Question

How to prove sub-additivity of the Lebesgue measure for open sets?

score 34 · Answer 1 · 2018-04-22 09:24:59

34

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How to prove sub-additivity of the Lebesgue measure for open sets?

Published on 22 Apr 2018 - 9:24

score 1.7k · Answer 2 · 2018-04-22 11:31:20

1.7k

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Failure of Fatou's lemma for limit supremum

Published on 22 Apr 2018 - 11:31

score 68 · Answer 3 · 2026-03-25 16:12:43

68

Views

Determine whether the following Lebesgue integral exists.

Published on 25 Mar 2026 - 16:12

score 54 · Answer 4 · 2026-03-24 22:11:45

54

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The function defined with the Lebesgue's measure is measurable

Published on 24 Mar 2026 - 22:11

score 31 · Answer 5 · 2018-04-24 14:36:47

31

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Does there exist a Lebesgue measurable set $[0,1] \subset D$ such that $D \neq [0,1]$ and $\lambda ^{*} (D) = 1$?

Published on 24 Apr 2018 - 14:36

score 87 · Answer 6 · 2026-03-25 07:38:50

87

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Equality between volume and surface

Published on 25 Mar 2026 - 7:38

score 145 · Answer 7 · 2026-03-25 17:44:08

145

Views

Approximating Lebesgue integrable functions with continuous $2\pi$ periodic functions and trig polynomials

Published on 25 Mar 2026 - 17:44

score 840 · Answer 8 · 2026-03-25 15:59:24

840

Views

Question on the proof that the Cantor set has measure zero

Published on 25 Mar 2026 - 15:59

score 336 · Answer 9 · 2018-04-26 15:55:03

336

Views

If $B \subset A \subset \mathbb{R}$ such that $m^(B)=0 $. Show that $m^(A \setminus B)=m^*(A)$.

Published on 26 Apr 2018 - 15:55

score 47 · Answer 10 · 2018-04-26 20:35:58

47

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If $A,B \subset \mathbb{R} $ such that $m^*(A)=0 $ and $A \cup B$ is Lebesgue measurable. Show that $B$ is Lebesgue measurable.

Published on 26 Apr 2018 - 20:35

score 98 · Answer 11 · 2026-03-25 17:44:44

98

Views

Prove these can't both be true: $\int_{0}^{\pi} [f(x) - \sin x] ^2 \, dx \le \frac{4}{9}$, $\int_{0}^{\pi} [f(x) - \cos x] ^2 \, dx \le \frac{1}{9}$

Published on 25 Mar 2026 - 17:44

score 31 · Answer 12 · 2026-03-25 17:38:04

31

Views

Two questions regarding Lebesgue measure and integral

Published on 25 Mar 2026 - 17:38

score 51 · Answer 13 · 2026-03-25 11:30:19

51

Views

Maximal domain of function and its derivative if both have Lebesgue > 0

Published on 25 Mar 2026 - 11:30

score 35 · Answer 14 · 2026-03-25 17:43:31

35

Views

application for Integration of non-negative measurable functions

Published on 25 Mar 2026 - 17:43

score 53 · Answer 15 · 2018-04-27 19:50:38

53

Views

Proving Measurable sets

Published on 27 Apr 2018 - 19:50

List Question

How to prove sub-additivity of the Lebesgue measure for open sets?

Failure of Fatou's lemma for limit supremum

Determine whether the following Lebesgue integral exists.

The function defined with the Lebesgue's measure is measurable

Does there exist a Lebesgue measurable set $[0,1] \subset D$ such that $D \neq [0,1]$ and $\lambda ^{*} (D) = 1$?

Equality between volume and surface

Approximating Lebesgue integrable functions with continuous $2\pi$ periodic functions and trig polynomials

Question on the proof that the Cantor set has measure zero

If $B \subset A \subset \mathbb{R}$ such that $m^(B)=0 $. Show that $m^(A \setminus B)=m^*(A)$.

If $A,B \subset \mathbb{R} $ such that $m^*(A)=0 $ and $A \cup B$ is Lebesgue measurable. Show that $B$ is Lebesgue measurable.

Prove these can't both be true: $\int_{0}^{\pi} [f(x) - \sin x] ^2 \, dx \le \frac{4}{9}$, $\int_{0}^{\pi} [f(x) - \cos x] ^2 \, dx \le \frac{1}{9}$

Two questions regarding Lebesgue measure and integral

Maximal domain of function and its derivative if both have Lebesgue > 0

application for Integration of non-negative measurable functions

Proving Measurable sets

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