Question

$\liminf_{r\to 0} f(r)/r^n=0$ implies $\mu=0$ if $\mu$ is a positive Borel measure on $\Bbb R^n$, $f(r)=\sup _{x\in \Bbb R^n} \mu(B(x,r))<\infty$

score 63 · Answer 1 · 2026-04-16 10:15:11

63

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$\liminf_{r\to 0} f(r)/r^n=0$ implies $\mu=0$ if $\mu$ is a positive Borel measure on $\Bbb R^n$, $f(r)=\sup _{x\in \Bbb R^n} \mu(B(x,r))<\infty$

Published on 16 Apr 2026 - 10:15

score 188 · Answer 2 · 2026-04-13 06:12:26

188

Views

The law induced by a norm $\rho$ on $\mathbb{R}^n$ with respect to Lebesgue's measure.

Published on 13 Apr 2026 - 6:12

score 99 · Answer 3 · 2026-04-11 09:50:42

99

Views

Hint for solving this problem about constant function on $\mathbb{R^2}$

Published on 11 Apr 2026 - 9:50

score 150 · Answer 4 · 2026-04-12 15:00:52

150

Views

theorem 10.14 Browder Mathematical Analysis

Published on 12 Apr 2026 - 15:00

score 224 · Answer 5 · 2026-04-15 17:12:19

224

Views

Twice continuously differentiable function $f:\Bbb R^2\to \Bbb R$ such that $f, f_{xx}, f_{yy}\in L^2$ is bounded

Published on 15 Apr 2026 - 17:12

score 61 · Answer 6 · 2026-03-25 13:52:01

61

Views

Finding dominating function

Published on 25 Mar 2026 - 13:52

score 54 · Answer 7 · 2026-04-17 05:22:08

54

Views

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score 43 · Answer 8 · 2026-04-09 00:42:49

43

Views

Writing an integral in terms of Lebesgue measure

Published on 09 Apr 2026 - 0:42

score 147 · Answer 9 · 2026-04-11 13:16:51

147

Views

Finding Radon-Nikodym derivative $d\mu/dm$ where $m$ is the Lebesgue measure on $[0,1]$, $f(x)=x^2$, and $\mu(E)=m(f(E))$

Published on 11 Apr 2026 - 13:16

score 44 · Answer 10 · 2026-04-15 06:02:54

44

Views

Are these characteristic functions Lebesgue measurable?

Published on 15 Apr 2026 - 6:02

score 41 · Answer 11 · 2026-04-13 09:27:54

41

Views

Sure vs almost sure convergence for a simple random variable

Published on 13 Apr 2026 - 9:27

score 397 · Answer 12 · 2026-04-14 02:31:55

397

Views

Prove the union of uncountably many closed intervals is measurable

Published on 14 Apr 2026 - 2:31

score 38 · Answer 13 · 2026-04-14 14:11:04

38

Views

non-existence of homeomorphism between open subsets of $\mathbb{S}^{n}$ and $B(0,1,\mathbb{R}^{n})$

Published on 14 Apr 2026 - 14:11

score 1.2k · Answer 14 · 2026-04-13 16:11:17

1.2k

Views

Lebesgue integral of $f$ is equal to the Lebesgue measure of the area under the curve of $f$

Published on 13 Apr 2026 - 16:11

score 67 · Answer 15 · 2026-04-16 03:08:31

67

Views

Meaning of $\int_{0}^{1} f(x) \mathrm{d}x $ and $\int_{0}^{1} \mathrm{d}f(x) $

Published on 16 Apr 2026 - 3:08

List Question

$\liminf_{r\to 0} f(r)/r^n=0$ implies $\mu=0$ if $\mu$ is a positive Borel measure on $\Bbb R^n$, $f(r)=\sup _{x\in \Bbb R^n} \mu(B(x,r))<\infty$

The law induced by a norm $\rho$ on $\mathbb{R}^n$ with respect to Lebesgue's measure.

Hint for solving this problem about constant function on $\mathbb{R^2}$

theorem 10.14 Browder Mathematical Analysis

Twice continuously differentiable function $f:\Bbb R^2\to \Bbb R$ such that $f, f_{xx}, f_{yy}\in L^2$ is bounded

Finding dominating function

Prove a function is Lebesgue measurable

Writing an integral in terms of Lebesgue measure

Finding Radon-Nikodym derivative $d\mu/dm$ where $m$ is the Lebesgue measure on $[0,1]$, $f(x)=x^2$, and $\mu(E)=m(f(E))$

Are these characteristic functions Lebesgue measurable?

Sure vs almost sure convergence for a simple random variable

Prove the union of uncountably many closed intervals is measurable

non-existence of homeomorphism between open subsets of $\mathbb{S}^{n}$ and $B(0,1,\mathbb{R}^{n})$

Lebesgue integral of $f$ is equal to the Lebesgue measure of the area under the curve of $f$

Meaning of $\int_{0}^{1} f(x) \mathrm{d}x $ and $\int_{0}^{1} \mathrm{d}f(x) $

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