Question

What is wrong with my application of Lebesgue Dominated Convergence Theorem in these two examples?

score 699 · Answer 1 · 2026-04-13 10:49:32

699

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What is wrong with my application of Lebesgue Dominated Convergence Theorem in these two examples?

Published on 13 Apr 2026 - 10:49

score 86 · Answer 2 · 2026-04-22 21:54:15

86

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Show $\int_{\mathbb{R}^n} \exp(-\|Ax\|^2) d\mu(x)=\frac{\pi^{\frac{n}{2}}}{|\det(A)|}$

Published on 22 Apr 2026 - 21:54

score 85 · Answer 3 · 2026-04-16 04:44:18

85

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Prove that $\sum_{k=0}^{\infty}\frac{1}{(k+p)^2}=-\int_0^1\frac{x^p}{1-x}\log x \mathrm{ d}x$

Published on 16 Apr 2026 - 4:44

score 47 · Answer 4 · 2026-04-17 07:46:42

47

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Evaluate $ \int_{\mathbb{R}^n} \! \exp(-||x||^2) d\mu$

Published on 17 Apr 2026 - 7:46

score 555 · Answer 5 · 2026-04-16 06:23:06

555

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Concerning existence of subsequence of converging integrals on subsets of $[0,1]$ of a sequence $(f_n)\in[0,1]$

Published on 16 Apr 2026 - 6:23

score 241 · Answer 6 · 2026-04-10 21:51:58

241

Views

Bounded Variation imply L^1

Published on 10 Apr 2026 - 21:51

score 81 · Answer 7 · 2026-04-13 03:58:49

81

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Does $\left\|u_n-u\right\|_{L^2(\Omega)}\stackrel{n\to\infty}{\to}0$ imply $u\in L^2(\Omega)$?

Published on 13 Apr 2026 - 3:58

score 625 · Answer 8 · 2026-04-11 20:16:09

625

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A classical solution of Poisson's equation is also a weak solution

Published on 11 Apr 2026 - 20:16

score 2.1k · Answer 9 · 2026-04-15 18:52:35

2.1k

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If $\mu$ is finite, then $\{f_n\}$ is uniformly integrable iff $\sup_n \int|f_n| d\mu<\infty$ and $\{f_n\}$ is uniformly absolutely continuous.

Published on 15 Apr 2026 - 18:52

score 145 · Answer 10 · 2026-04-10 09:50:40

145

Views

Complex Measures: Pushforward

Published on 10 Apr 2026 - 9:50

score 27 · Answer 11 · 2026-04-15 09:26:39

27

Views

$\|\sum_ia_i g(x-i)\|_{L^p(\mathbb{R})}\le (\sup_x \{\sum_k|g(x-k)|\})\|a\|_{L^p(\mathbb{Z})}$

Published on 15 Apr 2026 - 9:26

score 166 · Answer 12 · 2026-04-10 14:03:17

166

Views

Lebesgue integration

Published on 10 Apr 2026 - 14:03

score 1.3k · Answer 13 · 2026-04-16 01:12:18

1.3k

Views

Suppose $\mu$ is a finite measure and $\sup_n \int |f_n|^{1+\epsilon} \ d\mu<\infty$ for some $\epsilon$. Prove that $\{f_n\}$ is uniformly integrable

Published on 16 Apr 2026 - 1:12

score 740 · Answer 14 · 2026-04-16 00:43:53

740

Views

Collection is uniformly integrable, but individual is not integrable

Published on 16 Apr 2026 - 0:43

score 225 · Answer 15 · 2026-04-17 02:29:25

225

Views

A sequence tending to zero where each element cannot be bounded by an integrable function

Published on 17 Apr 2026 - 2:29

List Question

What is wrong with my application of Lebesgue Dominated Convergence Theorem in these two examples?

Show $\int_{\mathbb{R}^n} \exp(-\|Ax\|^2) d\mu(x)=\frac{\pi^{\frac{n}{2}}}{|\det(A)|}$

Prove that $\sum_{k=0}^{\infty}\frac{1}{(k+p)^2}=-\int_0^1\frac{x^p}{1-x}\log x \mathrm{ d}x$

Evaluate $ \int_{\mathbb{R}^n} \! \exp(-||x||^2) d\mu$

Concerning existence of subsequence of converging integrals on subsets of $[0,1]$ of a sequence $(f_n)\in[0,1]$

Bounded Variation imply L^1

Does $\left\|u_n-u\right\|_{L^2(\Omega)}\stackrel{n\to\infty}{\to}0$ imply $u\in L^2(\Omega)$?

A classical solution of Poisson's equation is also a weak solution

If $\mu$ is finite, then $\{f_n\}$ is uniformly integrable iff $\sup_n \int|f_n| d\mu<\infty$ and $\{f_n\}$ is uniformly absolutely continuous.

Complex Measures: Pushforward

$\|\sum_ia_i g(x-i)\|_{L^p(\mathbb{R})}\le (\sup_x \{\sum_k|g(x-k)|\})\|a\|_{L^p(\mathbb{Z})}$

Lebesgue integration

Suppose $\mu$ is a finite measure and $\sup_n \int |f_n|^{1+\epsilon} \ d\mu<\infty$ for some $\epsilon$. Prove that $\{f_n\}$ is uniformly integrable

Collection is uniformly integrable, but individual is not integrable

A sequence tending to zero where each element cannot be bounded by an integrable function

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