Questions tagged [measure-theory]
Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.
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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 ...
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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 ...
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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 $\...
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4
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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\|...
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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 ...
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7
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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 ...
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votes
3
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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 $x$?
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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}(...
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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 ...
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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
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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
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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 ...
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7
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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 ...
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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
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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, ...
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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)$? ...
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2
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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
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4
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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
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3
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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 ...
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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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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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4
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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 ...
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3
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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 $\...
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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 ) $ ...
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votes
3
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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
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5
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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 ...
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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'...
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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 = \...
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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 ...
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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 ...
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