All Questions
Tagged with measure-theory integration
693
questions with no upvoted or accepted answers
19
votes
0
answers
896
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Open problems in Federer's Geometric Measure Theory
I wanted to know if the problems mentionned in this book are solved. More specifically, at some places, the author says that he doesn't know the answer, for example :"I do not know whether this ...
- 301
12
votes
0
answers
505
views
A very useful lemma for Henstock-Stieltjes integration
I'd like to see a proof (or hints and outlines) for the following lemma, which is very useful to prove some interesting properties, including an Integration by Parts theorem for Henstock-Stieltjes ...
11
votes
0
answers
2k
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Cauchy-Formula for Repeated Lebesgue-Integration
Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given.
Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...
- 559
10
votes
0
answers
263
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Integral of a function over the Koch Curve. Is it rigourous enough?
(I want to investigate the validity of this approach, as I already know this is the correct result)
I present a proof that $$\int_{K} (x+y) \ \mu(x,y)={{9+\sqrt 3} \over 18}$$
Where the region of ...
- 8,361
9
votes
0
answers
2k
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On Bourbaki's "Integration" - Why/What/How to read it
I have a question concerning the Bourbaki's book on integration.
Whenever I find them referenced (in answers on this site as well), it looks like they aged more than volumes of the same series on ...
- 4,113
7
votes
0
answers
77
views
The difference between $d\mu(\omega)$ and $\mu(d\omega)$
Let $(\Omega,\mathcal{A},\mu)$ be measure space and $f:\Omega\to \mathbb{R}$ be a integrable function.
What is the difference between the following two formulas :
$$
(1) \int_{\Omega} f(\omega)d\mu(\...
- 129
6
votes
0
answers
126
views
How would one integrate over $SO(n)$?
Suppose you want to find the average of an $n\times n$ diagonal matrix $A$ over all possible rotations,
$$
\langle A\rangle = \int\limits_\text{SO($n$)} Q^T A Q \; dQ.
$$
It's easy enough to do this ...
- 61
6
votes
0
answers
135
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If $f$ is real-valued bounded measurable and $\mu$ a complex measure, then $\left | \int_X f \mathrm d \mu \right | \le \int_X |f| \mathrm d |\mu|$
Let $(X, \mathcal X)$ be a measurable space. Let $\mu$ be a complex measure on $X$ and $|\mu|$ its variation. Then $|\mu|$ is a non-negative finite measure. By definition, $|\mu(B)| \le |\mu| (B)$ for ...
- 5,817
6
votes
0
answers
470
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Why can't we integrate functions on a manifold?
Most books about manifolds say that "there is no way to integrate real-valued functions in a coordinate-independent way on a manifold". I've read the usual reasons but they don't seem to differ from ...
- 4,563
6
votes
0
answers
354
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Integral depending measurable on a parameter
Let $(\Omega, \Sigma, \mu)$ be a measurable space and let $f\colon [0,1] \times \Omega \to \mathbb R$ be such that $x\mapsto f(t,x)$ is integrable for all $t\in [0,1]$. Then $$F\colon [0,1] \to \...
- 594
6
votes
0
answers
153
views
Is $\sigma$-finiteness really a necessary condition for this problem?
Question: Let $(X, \mathcal A, \mu)$ be a measure space and suppose $\mu$ is $\sigma$-finite. Suppose $f$ is integrable. Prove that given any $\varepsilon$, there exists a $\delta >0$ such that $$\...
- 2,549
6
votes
0
answers
1k
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Lebesgue's criterion for Riemann integrability of functions of two variables
Lebesgue's criterion for Riemann integrability for functions of one variable states that:
$f:[a,b]\rightarrow \mathbb{R}$, with $a,b\in\mathbb{R},a\leq b$, is Riemann integrable if and only if $f$ is ...
- 6,249
6
votes
0
answers
2k
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"Simple" proof of Lebesgue's Differentiation Theorem for dimension 1?
Lebesgue Differentiation Theorem for $\mathbb{R}$: Let $f:[a,b]\to \mathbb{R}$ be intergable and $F(x)=\int_a^xf$. Then $F$ is differentiable almost everywhere in $[a,b]$ and $F'=f$ a.e.
Is there a (...
- 633
5
votes
0
answers
136
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Property of vector-valued measure
Let $B$ be a Banach space, let $(X,\mathcal{A})$ be a measurable space, and let $\mu:\mathcal{A}\to B$ be a vector-valued measure of bounded variation.
In general, if $B$ doesn't have the Radon-...
- 8,127
5
votes
1
answer
99
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An example of a function presented by Riemann in one of his paper
I came across this function which Riemann has constructed in his paper 'On the representation of a function by a trigonometric series' to prove that a function that is dicontinuous in infinitely many ...
- 63