Question

Prove the integral $\int_{1}^{\infty} \ln(1+ \frac{(\sin x)^n}{x^c})dx$ exist as a finite Lebesgue integral

score 256 · Answer 1 · 2026-04-16 12:20:28

256

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Prove the integral $\int_{1}^{\infty} \ln(1+ \frac{(\sin x)^n}{x^c})dx$ exist as a finite Lebesgue integral

Published on 16 Apr 2026 - 12:20

score 1.9k · Answer 2 · 2026-04-17 08:38:06

1.9k

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If $f$ is Lebesgue integrable on $ R$ then $\lim_{h\to 0} \int_R |f(x+h)-f(x)|dx=0? $

Published on 17 Apr 2026 - 8:38

score 59 · Answer 3 · 2026-04-17 11:11:05

59

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"$f\ge 0$, $g>0$, $fg\in L^1(\mathbb{R})$ and $g\notin L^1(\mathbb{R})$" implies "$f$ is integrable over $[-r,r]$"?

Published on 17 Apr 2026 - 11:11

score 976 · Answer 4 · 2026-04-13 05:21:11

976

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Question on Lebesgue point integration

Published on 13 Apr 2026 - 5:21

score 83 · Answer 5 · 2026-04-16 12:05:38

83

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If sum of a set of increasing functions defined on $[a,b]$ is convergent, will this sum be Lebesgue measurable?

Published on 16 Apr 2026 - 12:05

score 936 · Answer 6 · 2026-04-12 11:08:41

936

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Show that the Fubini Tonelli theorem does not work for this function

Published on 12 Apr 2026 - 11:08

score 376 · Answer 7 · 2026-04-14 08:08:32

376

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Is this theorem an extension of Scheffé Lemma

Published on 14 Apr 2026 - 8:08

score 72 · Answer 8 · 2026-04-16 11:34:25

72

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Show that the function $g(u)=\int_{-\infty}^{\infty} \frac{x^n e^{ux}}{e^x+1}dx$ is differentiable in $(0,1)$

Published on 16 Apr 2026 - 11:34

score 2.5k · Answer 9 · 2026-04-12 15:34:13

2.5k

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Sufficient conditions for the Hardy-Littlewood Maximal function $M(f)$ being continuous

Published on 12 Apr 2026 - 15:34

score 4.1k · Answer 10 · 2026-04-09 22:06:07

4.1k

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The Fundamental Theorem of Calculus (for the lebesgue integral)

Published on 09 Apr 2026 - 22:06

score 114 · Answer 11 · 2026-04-14 10:36:26

114

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I am seeking a solution using Lebesgue Differentiation Theorem to prove that $|S|=0$, and $|\cdot|$ means Lebesgue measure.

Published on 14 Apr 2026 - 10:36

score 60 · Answer 12 · 2026-04-12 14:20:56

60

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Exponential and convergence in $L^2$ bis

Published on 12 Apr 2026 - 14:20

score 57 · Answer 13 · 2026-04-10 20:02:28

57

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A question about Lebesgue Integrals

Published on 10 Apr 2026 - 20:02

score 226 · Answer 14 · 2026-04-10 22:44:57

226

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A bounded Lebesgue measurable function and integral.

Published on 10 Apr 2026 - 22:44

score 326 · Answer 15 · 2026-04-09 10:01:49

326

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Lebesgue Integral definition In $\mathbb{R}^n$

Published on 09 Apr 2026 - 10:01

List Question

Prove the integral $\int_{1}^{\infty} \ln(1+ \frac{(\sin x)^n}{x^c})dx$ exist as a finite Lebesgue integral

If $f$ is Lebesgue integrable on $ R$ then $\lim_{h\to 0} \int_R |f(x+h)-f(x)|dx=0? $

"$f\ge 0$, $g>0$, $fg\in L^1(\mathbb{R})$ and $g\notin L^1(\mathbb{R})$" implies "$f$ is integrable over $[-r,r]$"?

Question on Lebesgue point integration

If sum of a set of increasing functions defined on $[a,b]$ is convergent, will this sum be Lebesgue measurable?

Show that the Fubini Tonelli theorem does not work for this function

Is this theorem an extension of Scheffé Lemma

Show that the function $g(u)=\int_{-\infty}^{\infty} \frac{x^n e^{ux}}{e^x+1}dx$ is differentiable in $(0,1)$

Sufficient conditions for the Hardy-Littlewood Maximal function $M(f)$ being continuous

The Fundamental Theorem of Calculus (for the lebesgue integral)

I am seeking a solution using Lebesgue Differentiation Theorem to prove that $|S|=0$, and $|\cdot|$ means Lebesgue measure.

Exponential and convergence in $L^2$ bis

A question about Lebesgue Integrals

A bounded Lebesgue measurable function and integral.

Lebesgue Integral definition In $\mathbb{R}^n$

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