Question

Formal convergence of an integral over a descending sequence of domains involving a Lebesgue inegrable function.

score 37 · Answer 1 · 2020-04-15 20:45:13

37

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Formal convergence of an integral over a descending sequence of domains involving a Lebesgue inegrable function.

Published on 15 Apr 2020 - 20:45

score 766 · Answer 2 · 2020-04-15 23:20:52

766

Views

Is there a set $A \subset [0,1]$ such that $\int_{A \times A^\text{c}} \frac{\mathrm{d} x \, \mathrm{d} y}{\lvert x - y\vert}=\infty$?

Published on 15 Apr 2020 - 23:20

score 701 · Answer 3 · 2026-03-30 04:41:56

701

Views

why dominated convergence theorem is related to the superiority of Lebesgue integration (over Riemann integration)?

Published on 30 Mar 2026 - 4:41

score 31 · Answer 4 · 2020-04-16 08:43:07

31

Views

Prove that $f(x)=\sum_{n=1}^\infty \max(0, 1-2^n|x-n|)$ is square integrable on $\mathbb{R}$.

Published on 16 Apr 2020 - 8:43

score 355 · Answer 5 · 2020-04-16 09:15:49

355

Views

I need to prove that A is Lebesgue-measurable

Published on 16 Apr 2020 - 9:15

score 191 · Answer 6 · 2020-04-16 14:20:46

191

Views

Integrable functions with non-integrable limit

Published on 16 Apr 2020 - 14:20

score 271 · Answer 7 · 2020-04-17 03:22:12

271

Views

Integral of function equals integral of measure - Tonelli's Theorem

Published on 17 Apr 2020 - 3:22

score 242 · Answer 8 · 2020-04-17 13:37:56

242

Views

Lebesgue integral of $1/|x|$ on an open $n$-ball

Published on 17 Apr 2020 - 13:37

score 44 · Answer 9 · 2020-04-17 17:50:19

44

Views

Let $X$ be a finite measure space. Assume $0\leq f_n\leq 1$ and $f_n\to1$ in $L^1$ then $f_n \to 1$ a.e

Published on 17 Apr 2020 - 17:50

score 54 · Answer 10 · 2026-02-23 10:20:30

54

Views

Does the limit of this integrand (product of monotonic functions) exist, given the integral is bounded?

Published on 23 Feb 2026 - 10:20

score 35 · Answer 11 · 2020-04-18 00:18:45

35

Views

If $f \in L^1[0,1]$ then there exists $c \in [0,1/2)$ s.t $\int_{c}^{c+1/2}f=1/2\int_{0}^{1}f(x)dx$

Published on 18 Apr 2020 - 0:18

score 34 · Answer 12 · 2020-04-18 00:32:30

34

Views

Let $f$ be Lebesgue measurable on $[0,1]$. Show that $\|f(\sqrt{x})\|_1\leq\frac{7}{6}\|f\|_2 $

Published on 18 Apr 2020 - 0:32

score 45 · Answer 13 · 2020-04-18 01:32:25

45

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Show that $\int_Re^{|f(x)|}dx=1\implies \sup_{1\leq p<\infty}p^{-1}\|f\|_p<\infty$

Published on 18 Apr 2020 - 1:32

score 48 · Answer 14 · 2020-04-18 15:51:57

48

Views

$\int_a^b{f(x)dx}=0$ if $f$ is zero a.e.

Published on 18 Apr 2020 - 15:51

score 43 · Answer 15 · 2020-04-18 17:55:11

43

Views

When considering $\lim_{n \to \infty}\int_{0}^{g(n)}f_n(x)$, $g(n)\to \infty$ does $g$ matter when $f_n(x)$ is dominated?

Published on 18 Apr 2020 - 17:55

List Question

Formal convergence of an integral over a descending sequence of domains involving a Lebesgue inegrable function.

Is there a set $A \subset [0,1]$ such that $\int_{A \times A^\text{c}} \frac{\mathrm{d} x \, \mathrm{d} y}{\lvert x - y\vert}=\infty$?

why dominated convergence theorem is related to the superiority of Lebesgue integration (over Riemann integration)?

Prove that $f(x)=\sum_{n=1}^\infty \max(0, 1-2^n|x-n|)$ is square integrable on $\mathbb{R}$.

I need to prove that A is Lebesgue-measurable

Integrable functions with non-integrable limit

Integral of function equals integral of measure - Tonelli's Theorem

Lebesgue integral of $1/|x|$ on an open $n$-ball

Let $X$ be a finite measure space. Assume $0\leq f_n\leq 1$ and $f_n\to1$ in $L^1$ then $f_n \to 1$ a.e

Does the limit of this integrand (product of monotonic functions) exist, given the integral is bounded?

If $f \in L^1[0,1]$ then there exists $c \in [0,1/2)$ s.t $\int_{c}^{c+1/2}f=1/2\int_{0}^{1}f(x)dx$

Let $f$ be Lebesgue measurable on $[0,1]$. Show that $\|f(\sqrt{x})\|_1\leq\frac{7}{6}\|f\|_2 $

Show that $\int_Re^{|f(x)|}dx=1\implies \sup_{1\leq p<\infty}p^{-1}\|f\|_p<\infty$

$\int_a^b{f(x)dx}=0$ if $f$ is zero a.e.

When considering $\lim_{n \to \infty}\int_{0}^{g(n)}f_n(x)$, $g(n)\to \infty$ does $g$ matter when $f_n(x)$ is dominated?

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