Question

Let $f(x)=\int_{0}^{x}\frac{\sin(t)}{t^{3/2}}dt$ find $\lim_{k \to \infty} \int_{0}^{\infty}k^{3/2}f(x)e^{-kx} dx$

score 66 · Answer 1 · 2020-04-30 04:01:12

66

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Let $f(x)=\int_{0}^{x}\frac{\sin(t)}{t^{3/2}}dt$ find $\lim_{k \to \infty} \int_{0}^{\infty}k^{3/2}f(x)e^{-kx} dx$

Published on 30 Apr 2020 - 4:01

score 205 · Answer 2 · 2020-04-30 14:01:44

205

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Lebesgue Measure and Integration

Published on 30 Apr 2020 - 14:01

score 132 · Answer 3 · 2020-04-30 16:16:38

132

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$\mathbf{R}\cup\{\infty\}$ vs $[-\infty, +\infty]$ in the Lebesgue integration theory

Published on 30 Apr 2020 - 16:16

score 148 · Answer 4 · 2020-04-30 18:24:27

148

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Proposition $4.5$ Royden's Real Analysis

Published on 30 Apr 2020 - 18:24

score 83 · Answer 5 · 2020-04-30 18:50:50

83

Views

Proving that $f=1$ a.e. when its integral on $\left[a,b\right]$ is $b-a$

Published on 30 Apr 2020 - 18:50

score 27 · Answer 6 · 2020-05-01 01:03:42

27

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$f_n$ AC functions s.t $\sum\int_{0}^{1}|f'_n(x)|dx<\infty$ Prove $\sum f_n(x)=f(x)$ converges everywhere and that $f'(x)=\sum f'_n(x)$

Published on 01 May 2020 - 1:03

score 57 · Answer 7 · 2026-02-23 13:02:18

57

Views

Can you help on this question on Lebesgue Integrals

Published on 23 Feb 2026 - 13:02

score 199 · Answer 8 · 2020-05-01 06:04:59

199

Views

Interchanging an integrand and a measure

Published on 01 May 2020 - 6:04

score 85 · Answer 9 · 2020-05-01 09:26:29

85

Views

Use the monotone convergence theorem to calculate $\int_{-\infty}^{+\infty}e^{-|x|}d\bar{\lambda} $

Published on 01 May 2020 - 9:26

score 71 · Answer 10 · 2026-03-28 23:59:50

71

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Show that $f=0$ ae on [0,1] if $\int_E f \le m(E)^2$ for all measurable $E \subset [0,1]$.

Published on 28 Mar 2026 - 23:59

score 92 · Answer 11 · 2020-05-02 00:40:38

92

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Prove that $f_n \to 0$ in measure on $[0,1]$ $\iff$ $\lim_{n \to \infty}\int_{0}^{1}\mathbb{e}^{-|f_n(x)|^2} =1$

Published on 02 May 2020 - 0:40

score 55 · Answer 12 · 2026-03-29 05:12:08

55

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Prove "Measure theoretic" Fatous lemma. Let $f\geq0$ Prove that $\mu(x | \liminf f_n(x)>t)\leq\liminf \mu(x | f_n(x)>t)$ for each $t>0$

Published on 29 Mar 2026 - 5:12

score 276 · Answer 13 · 2020-05-02 04:19:37

276

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Weak convergence in $L^1$ $f_n \to f$ and bounded $L^2$ norm implies the limit is in $L^2$

Published on 02 May 2020 - 4:19

score 51 · Answer 14 · 2020-05-02 22:37:53

51

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Let $f$ be absolutely continuous on $[0,a]$ for all $a>0$ and $f(0)=0$ the show that $\int_{0}^{x}|f(t)f'(t)|\leq\frac{1}{2}(\int_{0}^{x}|f'(t)|)^2$.

Published on 02 May 2020 - 22:37

score 44 · Answer 15 · 2020-05-03 01:04:55

44

Views

if $f_n$ is point wise bounded measurable sequence then it is uniformly bounded on certain sets in $[0,1]$

Published on 03 May 2020 - 1:04

List Question

Let $f(x)=\int_{0}^{x}\frac{\sin(t)}{t^{3/2}}dt$ find $\lim_{k \to \infty} \int_{0}^{\infty}k^{3/2}f(x)e^{-kx} dx$

Lebesgue Measure and Integration

$\mathbf{R}\cup\{\infty\}$ vs $[-\infty, +\infty]$ in the Lebesgue integration theory

Proposition $4.5$ Royden's Real Analysis

Proving that $f=1$ a.e. when its integral on $\left[a,b\right]$ is $b-a$

$f_n$ AC functions s.t $\sum\int_{0}^{1}|f'_n(x)|dx<\infty$ Prove $\sum f_n(x)=f(x)$ converges everywhere and that $f'(x)=\sum f'_n(x)$

Can you help on this question on Lebesgue Integrals

Interchanging an integrand and a measure

Use the monotone convergence theorem to calculate $\int_{-\infty}^{+\infty}e^{-|x|}d\bar{\lambda} $

Show that $f=0$ ae on [0,1] if $\int_E f \le m(E)^2$ for all measurable $E \subset [0,1]$.

Prove that $f_n \to 0$ in measure on $[0,1]$ $\iff$ $\lim_{n \to \infty}\int_{0}^{1}\mathbb{e}^{-|f_n(x)|^2} =1$

Prove "Measure theoretic" Fatous lemma. Let $f\geq0$ Prove that $\mu(x | \liminf f_n(x)>t)\leq\liminf \mu(x | f_n(x)>t)$ for each $t>0$

Weak convergence in $L^1$ $f_n \to f$ and bounded $L^2$ norm implies the limit is in $L^2$

Let $f$ be absolutely continuous on $[0,a]$ for all $a>0$ and $f(0)=0$ the show that $\int_{0}^{x}|f(t)f'(t)|\leq\frac{1}{2}(\int_{0}^{x}|f'(t)|)^2$.

if $f_n$ is point wise bounded measurable sequence then it is uniformly bounded on certain sets in $[0,1]$

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