Question

How Lebesgue integration solved the problem of a function being integrable but its limit is not integrable?

score 223 · Answer 1 · 2020-06-18 15:23:16

223

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How Lebesgue integration solved the problem of a function being integrable but its limit is not integrable?

Published on 18 Jun 2020 - 15:23

score 94 · Answer 2 · 2026-04-06 15:43:25

94

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$f_{n} \overset{\left\lVert .\right\rVert{.}_{1}}{\longmapsto} 0$ in $L^{1}([0,1])$ but not converging to $0$ almost everywhere.

Published on 06 Apr 2026 - 15:43

score 1.4k · Answer 3 · 2020-06-18 20:51:33

1.4k

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Show that the function $f(x)g(x)$ is integrable.

Published on 18 Jun 2020 - 20:51

score 1.2k · Answer 4 · 2020-06-19 00:14:33

1.2k

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Proof that $f$ is Lebesgue-integrable in $[0,1]$.

Published on 19 Jun 2020 - 0:14

score 62 · Answer 5 · 2020-06-19 02:25:31

62

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Integrability of $f(x,y)=(1-x)^{a}$ on $D=\{(x,y):0<x^2+y^2<1\}$

Published on 19 Jun 2020 - 2:25

score 67 · Answer 6 · 2020-06-19 04:31:56

67

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A question in proof of Apostol ( Mathematical Analysis) in Theorem 10.27

Published on 19 Jun 2020 - 4:31

score 71 · Answer 7 · 2020-06-19 05:34:53

71

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A question in Proof of Theorem 10.27 of Apostol Mathematical Analysis

Published on 19 Jun 2020 - 5:34

score 54 · Answer 8 · 2020-06-19 06:20:21

54

Views

An integrable function $f$ on $\Bbb R$ satisfying $\lim_{h\to 0}\int_\Bbb R \frac{|f(x+h)-f(x)|}{h}dx=0$ must be constant

Published on 19 Jun 2020 - 6:20

score 14 · Answer 9 · 2026-03-03 05:22:31

14

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if $y,z$ converges to zero, does $yz \rightarrow 0 \,\, \text{in} \, L^2(\Omega)$ as well?

Published on 03 Mar 2026 - 5:22

score 1k · Answer 10 · 2026-04-03 05:24:38

1k

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Proving a subset of $H^1(\mathbb{R}^d)$ is compactly embedded in $L^2(\mathbb{R}^d)$.

Published on 03 Apr 2026 - 5:24

score 260 · Answer 11 · 2020-06-19 11:29:56

260

Views

Application of of Lebesgue integral properties in the scope of simple functions

Published on 19 Jun 2020 - 11:29

score 86 · Answer 12 · 2026-04-06 16:20:21

86

Views

Show $ f_n\xrightarrow{L^1}f\iff f_n\to f \text{ in measure} \iff f_n\to f \text{ almost uniformly} \iff f_n\to f \text{ a.e.}$

Published on 06 Apr 2026 - 16:20

score 59 · Answer 13 · 2026-04-01 11:59:54

59

Views

A question in functional equation of Gamma Function

Published on 01 Apr 2026 - 11:59

score 151 · Answer 14 · 2020-06-19 14:31:54

151

Views

Show that the function $f(x)g(x)$ is integrable(2).

Published on 19 Jun 2020 - 14:31

score 98 · Answer 15 · 2026-02-23 06:57:21

98

Views

Prove that $\lim_{\lambda\rightarrow\infty} \frac{1}{\lambda}\int_0^\lambda\int_0^xf(y)\,dy\,dx = \int_0^\infty f(x)\,dx$

Published on 23 Feb 2026 - 6:57

List Question

How Lebesgue integration solved the problem of a function being integrable but its limit is not integrable?

$f_{n} \overset{\left\lVert .\right\rVert{.}_{1}}{\longmapsto} 0$ in $L^{1}([0,1])$ but not converging to $0$ almost everywhere.

Show that the function $f(x)g(x)$ is integrable.

Proof that $f$ is Lebesgue-integrable in $[0,1]$.

Integrability of $f(x,y)=(1-x)^{a}$ on $D=\{(x,y):0<x^2+y^2<1\}$

A question in proof of Apostol ( Mathematical Analysis) in Theorem 10.27

A question in Proof of Theorem 10.27 of Apostol Mathematical Analysis

An integrable function $f$ on $\Bbb R$ satisfying $\lim_{h\to 0}\int_\Bbb R \frac{|f(x+h)-f(x)|}{h}dx=0$ must be constant

if $y,z$ converges to zero, does $yz \rightarrow 0 \,\, \text{in} \, L^2(\Omega)$ as well?

Proving a subset of $H^1(\mathbb{R}^d)$ is compactly embedded in $L^2(\mathbb{R}^d)$.

Application of of Lebesgue integral properties in the scope of simple functions

Show $ f_n\xrightarrow{L^1}f\iff f_n\to f \text{ in measure} \iff f_n\to f \text{ almost uniformly} \iff f_n\to f \text{ a.e.}$

A question in functional equation of Gamma Function

Show that the function $f(x)g(x)$ is integrable(2).

Prove that $\lim_{\lambda\rightarrow\infty} \frac{1}{\lambda}\int_0^\lambda\int_0^xf(y)\,dy\,dx = \int_0^\infty f(x)\,dx$

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