Question

Find $\lim_{k \to \infty}\int_{0}^{\infty}ke^{-kx^2}\arctan(x)dx$

score 195 · Answer 1 · 2026-04-14 20:05:16

195

Views

Find $\lim_{k \to \infty}\int_{0}^{\infty}ke^{-kx^2}\arctan(x)dx$

Published on 14 Apr 2026 - 20:05

score 395 · Answer 2 · 2026-04-13 20:03:32

395

Views

Image of a Lebesgue measurable set under an invertible linear map

Published on 13 Apr 2026 - 20:03

score 231 · Answer 3 · 2026-04-13 21:59:44

231

Views

Measure-dense subset of measurable set $E$

Published on 13 Apr 2026 - 21:59

score 23 · Answer 4 · 2026-04-05 23:28:29

23

Views

Let $g \in C^1[0,\infty)$ and assume $g'$ is bounded and $g(0)=0$. Find $\lim_{n \to \infty}\int_{0}^{\infty}\frac{g(x/n)}{x}e^{-x}$

Published on 05 Apr 2026 - 23:28

score 68 · Answer 5 · 2026-04-14 17:38:10

68

Views

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 14 Apr 2026 - 17:38

score 207 · Answer 6 · 2026-04-08 06:27:26

207

Views

Lebesgue Measure and Integration

Published on 08 Apr 2026 - 6:27

score 170 · Answer 7 · 2026-04-12 05:32:23

170

Views

Find the Lebesgue decomposition of of Lebesgue measure with respect to counting measure

Published on 12 Apr 2026 - 5:32

score 41 · Answer 8 · 2026-04-14 00:59:38

41

Views

Why must measurable sets behave well with all sets?

Published on 14 Apr 2026 - 0:59

score 151 · Answer 9 · 2026-04-14 02:22:54

151

Views

Proposition $4.5$ Royden's Real Analysis

Published on 14 Apr 2026 - 2:22

score 86 · Answer 10 · 2026-04-13 10:41:45

86

Views

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

Published on 13 Apr 2026 - 10:41

score 28 · Answer 11 · 2026-04-06 01:11:22

28

Views

$f_n$ 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 06 Apr 2026 - 1:11

score 58 · Answer 12 · 2026-04-11 10:55:23

58

Views

Can you help on this question on Lebesgue Integrals

Published on 11 Apr 2026 - 10:55

score 94 · Answer 13 · 2026-04-15 12:30:23

94

Views

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 15 Apr 2026 - 12:30

score 278 · Answer 14 · 2026-04-14 20:40:00

278

Views

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

Published on 14 Apr 2026 - 20:40

score 34 · Answer 15 · 2026-04-12 10:26:43

34

Views

How to write a proof for this?

Published on 12 Apr 2026 - 10:26

List Question

Find $\lim_{k \to \infty}\int_{0}^{\infty}ke^{-kx^2}\arctan(x)dx$

Image of a Lebesgue measurable set under an invertible linear map

Measure-dense subset of measurable set $E$

Let $g \in C^1[0,\infty)$ and assume $g'$ is bounded and $g(0)=0$. Find $\lim_{n \to \infty}\int_{0}^{\infty}\frac{g(x/n)}{x}e^{-x}$

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

Find the Lebesgue decomposition of of Lebesgue measure with respect to counting measure

Why must measurable sets behave well with all sets?

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

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

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

How to write a proof for this?

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