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Results tagged with lebesgue-integral
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user 818326
For questions about integration, where the theory is based on measures. It is almost always used together with the tag [measure-theory], and its aim is to specify questions about integrals, not only properties of the measure.
0
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0
answers
41
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Volume of unit sphere by iterative integrals
I want to calculate the volume of the unit sphere: $\{(x,y) \in\mathbb{R} \,|\, x^2+y^2 \leq 1\} = K$
I am supposed to use iterative integration. We have defined the volume as:
$\int_\mathbb{R^2}\,\, …
1
vote
2
answers
65
views
Function $f$ with $|f|$ is Lebesgue integrable but $f$ isn't locally Lebesgue integrable
I'm trying to find a function $f$ with $|f|$ is Lebesgue integrable but $f$ isn't locally Lebesgue integrable.
My approach:
Let $X = [0,1]$ and $A \subset X$ with A is the Vitali-set. So $A$ is not m …
4
votes
3
answers
363
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Proof for volume of n-ball with radius 1
I'm trying to prove this formula from here, that the volume of a n-ball with radius 1 (let's call it $B_n$) is: $$\frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}$$
However, I come to the wrong result and I ca …
1
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0
answers
37
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$E = \{(x,y) \in \mathbb{R} \, | \, x \leq y, \, xy \geq 1 \, y \leq 2\}$, Calculate $\int_E...
$E = \{(x,y) \in \mathbb{R} \, | \, x \leq y, \, xy \geq 1 \, y \leq 2\}$
Calculate $\int_E \frac{y^2}{x^2}$
I don't really know how to approach this. What we learned and what I normally do is to rew …
4
votes
1
answer
56
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Calculate $\lim_{j \rightarrow \infty} \int_0^j (1+\frac{x}{j})^j e^{-\pi x}dx$
My approach:
$$\lim_{j \rightarrow \infty} \int_0^j \left(1+\frac{x}{j}\right)^j e^{-\pi x}dx =
\lim_{j \rightarrow \infty} \int_0^{\infty} \left(1+\frac{x}{j}\right)^j e^{-\pi x}dx.$$
I'm not sure i …
2
votes
2
answers
62
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$\int_K |x|^m |y|^n dx dy$
Let $K=\{(x,y)\in \mathbb{R}^2 \,|\, x^2+y^2 \leq 1\}$.
I want to find $\int_K |x|^m |y|^n \,\,dx\,\, dy$.
My approach is:
$...\,\,=4\cdot \int_0^1 \int_0^{\sqrt{1-y^2}} |x|^m |y|^n\,dx \, dy = 4\cdo …
0
votes
1
answer
54
views
Counter example that Lebesgue is translation invariant
I seem to have a basic misunderstanding for what translation invariant means.
I thought it simply means that $\lambda(\varphi(x))=\lambda(\varphi(x+a))$ for some $a\in \mathbb{R}^n$. Here $\varphi$ is …
1
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Accepted
Counter example that Lebesgue is translation invariant
Okay I got it:
$\varphi(x) = \{x^2$ if $x\in [0,1], \,\,0 $ else
Then: $\int \varphi(x) dx = \frac{1}{3} = \int \varphi(x+1) dx = \int_{-1}^{0} (x+1)^2 dx = \frac{1}{3}$
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113
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Prove $\int_0^\infty t^n e^{-t} dt = n!$ by differentiation
I have to prove $\int_0^\infty t^n e^{-t} dt = n!$ by differentiating infinitely often under the integral. $(t, x > 0)$
I think I have proven that I can make use of a Lemma we have. It is differentia …
0
votes
1
answer
39
views
Is $M \times M \times\mathbb{Q} \subset \mathbb{R}^3$ measurable for $M \subset \mathbb R$?
Let $M \subset \mathbb{R}$. Is $M \times M \times\mathbb{Q} \subset \mathbb{R}^3$ measurable? (Lebesgue measurable)
I would say no, because by the definition the the Lebesgue measure, intuitively spe …
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0
answers
24
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Applying Theorem of dominated convergence (Lebesgue) on this task, possible?
a) $1<p<\infty, \,M\in\mathbb R,\,\,f,f_j\in L_p(\mathbb R^n)$ and $||f_j||_p\leq M \,\,\forall j$ and $f_j \rightarrow f$ almost everywhere. Proove: Then we have $||f||_p \leq M$. If $||f_j| …