Question

Hint needed on measure theory question: Approximation by semisimple functions

score 746 · Answer 1 · 2026-04-08 16:40:56

746

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Hint needed on measure theory question: Approximation by semisimple functions

Published on 08 Apr 2026 - 16:40

score 578 · Answer 2 · 2026-04-17 09:29:27

578

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Is the sequence of functions $f_n=\chi_{[n,n+1]}$ uniformly integrable?

Published on 17 Apr 2026 - 9:29

score 375 · Answer 3 · 2026-04-13 20:27:05

375

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Whether $\varphi(E)$ is a Jordan measurable set?

Published on 13 Apr 2026 - 20:27

score 1k · Answer 4 · 2026-04-11 13:38:54

1k

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Can we split up any lebesgue integral into sum of integrals?

Published on 11 Apr 2026 - 13:38

score 203 · Answer 5 · 2026-04-15 04:51:08

203

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proving $|\int_X{}fd\mu|\leq\int_X{}|f|d\mu$?

Published on 15 Apr 2026 - 4:51

score 1.2k · Answer 6 · 2026-04-14 10:40:48

1.2k

Views

If$f_n$ a sequence of integrable functions on a set of finite measure which converge to f uniformly, then the limit and integral can be swapped.

Published on 14 Apr 2026 - 10:40

score 186 · Answer 7 · 2026-04-16 01:13:31

186

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If $\psi$ is a nonnegative simple function check that $\int \psi = \sup\{\int\phi:0\le\phi\le\psi, \phi\, \text{simple and integrable}\}$

Published on 16 Apr 2026 - 1:13

score 46 · Answer 8 · 2026-04-16 11:59:29

46

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$\int_{-\infty}^\infty f=\lim_{n\to\infty} \int_{-n}^nf=\lim_{n\to\infty} \int_{\{f\ge(1/n)\}}f=\lim_{n\to\infty} \int_{\{f\le n\}}f$

Published on 16 Apr 2026 - 11:59

score 1.3k · Answer 9 · 2026-04-16 09:21:08

1.3k

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Let $f$ be measurable with $f$>0 a.e. If $ \int_E f$=0 for some measurable set E, show that m(E)=0.

Published on 16 Apr 2026 - 9:21

score 114 · Answer 10 · 2026-04-09 15:18:35

114

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Question about change of variables in limits of integrals using dominated convergence

Published on 09 Apr 2026 - 15:18

score 98 · Answer 11 · 2026-04-14 12:58:03

98

Views

About the sigma additivity of volume of half-open cubes in $\mathbb{R}^n$

Published on 14 Apr 2026 - 12:58

score 1.6k · Answer 12 · 2026-04-12 08:15:20

1.6k

Views

Correct definition of a measure.

Published on 12 Apr 2026 - 8:15

score 36 · Answer 13 · 2026-04-14 09:58:03

36

Views

Showing $e^{-n|1- \sin x|}$ converges to $0$ in measure

Published on 14 Apr 2026 - 9:58

score 7.5k · Answer 14 · 2026-04-12 08:41:04

7.5k

Views

What is the measure of the set of numbers in $[0,1]$ whose decimal expansions do not contain $5$?

Published on 12 Apr 2026 - 8:41

score 333 · Answer 15 · 2026-04-16 21:15:23

333

Views

Showing that the $L^1$-limit of a sequence of uniformly bounded A.C. functions is A.C.

Published on 16 Apr 2026 - 21:15

List Question

Hint needed on measure theory question: Approximation by semisimple functions

Is the sequence of functions $f_n=\chi_{[n,n+1]}$ uniformly integrable?

Whether $\varphi(E)$ is a Jordan measurable set?

Can we split up any lebesgue integral into sum of integrals?

proving $|\int_X{}fd\mu|\leq\int_X{}|f|d\mu$?

If$f_n$ a sequence of integrable functions on a set of finite measure which converge to f uniformly, then the limit and integral can be swapped.

If $\psi$ is a nonnegative simple function check that $\int \psi = \sup\{\int\phi:0\le\phi\le\psi, \phi\, \text{simple and integrable}\}$

$\int_{-\infty}^\infty f=\lim_{n\to\infty} \int_{-n}^nf=\lim_{n\to\infty} \int_{\{f\ge(1/n)\}}f=\lim_{n\to\infty} \int_{\{f\le n\}}f$

Let $f$ be measurable with $f$>0 a.e. If $ \int_E f$=0 for some measurable set E, show that m(E)=0.

Question about change of variables in limits of integrals using dominated convergence

About the sigma additivity of volume of half-open cubes in $\mathbb{R}^n$

Correct definition of a measure.

Showing $e^{-n|1- \sin x|}$ converges to $0$ in measure

What is the measure of the set of numbers in $[0,1]$ whose decimal expansions do not contain $5$?

Showing that the $L^1$-limit of a sequence of uniformly bounded A.C. functions is A.C.

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