mu$ is $L^\infty(\mu)$, given by the correspondence
$\phi \in L^\infty(\mu) \mapsto I_\phi$, where $I_\phi(f) = \int f \...
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votes
2
answers
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Convergence in measure implies convergence almost everywhere of a subsequence
How can I prove that if a sequence of functions $\{f_n\}$ that converges to $f$ in measure on a space of finite measure, then there exists a subsequence of $\{f_n\}$ that converges to $f$ almost ...
- 1,443
48
votes
2
answers
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What's the relationship between a measure space and a metric space?
Definition of Measurable Space:
An ordered pair $(\Omega, \mathcal{F})$ is a measurable space if $\mathcal{F}$ is a $\sigma$-algebra on $\Omega$.
Definition of Measure:
Let $(\Omega, \mathcal{F})$ be ...
- 3,615
48
votes
1
answer
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Example of a continuous function that is not Lebesgue measurable
Let $\mathcal{L}$ denote the $\sigma$-algebra of Lebesgue measurable sets on $\mathbb{R}$. Then, if memory serves, there is an example (and of course, if there is one, there are many) of a continuous ...
- 7,969
47
votes
3
answers
14k
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The set of differences for a set of positive Lebesgue measure
Quite a while ago, I heard about a statement in measure theory, that goes as follows:
Let $A \subset \mathbb R^n$ be a Lebesgue-measurable set of positive measure. Then we follow that $A-A = \{ x-y ...
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