Question

Continuity of a characteristic function of a translated set

score 198 · Answer 1 · 2026-03-28 03:30:50

198

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Continuity of a characteristic function of a translated set

Published on 28 Mar 2026 - 3:30

score 63 · Answer 2 · 2026-04-03 14:31:01

63

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Show: If $u \in W^{1,p}(B_4(0))$ then $\int_{B_4(0)} \frac{|\nabla u|^p}{|x-a|^{n-1}} dx < \infty$ for a.e. $a \in B_1(0)$.

Published on 03 Apr 2026 - 14:31

score 482 · Answer 3 · 2016-07-10 03:30:12

482

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Spectacular failure of Lebesgue differentiation for rectangles

Published on 10 Jul 2016 - 3:30

score 623 · Answer 4 · 2016-07-10 20:37:59

623

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$\int_X f^p d\mu = p\int_{[0,+\infty)} t^{p-1}\mu(\{x\in X: f(x)>t\}) d\mu_t$ for any natural $p\ge 1$

Published on 10 Jul 2016 - 20:37

score 190 · Answer 5 · 2016-07-11 16:24:20

190

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Prove that a limit of an integral function is finite

Published on 11 Jul 2016 - 16:24

score 375 · Answer 6 · 2026-03-26 21:27:47

375

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Change of variables, application of area and coarea formula

Published on 26 Mar 2026 - 21:27

score 76 · Answer 7 · 2026-03-30 23:00:03

76

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A neat characterization of measurable functions $f:\mathbb R\rightarrow \mathbb C$ for which $\lim_R\int_{-R}^R|f|dx<\infty$

Published on 30 Mar 2026 - 23:00

score 143 · Answer 8 · 2016-07-14 08:01:02

143

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Find an appropriate function such that the composition is Lebesgue integrable on $[0,1]$

Published on 14 Jul 2016 - 8:01

score 2.7k · Answer 9 · 2016-07-14 11:44:20

2.7k

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Lebesgue integration by substitution

Published on 14 Jul 2016 - 11:44

score 1.8k · Answer 10 · 2026-03-29 14:59:11

1.8k

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Show that $E=\bigcup_{k=1}^{\infty}E_k$, where for each index $k, E_k$ is measurable, and $(f_n)$ converges uniformly to $f$

Published on 29 Mar 2026 - 14:59

score 143 · Answer 11 · 2026-03-25 01:23:43

143

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Dealing with a Sequence of Sets with Two Indices and Simple Function based on that Sequence of Sets

Published on 25 Mar 2026 - 1:23

score 527 · Answer 12 · 2026-04-07 03:16:30

527

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How to prove that an $L^2$ function is also an $L^1$ function?

Published on 07 Apr 2026 - 3:16

score 70 · Answer 13 · 2016-07-16 05:39:54

70

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Do we have $\|x\|_p \le \|x\|_q$, where $x$ is bounded and $p>q$?

Published on 16 Jul 2016 - 5:39

score 62 · Answer 14 · 2016-07-16 16:24:49

62

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Evaluate $\left(\mbox{L}\right)\int_{0}^{1}\,\mathrm{ g}\left(x\right)\,\mathrm{d}x$

Published on 16 Jul 2016 - 16:24

score 731 · Answer 15 · 2016-07-17 08:57:48

731

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Intuition behind proof of bounded convergence theorem in Stein-Shakarchi

Published on 17 Jul 2016 - 8:57

List Question

Continuity of a characteristic function of a translated set

Show: If $u \in W^{1,p}(B_4(0))$ then $\int_{B_4(0)} \frac{|\nabla u|^p}{|x-a|^{n-1}} dx < \infty$ for a.e. $a \in B_1(0)$.

Spectacular failure of Lebesgue differentiation for rectangles

$\int_X f^p d\mu = p\int_{[0,+\infty)} t^{p-1}\mu(\{x\in X: f(x)>t\}) d\mu_t$ for any natural $p\ge 1$

Prove that a limit of an integral function is finite

Change of variables, application of area and coarea formula

A neat characterization of measurable functions $f:\mathbb R\rightarrow \mathbb C$ for which $\lim_R\int_{-R}^R|f|dx<\infty$

Find an appropriate function such that the composition is Lebesgue integrable on $[0,1]$

Lebesgue integration by substitution

Show that $E=\bigcup_{k=1}^{\infty}E_k$, where for each index $k, E_k$ is measurable, and $(f_n)$ converges uniformly to $f$

Dealing with a Sequence of Sets with Two Indices and Simple Function based on that Sequence of Sets

How to prove that an $L^2$ function is also an $L^1$ function?

Do we have $\|x\|_p \le \|x\|_q$, where $x$ is bounded and $p>q$?

Evaluate $\left(\mbox{L}\right)\int_{0}^{1}\,\mathrm{ g}\left(x\right)\,\mathrm{d}x$

Intuition behind proof of bounded convergence theorem in Stein-Shakarchi

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