Question

Prove that $\cos(f(x))$ is not Lebesgue-integrable if $\lim_{\lvert x\rvert\to\infty}f(x)=0$.

score 373 · Answer 1 · 2016-10-07 16:25:15

373

Views

Prove that $\cos(f(x))$ is not Lebesgue-integrable if $\lim_{\lvert x\rvert\to\infty}f(x)=0$.

Published on 07 Oct 2016 - 16:25

score 94 · Answer 2 · 2016-10-07 18:12:46

94

Views

$f \in L_1([0, \infty), \lambda)$, then $\sum_{n=0}^{\infty}f(nt)< \infty$

Published on 07 Oct 2016 - 18:12

score 37 · Answer 3 · 2016-10-07 19:21:14

37

Views

Proof that $ \lim_{r \to 0^+} \frac{m(f(I[a,r]))}{r^n} = |\, \det f'(a)\, | $

Published on 07 Oct 2016 - 19:21

score 516 · Answer 4 · 2016-10-07 23:49:38

516

Views

For lebesgue integrable function $f$, show that there exists a sequence $\{x_n\} \rightarrow \infty $ such that $x_n| f(x_n) | \rightarrow 0$

Published on 07 Oct 2016 - 23:49

score 36 · Answer 5 · 2016-10-08 16:36:37

36

Views

Non-absolutely convergent series are integrals

Published on 08 Oct 2016 - 16:36

score 54 · Answer 6 · 2016-10-09 18:53:47

54

Views

Proving a limit of an integral.

Published on 09 Oct 2016 - 18:53

score 393 · Answer 7 · 2016-10-09 20:53:11

393

Views

Differences (or similarity) between Radon-Nikodym and the Lebesgue-Stieltjes integral?

Published on 09 Oct 2016 - 20:53

score 64 · Answer 8 · 2016-10-11 17:25:13

64

Views

Sequence of functions that converge to zero a.e.

Published on 11 Oct 2016 - 17:25

score 1.7k · Answer 9 · 2016-10-11 21:29:20

1.7k

Views

How do I show that convergence in measure is equivalent to convergence of a certain integral?

Published on 11 Oct 2016 - 21:29

score 141 · Answer 10 · 2016-10-12 16:59:40

141

Views

Equality of integrals of periodic functions for general measure space

Published on 12 Oct 2016 - 16:59

score 34 · Answer 11 · 2016-10-13 14:50:57

34

Views

Strengthen a condition (something like uniform integrability)

Published on 13 Oct 2016 - 14:50

score 220 · Answer 12 · 2026-04-03 23:38:18

220

Views

Equivalence between these two definitions of measurable set and Lebesgue integral constructions

Published on 03 Apr 2026 - 23:38

score 138 · Answer 13 · 2016-10-15 06:02:13

138

Views

A definite integral involving the product of a rational function and a logarithm

Published on 15 Oct 2016 - 6:02

score 214 · Answer 14 · 2016-10-16 11:56:09

214

Views

$\int_{\mathbb{R}^n} \frac{1}{(\lVert x-y \rVert)^{n- \alpha}} d\mu (y) = (n-\alpha)\int _{[0,+\infty[} r^{\alpha - n -1} \mu(B(x,r)) d\lambda (r)$

Published on 16 Oct 2016 - 11:56

score 60 · Answer 15 · 2016-10-17 01:52:17

60

Views

L2 norm in measurable set

Published on 17 Oct 2016 - 1:52

List Question

Prove that $\cos(f(x))$ is not Lebesgue-integrable if $\lim_{\lvert x\rvert\to\infty}f(x)=0$.

$f \in L_1([0, \infty), \lambda)$, then $\sum_{n=0}^{\infty}f(nt)< \infty$

Proof that $ \lim_{r \to 0^+} \frac{m(f(I[a,r]))}{r^n} = |\, \det f'(a)\, | $

For lebesgue integrable function $f$, show that there exists a sequence $\{x_n\} \rightarrow \infty $ such that $x_n| f(x_n) | \rightarrow 0$

Non-absolutely convergent series are integrals

Proving a limit of an integral.

Differences (or similarity) between Radon-Nikodym and the Lebesgue-Stieltjes integral?

Sequence of functions that converge to zero a.e.

How do I show that convergence in measure is equivalent to convergence of a certain integral?

Equality of integrals of periodic functions for general measure space

Strengthen a condition (something like uniform integrability)

Equivalence between these two definitions of measurable set and Lebesgue integral constructions

A definite integral involving the product of a rational function and a logarithm

$\int_{\mathbb{R}^n} \frac{1}{(\lVert x-y \rVert)^{n- \alpha}} d\mu (y) = (n-\alpha)\int _{[0,+\infty[} r^{\alpha - n -1} \mu(B(x,r)) d\lambda (r)$

L2 norm in measurable set

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