Question

A condition for the additivity $\mu^{}(X \cup Y)=\mu^{}(X)+\mu^{*}(Y)$ in the outer-measure

score 73 · Answer 1 · 2020-11-18 14:58:02

73

Views

A condition for the additivity $\mu^{}(X \cup Y)=\mu^{}(X)+\mu^{*}(Y)$ in the outer-measure

Published on 18 Nov 2020 - 14:58

score 4.2k · Answer 2 · 2020-11-20 15:27:45

4.2k

Views

Union of two measurable sets is measurable

Published on 20 Nov 2020 - 15:27

score 53 · Answer 3 · 2020-11-22 10:32:04

53

Views

Non-trivial subsets $A$ satisfying $m^{}(A) = m^{}(A \cap E) + m^{*}(A \cap E^c)$ for all $ E \subset \mathbb{R}$

Published on 22 Nov 2020 - 10:32

score 56 · Answer 4 · 2020-11-30 09:50:00

56

Views

Outer measure question (disjoint set st $\lambda^(A \cup B) < \lambda^(A) + \lambda^*(B)$)

Published on 30 Nov 2020 - 9:50

score 55 · Answer 5 · 2020-12-03 03:32:13

55

Views

Show that for any $r\in(0,1]$, there is a Vitali set $\mathbb{V}\subseteq[0,1]$ such that $\mu^*(\mathbb{V})=r$

Published on 03 Dec 2020 - 3:32

score 116 · Answer 6 · 2020-12-07 08:15:48

116

Views

Example of a non-measurable set such that inner measure=otuer measure.

Published on 07 Dec 2020 - 8:15

score 112 · Answer 7 · 2020-12-07 14:53:59

112

Views

Show that an integral function is a Borel function

Published on 07 Dec 2020 - 14:53

score 90 · Answer 8 · 2020-12-11 04:04:48

90

Views

Prove $\mu_{\ast}(A\cap E)=0$ and $\mu^{\ast}(A\cap E)=\mu(E)$

Published on 11 Dec 2020 - 4:04

score 1.9k · Answer 9 · 2020-12-13 14:11:34

1.9k

Views

Proving a bounded set is Lebesgue Measurable

Published on 13 Dec 2020 - 14:11

score 35 · Answer 10 · 2020-12-17 21:48:28

35

Views

How to write any member of a sigma algebra as a subset of a countable union of members of the generating algebra

Published on 17 Dec 2020 - 21:48

score 162 · Answer 11 · 2020-12-22 01:21:57

162

Views

Approximation of subsets of $\mathbb{R}$ with Borel subsets?

Published on 22 Dec 2020 - 1:21

score 39 · Answer 12 · 2020-12-23 05:45:25

39

Views

A family of sets whose outer and inner measures are equal is a $\sigma$-algebra

Published on 23 Dec 2020 - 5:45

score 112 · Answer 13 · 2020-12-24 11:22:00

112

Views

If $A$ is bounded then the outer measure of $A$ is finite

Published on 24 Dec 2020 - 11:22

score 23 · Answer 14 · 2020-12-26 10:27:42

23

Views

Is selecting a subset of vectors a measure theory problem?

Published on 26 Dec 2020 - 10:27

score 159 · Answer 15 · 2021-01-21 00:54:57

159

Views

A slight twist to Folland 1.4.18 b)

Published on 21 Jan 2021 - 0:54

List Question

A condition for the additivity $\mu^{}(X \cup Y)=\mu^{}(X)+\mu^{*}(Y)$ in the outer-measure

Union of two measurable sets is measurable

Non-trivial subsets $A$ satisfying $m^{}(A) = m^{}(A \cap E) + m^{*}(A \cap E^c)$ for all $ E \subset \mathbb{R}$

Outer measure question (disjoint set st $\lambda^(A \cup B) < \lambda^(A) + \lambda^*(B)$)

Show that for any $r\in(0,1]$, there is a Vitali set $\mathbb{V}\subseteq[0,1]$ such that $\mu^*(\mathbb{V})=r$

Example of a non-measurable set such that inner measure=otuer measure.

Show that an integral function is a Borel function

Prove $\mu_{\ast}(A\cap E)=0$ and $\mu^{\ast}(A\cap E)=\mu(E)$

Proving a bounded set is Lebesgue Measurable

How to write any member of a sigma algebra as a subset of a countable union of members of the generating algebra

Approximation of subsets of $\mathbb{R}$ with Borel subsets?

A family of sets whose outer and inner measures are equal is a $\sigma$-algebra

If $A$ is bounded then the outer measure of $A$ is finite

Is selecting a subset of vectors a measure theory problem?

A slight twist to Folland 1.4.18 b)

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