Question

If $f_n \rightarrow f$ is a sequence of $L^p$ and $g_n \rightarrow g$ a bounded sequence of $L^{\infty}$ then $f_ng_n \rightarrow fg$ in $L^p$

score 390 · Answer 1 · 2026-02-23 01:23:46

390

Views

If $f_n \rightarrow f$ is a sequence of $L^p$ and $g_n \rightarrow g$ a bounded sequence of $L^{\infty}$ then $f_ng_n \rightarrow fg$ in $L^p$

Published on 23 Feb 2026 - 1:23

score 320 · Answer 2 · 2018-05-12 16:00:13

320

Views

Show that $\lim_{n\to \infty}\int_{0}^{1}g(x)f(nx)dx = \Big(\int_{0}^{1}g(x)dx\Big)\Big(\int_{0}^{1}f(x)dx\Big)$

Published on 12 May 2018 - 16:00

score 233 · Answer 3 · 2018-05-12 19:15:53

233

Views

Defined integral over a closed and bounded subset of $\mathbb{R}$

Published on 12 May 2018 - 19:15

score 879 · Answer 4 · 2018-05-14 07:29:21

879

Views

Absolute continuity of probability measure wrt Lebesgue measure

Published on 14 May 2018 - 7:29

score 182 · Answer 5 · 2018-05-14 10:13:18

182

Views

Does an integrable function have to have a point of continuity?

Published on 14 May 2018 - 10:13

score 90 · Answer 6 · 2026-03-25 20:42:05

90

Views

What should I use in order to get the required function $g \in L^1$?

Published on 25 Mar 2026 - 20:42

score 57 · Answer 7 · 2018-05-17 16:50:43

57

Views

Show that $ \sum n|A_n| $ converges

Published on 17 May 2018 - 16:50

score 78 · Answer 8 · 2018-05-18 01:47:58

78

Views

How the function $f_n$ is a measurable function?

Published on 18 May 2018 - 1:47

score 71 · Answer 9 · 2018-05-18 19:04:57

71

Views

The measure of a function defined on digits

Published on 18 May 2018 - 19:04

score 52 · Answer 10 · 2018-05-18 23:47:24

52

Views

If $f : R \rightarrow R \cup \{- ∞, ∞\}$ is continuous then f is measurable.

Published on 18 May 2018 - 23:47

score 855 · Answer 11 · 2026-03-25 19:04:39

855

Views

Example of positive measure compact set that contains no interval.

Published on 25 Mar 2026 - 19:04

score 88 · Answer 12 · 2018-05-20 21:41:17

88

Views

Measure of sets on $\mathbb R^{n+m}$ isomorphic to $\mathbb R^n\times \mathbb R^m$

Published on 20 May 2018 - 21:41

score 1.8k · Answer 13 · 2018-05-21 00:25:52

1.8k

Views

Lebesgue measurable set whose intersection has positive measure

Published on 21 May 2018 - 0:25

score 53 · Answer 14 · 2026-03-25 12:28:06

53

Views

Is the set of numbers $x$ with $\frac{x}{\pi} = \frac{m}{n} $ dense in $[0, \pi]$?

Published on 25 Mar 2026 - 12:28

score 54 · Answer 15 · 2026-03-25 18:50:37

54

Views

Given $F(x) = \int_a^xf(t)dt$, for $f \in L^p([a,b])$, prove a uniform bound of $\sum_{i=0}^{n-1} \frac{|F(x_{i+1})-F(x_i)|^p}{(x_{i+1}-x_i)^{p-1}}$

Published on 25 Mar 2026 - 18:50

List Question

If $f_n \rightarrow f$ is a sequence of $L^p$ and $g_n \rightarrow g$ a bounded sequence of $L^{\infty}$ then $f_ng_n \rightarrow fg$ in $L^p$

Show that $\lim_{n\to \infty}\int_{0}^{1}g(x)f(nx)dx = \Big(\int_{0}^{1}g(x)dx\Big)\Big(\int_{0}^{1}f(x)dx\Big)$

Defined integral over a closed and bounded subset of $\mathbb{R}$

Absolute continuity of probability measure wrt Lebesgue measure

Does an integrable function have to have a point of continuity?

What should I use in order to get the required function $g \in L^1$?

Show that $ \sum n|A_n| $ converges

How the function $f_n$ is a measurable function?

The measure of a function defined on digits

If $f : R \rightarrow R \cup \{- ∞, ∞\}$ is continuous then f is measurable.

Example of positive measure compact set that contains no interval.

Measure of sets on $\mathbb R^{n+m}$ isomorphic to $\mathbb R^n\times \mathbb R^m$

Lebesgue measurable set whose intersection has positive measure

Is the set of numbers $x$ with $\frac{x}{\pi} = \frac{m}{n} $ dense in $[0, \pi]$?

Given $F(x) = \int_a^xf(t)dt$, for $f \in L^p([a,b])$, prove a uniform bound of $\sum_{i=0}^{n-1} \frac{|F(x_{i+1})-F(x_i)|^p}{(x_{i+1}-x_i)^{p-1}}$

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