Question

Find Lebesgue measure of $\{(x_1,\dots.x_n)\in\mathbb{R}^n:\sum_{i=1}^{n}|x_i|\notin\mathbb{Q};\forall_i|x_i|\leq1\}$

score 169 · Answer 1 · 2026-04-17 00:40:18

169

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Find Lebesgue measure of $\{(x_1,\dots.x_n)\in\mathbb{R}^n:\sum_{i=1}^{n}|x_i|\notin\mathbb{Q};\forall_i|x_i|\leq1\}$

Published on 17 Apr 2026 - 0:40

score 84 · Answer 2 · 2026-04-14 02:40:26

84

Views

Radon-Nikodyn derivative identity

Published on 14 Apr 2026 - 2:40

score 129 · Answer 3 · 2026-04-17 14:49:38

129

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For $K := \{(x, y, z) \in [a, b] \times \Bbb R^2: y^2 + z^2 \le (f(x))^2\}$, prove that $\lambda^3(K) = \pi \int_a^b (f(x))^2 dx$

Published on 17 Apr 2026 - 14:49

score 914 · Answer 4 · 2026-04-14 17:51:53

914

Views

A simple proof that there exixsts lebesgue measurable sets, not borel measurable

Published on 14 Apr 2026 - 17:51

score 1.8k · Answer 5 · 2026-04-17 12:50:40

1.8k

Views

If $f'(x)>0$ on $E$ , where $m(E)>0,$ then $m(f(E))>0$

Published on 17 Apr 2026 - 12:50

score 476 · Answer 6 · 2026-04-11 10:38:24

476

Views

Pushforward of a measure

Published on 11 Apr 2026 - 10:38

score 34 · Answer 7 · 2026-04-24 11:27:39

34

Views

Show that $N_p^p (f+g) \le N_p^p(f) + N_p^p (g)$ for $p\in ]0,1]$

Published on 24 Apr 2026 - 11:27

score 168 · Answer 8 · 2026-04-09 04:14:46

168

Views

Integrability of Higher-Dimensional Functions

Published on 09 Apr 2026 - 4:14

score 1.6k · Answer 9 · 2026-03-25 11:08:20

1.6k

Views

Construct a random variable with a given distribution

Published on 25 Mar 2026 - 11:08

score 209 · Answer 10 · 2026-04-15 19:12:17

209

Views

Construct $f$ such that $f\in L^{p_0}$ but $f\notin L_p$ for $p\ne p_0$

Published on 15 Apr 2026 - 19:12

score 640 · Answer 11 · 2026-04-11 00:13:44

640

Views

Equality of push-forward measure and borel measure

Published on 11 Apr 2026 - 0:13

score 250 · Answer 12 · 2026-04-15 05:27:40

250

Views

Is there an orthonormal basis of the Sobolev space $H^{1}(\Omega)$ with uniformly bounded elements?

Published on 15 Apr 2026 - 5:27

score 558 · Answer 13 · 2026-04-13 22:06:53

558

Views

Subset of plane with measure $1$ in all lines

Published on 13 Apr 2026 - 22:06

score 2.3k · Answer 14 · 2026-04-18 22:41:54

2.3k

Views

union of a collection of open interval containing the rationals in $[0,1]$

Published on 18 Apr 2026 - 22:41

score 350 · Answer 15 · 2026-04-11 14:31:48

350

Views

Why do we prefer countable subadditivity to countable additivity?

Published on 11 Apr 2026 - 14:31

List Question

Find Lebesgue measure of $\{(x_1,\dots.x_n)\in\mathbb{R}^n:\sum_{i=1}^{n}|x_i|\notin\mathbb{Q};\forall_i|x_i|\leq1\}$

Radon-Nikodyn derivative identity

For $K := \{(x, y, z) \in [a, b] \times \Bbb R^2: y^2 + z^2 \le (f(x))^2\}$, prove that $\lambda^3(K) = \pi \int_a^b (f(x))^2 dx$

A simple proof that there exixsts lebesgue measurable sets, not borel measurable

If $f'(x)>0$ on $E$ , where $m(E)>0,$ then $m(f(E))>0$

Pushforward of a measure

Show that $N_p^p (f+g) \le N_p^p(f) + N_p^p (g)$ for $p\in ]0,1]$

Integrability of Higher-Dimensional Functions

Construct a random variable with a given distribution

Construct $f$ such that $f\in L^{p_0}$ but $f\notin L_p$ for $p\ne p_0$

Equality of push-forward measure and borel measure

Is there an orthonormal basis of the Sobolev space $H^{1}(\Omega)$ with uniformly bounded elements?

Subset of plane with measure $1$ in all lines

union of a collection of open interval containing the rationals in $[0,1]$

Why do we prefer countable subadditivity to countable additivity?

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