Question

Let $A \subset \mathbb{R}^n, k> 0$. Define $kA = \{kx : x \in A \}$. Show that $m^\ast(kA)=km^\ast(A).$

score 259 · Answer 1 · 2021-05-24 18:29:31

259

Views

Let $A \subset \mathbb{R}^n, k> 0$. Define $kA = \{kx : x \in A \}$. Show that $m^\ast(kA)=km^\ast(A).$

Published on 24 May 2021 - 18:29

score 109 · Answer 2 · 2021-05-24 19:04:42

109

Views

Prob. 19, Chap. 2, in Royden's REAL ANALYSIS: For a nonmeasurable set $E$ of finite outer measure there exists an open set $O \supset E$ such that ...

Published on 24 May 2021 - 19:04

score 66 · Answer 3 · 2021-05-25 08:20:34

66

Views

Positive measure of a Lebesgue measurable set

Published on 25 May 2021 - 8:20

score 40 · Answer 4 · 2021-05-26 19:15:34

40

Views

Question on weak finite additivity of outer measure.

Published on 26 May 2021 - 19:15

score 159 · Answer 5 · 2021-06-03 02:02:20

159

Views

Find a Borel subset $E$ of $[-1,1]$ s.t. $\lim_{r\to 0^{+}} \frac{m(E\cap [-r,r])}{2r}=\alpha$ -- following up a response.

Published on 03 Jun 2021 - 2:02

score 303 · Answer 6 · 2021-06-11 09:54:16

303

Views

Prob. 26, Chap. 2, in Royden's REAL ANALYSIS: Proof of $m^\left(A\cap\bigcup_{k=1}^\infty E_k\right)=\sum_{k=1}^\infty m^\left(A\cap B_k\right)$

Published on 11 Jun 2021 - 9:54

score 254 · Answer 7 · 2021-06-13 06:28:45

254

Views

Why the Lebesgue outer measure of the boundary of rectangle in $\mathbb{R^n}$ is zero?

Published on 13 Jun 2021 - 6:28

score 366 · Answer 8 · 2021-06-14 19:10:39

366

Views

Does my rigorous definiiton make sense and give what I want? How do we simplify my definiton?

Published on 14 Jun 2021 - 19:10

score 176 · Answer 9 · 2021-06-15 23:13:01

176

Views

The property of Lebesgue inner measure.

Published on 15 Jun 2021 - 23:13

score 150 · Answer 10 · 2021-06-16 16:19:51

150

Views

Proof from the definition that the outer measure of any open set is the sum over its interval lengths.

Published on 16 Jun 2021 - 16:19

score 43 · Answer 11 · 2021-06-18 05:42:00

43

Views

The property of Lebesgue-measurable set.

Published on 18 Jun 2021 - 5:42

score 41 · Answer 12 · 2021-06-22 04:44:44

41

Views

Proof of $m^*(\mathbb{R}^n)=\infty.$

Published on 22 Jun 2021 - 4:44

score 44 · Answer 13 · 2021-07-05 15:09:23

44

Views

Find the outer measure of $[-1,2]\cup \{3\}$.

Published on 05 Jul 2021 - 15:09

score 125 · Answer 14 · 2021-07-05 16:43:17

125

Views

Question on extension of a finitely additive set function.

Published on 05 Jul 2021 - 16:43

score 259 · Answer 15 · 2021-07-05 23:00:33

259

Views

Where is the axiom of choice used in the measure-theoretic proof of the uncountability of $[0,1]$?

Published on 05 Jul 2021 - 23:00

List Question

Let $A \subset \mathbb{R}^n, k> 0$. Define $kA = \{kx : x \in A \}$. Show that $m^\ast(kA)=km^\ast(A).$

Prob. 19, Chap. 2, in Royden's REAL ANALYSIS: For a nonmeasurable set $E$ of finite outer measure there exists an open set $O \supset E$ such that ...

Positive measure of a Lebesgue measurable set

Question on weak finite additivity of outer measure.

Find a Borel subset $E$ of $[-1,1]$ s.t. $\lim_{r\to 0^{+}} \frac{m(E\cap [-r,r])}{2r}=\alpha$ -- following up a response.

Prob. 26, Chap. 2, in Royden's REAL ANALYSIS: Proof of $m^\left(A\cap\bigcup_{k=1}^\infty E_k\right)=\sum_{k=1}^\infty m^\left(A\cap B_k\right)$

Why the Lebesgue outer measure of the boundary of rectangle in $\mathbb{R^n}$ is zero?

Does my rigorous definiiton make sense and give what I want? How do we simplify my definiton?

The property of Lebesgue inner measure.

Proof from the definition that the outer measure of any open set is the sum over its interval lengths.

The property of Lebesgue-measurable set.

Proof of $m^*(\mathbb{R}^n)=\infty.$

Find the outer measure of $[-1,2]\cup \{3\}$.

Question on extension of a finitely additive set function.

Where is the axiom of choice used in the measure-theoretic proof of the uncountability of $[0,1]$?

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