Question

Why is $f(x) = \sin(x)$ an element of $L^2(-\pi, \pi)$ not $L^2(a,b)$

score 1.1k · Answer 1 · 2026-04-17 14:29:37

1.1k

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Why is $f(x) = \sin(x)$ an element of $L^2(-\pi, \pi)$ not $L^2(a,b)$

Published on 17 Apr 2026 - 14:29

score 137 · Answer 2 · 2026-04-16 09:04:08

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Show $\int_E {(f_1 + f_2)d\mu } = \int_E {f_1 d\mu } + \int_E {f_2 d\mu } $

Published on 16 Apr 2026 - 9:04

score 1.1k · Answer 3 · 2026-04-11 01:02:27

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What is the cardinality of $L^p(\mathbb R)$, $1 \le p < \infty$?

Published on 11 Apr 2026 - 1:02

score 1.4k · Answer 4 · 2026-04-16 19:44:14

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Let $m$ be Lebesgue measure and $a \in R$. Suppose that $f : R \to R$ is integrable, and $\int_a^xf(y) \, dy = 0$ for all $x$. Then $f = 0$ a.e.

Published on 16 Apr 2026 - 19:44

score 1.3k · Answer 5 · 2026-04-10 00:19:46

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How to prove an inequality of Lebesgue integral?

Published on 10 Apr 2026 - 0:19

score 542 · Answer 6 · 2026-04-07 23:25:39

542

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New definition of Lebesgue integral

Published on 07 Apr 2026 - 23:25

score 68 · Answer 7 · 2026-04-14 22:19:15

68

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How to prove that $\phi_n\uparrow f$, where $f\in R[a,b]$ and $\phi_n$ is like below?

Published on 14 Apr 2026 - 22:19

score 8.4k · Answer 8 · 2026-04-14 23:28:46

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Prove that $f$ is integrable if and only if $\sum^\infty_{n=1} \mu(\{x \in X : f(x) \ge n\}) < \infty$

Published on 14 Apr 2026 - 23:28

score 1.3k · Answer 9 · 2026-04-12 00:51:55

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Questions of an exercise in Lebesgue integral

Published on 12 Apr 2026 - 0:51

score 192 · Answer 10 · 2026-04-12 03:22:18

192

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Questions of 2nd exercise in Lebesgue integral

Published on 12 Apr 2026 - 3:22

score 545 · Answer 11 · 2026-04-10 22:27:44

545

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counter examples for Riemann and Lebesgue integrabilities

Published on 10 Apr 2026 - 22:27

score 35 · Answer 12 · 2026-04-14 20:50:35

35

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Non-Integrability by the comparaison theorem and contradiction

Published on 14 Apr 2026 - 20:50

score 185 · Answer 13 · 2026-04-13 13:25:29

185

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Calculating Lebesgue integral 2

Published on 13 Apr 2026 - 13:25

score 207 · Answer 14 · 2026-04-09 14:28:19

207

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Questions about Lebesgue integral of increasing measurable function converges in measure

Published on 09 Apr 2026 - 14:28

score 186 · Answer 15 · 2026-04-17 13:45:06

186

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Prove that $\limsup_{n \rightarrow \infty} (f_n(x))^{1/n} \leq1 $

Published on 17 Apr 2026 - 13:45

List Question

Why is $f(x) = \sin(x)$ an element of $L^2(-\pi, \pi)$ not $L^2(a,b)$

Show $\int_E {(f_1 + f_2)d\mu } = \int_E {f_1 d\mu } + \int_E {f_2 d\mu } $

What is the cardinality of $L^p(\mathbb R)$, $1 \le p < \infty$?

Let $m$ be Lebesgue measure and $a \in R$. Suppose that $f : R \to R$ is integrable, and $\int_a^xf(y) \, dy = 0$ for all $x$. Then $f = 0$ a.e.

How to prove an inequality of Lebesgue integral?

New definition of Lebesgue integral

How to prove that $\phi_n\uparrow f$, where $f\in R[a,b]$ and $\phi_n$ is like below?

Prove that $f$ is integrable if and only if $\sum^\infty_{n=1} \mu(\{x \in X : f(x) \ge n\}) < \infty$

Questions of an exercise in Lebesgue integral

Questions of 2nd exercise in Lebesgue integral

counter examples for Riemann and Lebesgue integrabilities

Non-Integrability by the comparaison theorem and contradiction

Calculating Lebesgue integral 2

Questions about Lebesgue integral of increasing measurable function converges in measure

Prove that $\limsup_{n \rightarrow \infty} (f_n(x))^{1/n} \leq1 $

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