List questions from lebesgue-integral

Published on 27 Aug 2014 - 9:55

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About a $\sigma$-finite measure

Published on 28 Aug 2014 - 14:15

140

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$L^{1}$ norm of a horizontally shifted measurable function

Published on 29 Aug 2014 - 3:27

592

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Show that there exists an $\sum$-measurable simple function $\phi$ such that: $\int |f-\phi| d\mu <\epsilon$

Published on 29 Aug 2014 - 17:47

84

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Does it make sense to talk about the integral of measurable functions that are not absolutely integrable?

Published on 29 Aug 2014 - 18:28

447

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Fubini's theorem for complete $\sigma$-algebras vs. non-complete $\sigma$-algebras

Published on 30 Aug 2014 - 3:49

174

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Consider the sequence $f_n(x) = (\sin(πnx))^n , n = 1, 2, ...,$ on the interval $[0,1].$

Published on 31 Aug 2014 - 1:01

837

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Proof of FTC, continuity part, for Lebesgue integrable functions

Published on 31 Aug 2014 - 6:02

93

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Prove that there exists a continuous increasing function $\sigma(x)$ on $I$ such that $\sigma'(x) = + \infty$ for every $x_0 \in E$.

Published on 01 Sep 2014 - 18:00

73

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(a) Prove that $f_n(x) → f(x)$ uniformly on $E$ as $n → ∞.$

Published on 02 Sep 2014 - 0:23

1k

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Example with almost every convergence where the dominated convergence theorem fails

Published on 28 Mar 2026 - 3:00

237

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Prove that $\int_0^1f'(x)dx \leq f(1) - f(0)$.

Published on 03 Sep 2014 - 21:58

583

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Proof monotone convergence theorem, why do they use this lim sup?

Published on 04 Sep 2014 - 9:40

74

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Lebesgue's integral and measure

Published on 04 Sep 2014 - 17:06

1.1k

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Holder inequality (reverse or equality?)

Published on 05 Sep 2014 - 17:32

List Question

Lebesuge integrable function always bounded?

About a $\sigma$-finite measure

$L^{1}$ norm of a horizontally shifted measurable function

Show that there exists an $\sum$-measurable simple function $\phi$ such that: $\int |f-\phi| d\mu <\epsilon$

Does it make sense to talk about the integral of measurable functions that are not absolutely integrable?

Fubini's theorem for complete $\sigma$-algebras vs. non-complete $\sigma$-algebras

Consider the sequence $f_n(x) = (\sin(πnx))^n , n = 1, 2, ...,$ on the interval $[0,1].$

Proof of FTC, continuity part, for Lebesgue integrable functions

Prove that there exists a continuous increasing function $\sigma(x)$ on $I$ such that $\sigma'(x) = + \infty$ for every $x_0 \in E$.

(a) Prove that $f_n(x) → f(x)$ uniformly on $E$ as $n → ∞.$

Example with almost every convergence where the dominated convergence theorem fails

Prove that $\int_0^1f'(x)dx \leq f(1) - f(0)$.

Proof monotone convergence theorem, why do they use this lim sup?

Lebesgue's integral and measure

Holder inequality (reverse or equality?)

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