Question

Prove that $f = g$ almost everywhere on $\mathbb{R}$

score 86 · Answer 1 · 2026-04-14 16:20:17

86

Views

Prove that $f = g$ almost everywhere on $\mathbb{R}$

Published on 14 Apr 2026 - 16:20

score 23 · Answer 2 · 2026-03-27 07:50:55

23

Views

$f_j \rightharpoonup f$ and $g_j \rightharpoonup g$ weakly in $H^1(\Omega)$ $\Rightarrow$ $\nabla(f_kg_k)\rightharpoonup \nabla(fg)$ in $L^p(\Omega)$

Published on 27 Mar 2026 - 7:50

score 32 · Answer 3 · 2024-03-28 18:21:30

32

Views

Prove that the function $f(x) = \frac{1}{x^p}$ belongs to $L_{1}(1,\infty)$ if and only if $p > 1$.

Published on 28 Mar 2024 - 18:21

score 41 · Answer 4 · 2024-03-28 20:48:00

41

Views

Let $f(x) = x^{-1}\sin(x^{-1})+\cos(x^{-1})$. How can one prove that $\int f^+ = \infty$?

Published on 28 Mar 2024 - 20:48

score 25 · Answer 5 · 2026-03-26 20:16:47

25

Views

How to find the m such that $(1+|x|)^{-m} f(x) \in L^1\left(\mathbb{R}^n\right) $

Published on 26 Mar 2026 - 20:16

score 60 · Answer 6 · 2024-03-30 15:05:36

60

Views

$A$ is a borel subset of $\mathbb{R}^2$ and we know that $\lambda_2(A) = \pi$. Prove that: $\int_A (x^2 + y^2) \ d \lambda_2(x,y) \geq \frac{\pi}{2}$

Published on 30 Mar 2024 - 15:05

score 31 · Answer 7 · 2024-03-31 15:54:06

31

Views

If $\int_0^1 x^nf(x)=0$, then $f(x)=0$ a.e.

Published on 31 Mar 2024 - 15:54

score 230 · Answer 8 · 2026-04-03 00:52:40

230

Views

Distributions defined by $C_0^\infty(\mathbb{R})$ enough to distinguish $f_1,f_2\in L^1(\mathbb{R})$?

Published on 03 Apr 2026 - 0:52

score 94 · Answer 9 · 2026-04-13 09:26:06

94

Views

how to show that $\int_A f \, d\mu =\mu(A) \int_X f \, d\mu$ when $\mu(E\cap A)=\mu(E) \mu(A)$

Published on 13 Apr 2026 - 9:26

score 2.2k · Answer 10 · 2026-04-13 00:25:13

2.2k

Views

Integral of a simple function

Published on 13 Apr 2026 - 0:25

score 30 · Answer 11 · 2026-03-27 17:37:16

30

Views

$\iint |x-y|^{-t} \,d\mu\, d\mu < \infty$ iff $t<1$

Published on 27 Mar 2026 - 17:37

score 1.1k · Answer 12 · 2026-04-14 17:48:44

1.1k

Views

A function $f$ such that $f \in L_1$ but $f \notin L_p$ for $p>1$

Published on 14 Apr 2026 - 17:48

score 511 · Answer 13 · 2015-01-05 03:09:13

511

Views

weakly convergence imply strong convergence when $ \|f_n\| \rightarrow \|f\| $ in $l^2([0,1])$?

Published on 05 Jan 2015 - 3:09

score 271 · Answer 14 · 2015-01-05 11:56:02

271

Views

Show that $\int|f(x)|dx=\int_0^\infty m(E_\alpha)d\alpha$

Published on 05 Jan 2015 - 11:56

score 368 · Answer 15 · 2026-04-09 08:38:37

368

Views

Borel-Stieltjes measure problem with floor functions.

Published on 09 Apr 2026 - 8:38

List Question

Prove that $f = g$ almost everywhere on $\mathbb{R}$

$f_j \rightharpoonup f$ and $g_j \rightharpoonup g$ weakly in $H^1(\Omega)$ $\Rightarrow$ $\nabla(f_kg_k)\rightharpoonup \nabla(fg)$ in $L^p(\Omega)$

Prove that the function $f(x) = \frac{1}{x^p}$ belongs to $L_{1}(1,\infty)$ if and only if $p > 1$.

Let $f(x) = x^{-1}\sin(x^{-1})+\cos(x^{-1})$. How can one prove that $\int f^+ = \infty$?

How to find the m such that $(1+|x|)^{-m} f(x) \in L^1\left(\mathbb{R}^n\right) $

$A$ is a borel subset of $\mathbb{R}^2$ and we know that $\lambda_2(A) = \pi$. Prove that: $\int_A (x^2 + y^2) \ d \lambda_2(x,y) \geq \frac{\pi}{2}$

If $\int_0^1 x^nf(x)=0$, then $f(x)=0$ a.e.

Distributions defined by $C_0^\infty(\mathbb{R})$ enough to distinguish $f_1,f_2\in L^1(\mathbb{R})$?

how to show that $\int_A f \, d\mu =\mu(A) \int_X f \, d\mu$ when $\mu(E\cap A)=\mu(E) \mu(A)$

Integral of a simple function

$\iint |x-y|^{-t} \,d\mu\, d\mu < \infty$ iff $t<1$

A function $f$ such that $f \in L_1$ but $f \notin L_p$ for $p>1$

weakly convergence imply strong convergence when $ \|f_n\| \rightarrow \|f\| $ in $l^2([0,1])$?

Show that $\int|f(x)|dx=\int_0^\infty m(E_\alpha)d\alpha$

Borel-Stieltjes measure problem with floor functions.

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