Question

Could you please recommend some open courses on Real Analysis, mainly about Lebesgue measure and Lebesgue Integral?

score 681 · Answer 1 · 2019-07-15 08:25:46

681

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Could you please recommend some open courses on Real Analysis, mainly about Lebesgue measure and Lebesgue Integral?

Published on 15 Jul 2019 - 8:25

score 63 · Answer 2 · 2019-07-15 14:44:37

63

Views

Showing that $ \int_{0}^{1} |f(x)|^4 \: dx $ is finite.

Published on 15 Jul 2019 - 14:44

score 60 · Answer 3 · 2019-07-15 16:03:51

60

Views

$h_c(x) = \sup \{n^c f(nx):n=1,2,3,\dots\}$ is in $L^1(R)$ if $0<c<1$.

Published on 15 Jul 2019 - 16:03

score 232 · Answer 4 · 2019-07-15 16:09:33

232

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Rudin RCA 7.22. If both $f$ and the maximal function $Mf$ are integrable in $R^k$ then $f=0$ a.e.

Published on 15 Jul 2019 - 16:09

score 135 · Answer 5 · 2019-07-15 19:43:27

135

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Rudin RCA exercise 7.22. $f(x)=x^{-1}(\log x)^{-2}$ is integrable but $\int_0^1 Mf = \infty$.

Published on 15 Jul 2019 - 19:43

score 114 · Answer 6 · 2019-07-16 06:59:21

114

Views

Application of Dominated Convergence

Published on 16 Jul 2019 - 6:59

score 206 · Answer 7 · 2019-07-17 12:59:20

206

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Non-negative integrable functions converging in measure on $[0,1]$ with $ \lim_{k\to\infty}\int_0^1 f_k(x)dx=\int_0^1 f(x)dx .$

Published on 17 Jul 2019 - 12:59

score 235 · Answer 8 · 2019-07-18 22:52:59

235

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Prove for $\epsilon >0$ exists a finite interval $[a,b]$ such that $|\int{f(x)}dx-\int_{a}^b f(x)dx|<\epsilon$

Published on 18 Jul 2019 - 22:52

score 302 · Answer 9 · 2019-07-19 01:26:38

302

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Prove For $\epsilon >0$ exists a $g$ function bounded such that $g$ is integrable and $|f(x)dx-\int{g(x)}dx|<\epsilon$

Published on 19 Jul 2019 - 1:26

score 281 · Answer 10 · 2019-07-19 03:06:17

281

Views

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ Lebesgue integrable. Prove $\int{f(x)dx}=\int{f(x+t)}dx$

Published on 19 Jul 2019 - 3:06

score 37 · Answer 11 · 2019-07-21 04:48:37

37

Views

A 1st order "intermittent" differential equation defined on a disconnected open set

Published on 21 Jul 2019 - 4:48

score 64 · Answer 12 · 2019-07-22 07:18:34

64

Views

If $f$ is minorised by $g$, yet their Lebesgue integrals are equal, then both function must be equal almost everywhere

Published on 22 Jul 2019 - 7:18

score 96 · Answer 13 · 2026-02-23 17:21:08

96

Views

About Monotone Convergence Theorem

Published on 23 Feb 2026 - 17:21

score 154 · Answer 14 · 2019-07-23 15:29:18

154

Views

Limit of an definite integral for parameter going to zero

Published on 23 Jul 2019 - 15:29

score 115 · Answer 15 · 2019-07-23 19:54:07

115

Views

Let $\varepsilon >0$ exist a finite interval $[a,b]$ and a bounded function $h$ such that then $\int_{-\infty}^{\infty}{|f(x)-h(x)|}dx<\varepsilon$

Published on 23 Jul 2019 - 19:54

List Question

Could you please recommend some open courses on Real Analysis, mainly about Lebesgue measure and Lebesgue Integral?

Showing that $ \int_{0}^{1} |f(x)|^4 \: dx $ is finite.

$h_c(x) = \sup \{n^c f(nx):n=1,2,3,\dots\}$ is in $L^1(R)$ if $0<c<1$.

Rudin RCA 7.22. If both $f$ and the maximal function $Mf$ are integrable in $R^k$ then $f=0$ a.e.

Rudin RCA exercise 7.22. $f(x)=x^{-1}(\log x)^{-2}$ is integrable but $\int_0^1 Mf = \infty$.

Application of Dominated Convergence

Non-negative integrable functions converging in measure on $[0,1]$ with $ \lim_{k\to\infty}\int_0^1 f_k(x)dx=\int_0^1 f(x)dx .$

Prove for $\epsilon >0$ exists a finite interval $[a,b]$ such that $|\int{f(x)}dx-\int_{a}^b f(x)dx|<\epsilon$

Prove For $\epsilon >0$ exists a $g$ function bounded such that $g$ is integrable and $|f(x)dx-\int{g(x)}dx|<\epsilon$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ Lebesgue integrable. Prove $\int{f(x)dx}=\int{f(x+t)}dx$

A 1st order "intermittent" differential equation defined on a disconnected open set

If $f$ is minorised by $g$, yet their Lebesgue integrals are equal, then both function must be equal almost everywhere

About Monotone Convergence Theorem

Limit of an definite integral for parameter going to zero

Let $\varepsilon >0$ exist a finite interval $[a,b]$ and a bounded function $h$ such that then $\int_{-\infty}^{\infty}{|f(x)-h(x)|}dx<\varepsilon$

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