Question

Checking if a parameter dependent integral is differentiable

score 172 · Answer 1 · 2018-04-10 18:10:22

172

Views

Checking if a parameter dependent integral is differentiable

Published on 10 Apr 2018 - 18:10

score 123 · Answer 2 · 2018-04-10 18:59:02

123

Views

Lebesgue integral of non-negative measurable sequence of functions (not monotone)

Published on 10 Apr 2018 - 18:59

score 117 · Answer 3 · 2018-04-10 19:03:07

117

Views

Prove $ \ \int_E f(x)dx=\int_E f(x+t)dx \ $

Published on 10 Apr 2018 - 19:03

score 79 · Answer 4 · 2018-04-11 17:00:17

79

Views

Find $\lim_{n\to \infty}\int_1^{\infty}\frac{n}{1+x^n}dx$

Published on 11 Apr 2018 - 17:00

score 97 · Answer 5 · 2018-04-11 19:18:53

97

Views

Showing that $u(x)=(1+x^2)^{-k/2} (\ln(2+x^2))^{-1}$ belongs to $\mathcal W^{1,p}(\Bbb R)$.

Published on 11 Apr 2018 - 19:18

score 113 · Answer 6 · 2018-04-12 14:56:14

113

Views

Prove that $\frac{g(x)}{\ln(1+x)}$ is integrable over $[0,1].$

Published on 12 Apr 2018 - 14:56

score 277 · Answer 7 · 2018-04-12 15:41:55

277

Views

How to change order of integration with changing limits (Lebesgue integral)

Published on 12 Apr 2018 - 15:41

score 64 · Answer 8 · 2018-04-12 20:24:36

64

Views

If $|f(x+y) - f(x)| \leq g(x)|y|$ for some $g \in L^1(\mathbb{R})$, then $\int_a^b f'(x) dx = f(b) - f(a)$ if $f'(a)$ and $f'(b)$ exist

Published on 12 Apr 2018 - 20:24

score 361 · Answer 9 · 2018-04-13 12:04:16

361

Views

Stein-Shakarchi Real ananlysis problem 2.1: a function has integration 0 in any subset of a given interval.

Published on 13 Apr 2018 - 12:04

score 69 · Answer 10 · 2018-04-13 21:46:42

69

Views

$\lim_{N \rightarrow \infty} \int_{\{x \in E : |f_n(x)|>N\}}|f_n(x)|dx = 0.$

Published on 13 Apr 2018 - 21:46

score 108 · Answer 11 · 2018-04-14 17:52:40

108

Views

$F(x):=\int_{[a,x)} f(t) \lambda(dt)$ is absolutely continuous for all $f \in L^1(\lambda)$.

Published on 14 Apr 2018 - 17:52

score 12.2k · Answer 12 · 2018-04-14 23:00:58

12.2k

Views

Prove that $ f = 0 $ almost everywhere on $ E $ iff $ \int_{E} f = 0 $

Published on 14 Apr 2018 - 23:00

score 116 · Answer 13 · 2018-04-15 11:04:22

116

Views

Properity about Lebesgue integral with measurable and non-negative functions on measurable and bounded set.

Published on 15 Apr 2018 - 11:04

score 42 · Answer 14 · 2018-04-16 12:01:05

42

Views

Cauchy functional with local Lebesgue integrability

Published on 16 Apr 2018 - 12:01

score 303 · Answer 15 · 2026-03-22 20:13:17

303

Views

Inequality of p-norm for Hardy-Littlewood maximum function

Published on 22 Mar 2026 - 20:13

List Question

Checking if a parameter dependent integral is differentiable

Lebesgue integral of non-negative measurable sequence of functions (not monotone)

Prove $ \ \int_E f(x)dx=\int_E f(x+t)dx \ $

Find $\lim_{n\to \infty}\int_1^{\infty}\frac{n}{1+x^n}dx$

Showing that $u(x)=(1+x^2)^{-k/2} (\ln(2+x^2))^{-1}$ belongs to $\mathcal W^{1,p}(\Bbb R)$.

Prove that $\frac{g(x)}{\ln(1+x)}$ is integrable over $[0,1].$

How to change order of integration with changing limits (Lebesgue integral)

If $|f(x+y) - f(x)| \leq g(x)|y|$ for some $g \in L^1(\mathbb{R})$, then $\int_a^b f'(x) dx = f(b) - f(a)$ if $f'(a)$ and $f'(b)$ exist

Stein-Shakarchi Real ananlysis problem 2.1: a function has integration 0 in any subset of a given interval.

$\lim_{N \rightarrow \infty} \int_{\{x \in E : |f_n(x)|>N\}}|f_n(x)|dx = 0.$

$F(x):=\int_{[a,x)} f(t) \lambda(dt)$ is absolutely continuous for all $f \in L^1(\lambda)$.

Prove that $ f = 0 $ almost everywhere on $ E $ iff $ \int_{E} f = 0 $

Properity about Lebesgue integral with measurable and non-negative functions on measurable and bounded set.

Cauchy functional with local Lebesgue integrability

Inequality of p-norm for Hardy-Littlewood maximum function

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