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 30 · Answer 1 · 2016-10-04 21:11:00

30

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 04 Oct 2016 - 21:11

score 80 · Answer 2 · 2016-10-05 06:47:01

80

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 05 Oct 2016 - 6:47

score 1.2k · Answer 3 · 2016-10-05 14:44:38

1.2k

Views

Integral Less than Epsilon on Complement of Compact Set

Published on 05 Oct 2016 - 14:44

score 46 · Answer 4 · 2016-10-05 15:07:53

46

Views

Is there still research going on for Lebesgue Integral / Measure

Published on 05 Oct 2016 - 15:07

score 70 · Answer 5 · 2026-03-25 18:48:06

70

Views

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

Published on 25 Mar 2026 - 18:48

score 1.4k · Answer 6 · 2026-03-25 18:48:10

1.4k

Views

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

Published on 25 Mar 2026 - 18:48

score 93 · Answer 7 · 2016-10-06 16:12:10

93

Views

pointwise convergence and $L^1$ norm-boundedness implies what...?

Published on 06 Oct 2016 - 16:12

score 32 · Answer 8 · 2016-10-06 16:48:37

32

Views

$f\in L^2(\mathbb{R})\Rightarrow f\to 0, x\to\pm\infty$?

Published on 06 Oct 2016 - 16:48

score 232 · Answer 9 · 2016-10-06 16:49:08

232

Views

Definite Exponential Integral evaluation

Published on 06 Oct 2016 - 16:49

score 76 · Answer 10 · 2026-03-26 21:10:32

76

Views

Does there exists $g\colon (0,1)^2 \to \mathbb{R}$ such that $\int_0^1 g(x,y) dy = 1$ and there exists $f$ satisfying $f(y) = \int_0^1 g(x,y)f(x) dx$

Published on 26 Mar 2026 - 21:10

score 44 · Answer 11 · 2016-10-07 00:35:54

44

Views

Analysis Convergent Series and Integrability

Published on 07 Oct 2016 - 0:35

score 294 · Answer 12 · 2026-04-06 09:43:48

294

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 06 Apr 2026 - 9:43

score 78 · Answer 13 · 2026-03-25 04:43:33

78

Views

Is the operator $Af = \int_0^1 k(x,y) f(x) dx$ irreducible provided that $Af = f$ has unique solution such that $\int_0^1 f = 1$

Published on 25 Mar 2026 - 4:43

score 82 · Answer 14 · 2026-03-25 18:50:10

82

Views

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

Published on 25 Mar 2026 - 18:50

score 271 · Answer 15 · 2016-10-07 13:27:01

271

Views

Monotone Convergence Theorem for nonpositive functions

Published on 07 Oct 2016 - 13:27

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?

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

pointwise convergence and $L^1$ norm-boundedness implies what...?

$f\in L^2(\mathbb{R})\Rightarrow f\to 0, x\to\pm\infty$?

Definite Exponential Integral evaluation

Does there exists $g\colon (0,1)^2 \to \mathbb{R}$ such that $\int_0^1 g(x,y) dy = 1$ and there exists $f$ satisfying $f(y) = \int_0^1 g(x,y)f(x) dx$

Analysis Convergent Series and Integrability

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

Is the operator $Af = \int_0^1 k(x,y) f(x) dx$ irreducible provided that $Af = f$ has unique solution such that $\int_0^1 f = 1$

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

Monotone Convergence Theorem for nonpositive functions

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