Question

Let $f_n \in L^2[0,1]$ and $\|f_n\|_2=1$ and suppose $\lim_{m,n \to \infty}\|f_m+f_n\|_2=2$, prove that there is $f \in L^2$ s.t $f_n \to f$ in $L^2$

score 45 · Answer 1 · 2026-04-16 21:46:50

45

Views

Let $f_n \in L^2[0,1]$ and $\|f_n\|_2=1$ and suppose $\lim_{m,n \to \infty}\|f_m+f_n\|_2=2$, prove that there is $f \in L^2$ s.t $f_n \to f$ in $L^2$

Published on 16 Apr 2026 - 21:46

score 78 · Answer 2 · 2026-04-16 18:35:37

78

Views

Let $f \in L^1(\mathbb{R})$ and $\|f\|_1=1$, show that for $\delta>0$ $\lim_{n \to \infty}\int_{|x|\geq \delta}nf(nx)dx=0$

Published on 16 Apr 2026 - 18:35

score 52 · Answer 3 · 2026-04-17 15:35:55

52

Views

if $A\subseteq B\subseteq C$ and $m(A)=m(C)$, can we get $m(A)=m(B)=m(C)$

Published on 17 Apr 2026 - 15:35

score 204 · Answer 4 · 2026-04-14 08:12:55

204

Views

Lebesgue Measurable function with domain that is finite a.e. and has finite measure, then $\exists$measurable and bounded subset.

Published on 14 Apr 2026 - 8:12

score 93 · Answer 5 · 2026-04-16 05:42:13

93

Views

Is $F(x)=\int_{\mathbb{R}}f(t)\frac{\sin(xt)}{t}dt$ differentiable for $f\in L^1$

Published on 16 Apr 2026 - 5:42

score 64 · Answer 6 · 2026-04-16 06:51:21

64

Views

Small open set containing a set of measure zero

Published on 16 Apr 2026 - 6:51

score 582 · Answer 7 · 2026-04-14 09:34:33

582

Views

Weak convergence in $L^{p}$ examples

Published on 14 Apr 2026 - 9:34

score 97 · Answer 8 · 2026-04-10 15:25:34

97

Views

Show that $\mu^(E\Delta A)<\epsilon$ then $E\in\mathcal{A}^$

Published on 10 Apr 2026 - 15:25

score 281 · Answer 9 · 2026-03-25 13:57:51

281

Views

Equivalence of Hausdorff and Spherical measure on $\mathbb{S}^{n}$

Published on 25 Mar 2026 - 13:57

score 25 · Answer 10 · 2026-04-16 01:36:00

25

Views

Let $[0,1]=\cup A_n$ and assume that $\lambda(A_i\cap E_j) \to 0$ as $j\to \infty$ for all $i$. Show that $\lambda(E_j)\to 0$ ($E_j\subseteq [0,1]$)

Published on 16 Apr 2026 - 1:36

score 609 · Answer 11 · 2026-04-15 04:46:26

609

Views

$L_p(X)$ separable if $(X,\mu)$ is separable measure space.

Published on 15 Apr 2026 - 4:46

score 62 · Answer 12 · 2026-04-16 10:51:26

62

Views

Calculate $\lim\limits_{n \to \infty}\int\limits_{0}^{1} \frac{f_n(x)}{\sqrt{|x-\frac{1}{n}|}}$ given that $|f_n(x)|\leq 1$ and $f_n \to f$ pointwise.

Published on 16 Apr 2026 - 10:51

score 26 · Answer 13 · 2026-04-14 18:00:30

26

Views

Assume that $\lambda(A+t-A)=0$ for all $t \in Q$ show that either $A$ is a null set, or full measure.

Published on 14 Apr 2026 - 18:00

score 54 · Answer 14 · 2026-04-06 04:40:39

54

Views

$\lim_{n\to \infty} \int_n^{n+1}f(x)~dx=0$ if $f\in L^2(\Bbb R)$

Published on 06 Apr 2026 - 4:40

score 192 · Answer 15 · 2026-03-26 17:52:55

192

Views

Simple question about Lebesgue density points of open sets

Published on 26 Mar 2026 - 17:52

List Question

Let $f_n \in L^2[0,1]$ and $\|f_n\|_2=1$ and suppose $\lim_{m,n \to \infty}\|f_m+f_n\|_2=2$, prove that there is $f \in L^2$ s.t $f_n \to f$ in $L^2$

Let $f \in L^1(\mathbb{R})$ and $\|f\|_1=1$, show that for $\delta>0$ $\lim_{n \to \infty}\int_{|x|\geq \delta}nf(nx)dx=0$

if $A\subseteq B\subseteq C$ and $m(A)=m(C)$, can we get $m(A)=m(B)=m(C)$

Lebesgue Measurable function with domain that is finite a.e. and has finite measure, then $\exists$measurable and bounded subset.

Is $F(x)=\int_{\mathbb{R}}f(t)\frac{\sin(xt)}{t}dt$ differentiable for $f\in L^1$

Small open set containing a set of measure zero

Weak convergence in $L^{p}$ examples

Show that $\mu^(E\Delta A)<\epsilon$ then $E\in\mathcal{A}^$

Equivalence of Hausdorff and Spherical measure on $\mathbb{S}^{n}$

Let $[0,1]=\cup A_n$ and assume that $\lambda(A_i\cap E_j) \to 0$ as $j\to \infty$ for all $i$. Show that $\lambda(E_j)\to 0$ ($E_j\subseteq [0,1]$)

$L_p(X)$ separable if $(X,\mu)$ is separable measure space.

Calculate $\lim\limits_{n \to \infty}\int\limits_{0}^{1} \frac{f_n(x)}{\sqrt{|x-\frac{1}{n}|}}$ given that $|f_n(x)|\leq 1$ and $f_n \to f$ pointwise.

Assume that $\lambda(A+t-A)=0$ for all $t \in Q$ show that either $A$ is a null set, or full measure.

$\lim_{n\to \infty} \int_n^{n+1}f(x)~dx=0$ if $f\in L^2(\Bbb R)$

Simple question about Lebesgue density points of open sets

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