Question

Show that the family $\left\{A\in\mathcal{S}:\mu(A)=0 \text{ or } \mu(A^c)=0\right\}$ is the $\mu$-trivial $\sigma$-algebra.

score 32 · Answer 1 · 2026-04-14 18:47:09

32

Views

Show that the family $\left\{A\in\mathcal{S}:\mu(A)=0 \text{ or } \mu(A^c)=0\right\}$ is the $\mu$-trivial $\sigma$-algebra.

Published on 14 Apr 2026 - 18:47

score 83 · Answer 2 · 2026-04-16 09:51:45

83

Views

If $f\ge0$ and $ \int_{0}^{1} f(x)^kdx$ does not depend on $k\geq 1$ then $f=\mathbf 1_A$ almost everywhere, for some measurable subset $A$ of $[0,1]$

Published on 16 Apr 2026 - 9:51

score 1.2k · Answer 3 · 2026-04-12 06:51:54

1.2k

Views

Integral Less than Epsilon on Complement of Compact Set

Published on 12 Apr 2026 - 6:51

score 49 · Answer 4 · 2026-04-11 12:58:17

49

Views

Is there still research going on for Lebesgue Integral / Measure

Published on 11 Apr 2026 - 12:58

score 72 · Answer 5 · 2026-04-12 04:51:21

72

Views

Why does $f\in \mathcal{L}_+^0$ and $\int fd\mu=0\implies f=0$ almost everywhere?

Published on 12 Apr 2026 - 4:51

score 89 · Answer 6 · 2026-04-15 14:49:11

89

Views

If $\int _c^d f(x)\text{dx}=0$ for all $a\le c<d\le b$ then $f=0a.e.$

Published on 15 Apr 2026 - 14:49

score 85 · Answer 7 · 2026-04-15 19:28:00

85

Views

Does $f$ measurable iff there is a measurable $G$ s.t. $f=1_G$?

Published on 15 Apr 2026 - 19:28

score 1.4k · Answer 8 · 2026-04-16 07:43:59

1.4k

Views

version of the dominated convergence theorem where the almost-everywhere convergence is used

Published on 16 Apr 2026 - 7:43

score 528 · Answer 9 · 2026-04-10 15:18:14

528

Views

Questions about construction of Lebesgue measure in Rudin's RCA

Published on 10 Apr 2026 - 15:18

score 47 · Answer 10 · 2026-04-09 17:33:03

47

Views

Analysis Convergent Series and Integrability

Published on 09 Apr 2026 - 17:33

score 73 · Answer 11 · 2026-04-17 06:32:03

73

Views

For a d-dimensional d.f. $F$ prove $F(x_1,x_2,...,x_d)$ is nondecreasing in each variable?

Published on 17 Apr 2026 - 6:32

score 295 · Answer 12 · 2026-04-17 02:46:18

295

Views

Prove that the sequence $cos(nx+a_n)$ can not tend to $0$ for all x on a set of positive measure E

Published on 17 Apr 2026 - 2:46

score 58 · Answer 13 · 2026-04-17 16:03:37

58

Views

Prove that there exists $c ∈ [0, 1]$ such that $A ∩ (c + J) = ∅,$

Published on 17 Apr 2026 - 16:03

score 84 · Answer 14 · 2026-04-15 23:18:50

84

Views

If $f\in \mathcal{L}_+^0$ and $\int fd\mu=0$ show that $f=0$ almost everywhere

Published on 15 Apr 2026 - 23:18

score 72 · Answer 15 · 2026-04-15 09:59:11

72

Views

An estimate of measure of a set

Published on 15 Apr 2026 - 9:59

List Question

Show that the family $\left\{A\in\mathcal{S}:\mu(A)=0 \text{ or } \mu(A^c)=0\right\}$ is the $\mu$-trivial $\sigma$-algebra.

If $f\ge0$ and $ \int_{0}^{1} f(x)^kdx$ does not depend on $k\geq 1$ then $f=\mathbf 1_A$ almost everywhere, for some measurable subset $A$ of $[0,1]$

Integral Less than Epsilon on Complement of Compact Set

Is there still research going on for Lebesgue Integral / Measure

Why does $f\in \mathcal{L}_+^0$ and $\int fd\mu=0\implies f=0$ almost everywhere?

If $\int _c^d f(x)\text{dx}=0$ for all $a\le c<d\le b$ then $f=0a.e.$

Does $f$ measurable iff there is a measurable $G$ s.t. $f=1_G$?

version of the dominated convergence theorem where the almost-everywhere convergence is used

Questions about construction of Lebesgue measure in Rudin's RCA

Analysis Convergent Series and Integrability

For a d-dimensional d.f. $F$ prove $F(x_1,x_2,...,x_d)$ is nondecreasing in each variable?

Prove that the sequence $cos(nx+a_n)$ can not tend to $0$ for all x on a set of positive measure E

Prove that there exists $c ∈ [0, 1]$ such that $A ∩ (c + J) = ∅,$

If $f\in \mathcal{L}_+^0$ and $\int fd\mu=0$ show that $f=0$ almost everywhere

An estimate of measure of a set

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