Question

Applying Lebesgue Dominated Convergence Theorem

score 42 · Answer 1 · 2026-04-11 03:08:06

42

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Applying Lebesgue Dominated Convergence Theorem

Published on 11 Apr 2026 - 3:08

score 99 · Answer 2 · 2026-04-08 21:09:08

99

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Properties of measures

Published on 08 Apr 2026 - 21:09

score 187 · Answer 3 · 2026-04-16 10:21:37

187

Views

Show that $f \in L^1(\Bbb R) \Longleftrightarrow g \in L^1 (\Bbb R \times \Bbb R)$ where $g(x,y)=f(x+y)f(x).$

Published on 16 Apr 2026 - 10:21

score 60 · Answer 4 · 2026-04-29 09:23:03

60

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Assume that $f,g \in L_{10}(\mu)$. Prove that their product $fg$ belongs to $L_5(\mu)$.

Published on 29 Apr 2026 - 9:23

score 44 · Answer 5 · 2026-04-29 10:22:11

44

Views

Let $g_n(x)=\frac{1}{x} \chi_{(1,n)} (x) : \mathbb{R} \rightarrow \mathbb{R}$. Show $\forall$ $n>1$, $\exists$ a m>1 such that $||g_n - g_m||_1>1$.

Published on 29 Apr 2026 - 10:22

score 8k · Answer 6 · 2026-04-12 15:15:51

8k

Views

Riemann Integrable implies Lebesgue Integrable

Published on 12 Apr 2026 - 15:15

score 299 · Answer 7 · 2026-04-18 15:17:56

299

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Calculate $\lim_{n\to\infty}\int_{[0,1]}\frac{nx}{1+nx^2}$.

Published on 18 Apr 2026 - 15:17

score 160 · Answer 8 · 2026-04-18 20:04:00

160

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Show that $\lim_{n\to\infty}\int_{[0,1]}\frac{nx}{1+n^2x^2}=0$.

Published on 18 Apr 2026 - 20:04

score 62 · Answer 9 · 2026-04-14 04:32:23

62

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Explain the Equality

Published on 14 Apr 2026 - 4:32

score 410 · Answer 10 · 2026-04-23 06:41:55

410

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Example of decreasing sequence of non-negative measurable functions such that $\int f_n \ d\mu= +\infty \ \text{ and } \ \int f \ d\mu=0.$

Published on 23 Apr 2026 - 6:41

score 475 · Answer 11 · 2026-04-15 01:12:48

475

Views

If a function is Riemann integrable and Lebesgue integrable, the two integrals are the same?

Published on 15 Apr 2026 - 1:12

score 158 · Answer 12 · 2026-04-17 04:30:57

158

Views

Prove that if f $\in L^1$, then $\forall t \in \mathbb{R}: \int_{\mathbb{R}} f(x)dx = \int_{\mathbb{R}}f(x+t)dx$

Published on 17 Apr 2026 - 4:30

score 74 · Answer 13 · 2026-04-11 15:00:43

74

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Inequality of Schmidt: $\int_S |x-y|^{-s} dy \leq \omega_n (n-s)^{-1} R^{n-s},$ where $|S| = e_n R^n = | B_R(x) |$ and $e_n = \frac{\omega_n}{n}$

Published on 11 Apr 2026 - 15:00

score 40 · Answer 14 · 2026-04-12 22:44:05

40

Views

Lebesgue theory and equivalence integral-series

Published on 12 Apr 2026 - 22:44

score 87 · Answer 15 · 2026-04-09 23:14:42

87

Views

Measurability of continuous function in R

Published on 09 Apr 2026 - 23:14

List Question

Applying Lebesgue Dominated Convergence Theorem

Properties of measures

Show that $f \in L^1(\Bbb R) \Longleftrightarrow g \in L^1 (\Bbb R \times \Bbb R)$ where $g(x,y)=f(x+y)f(x).$

Assume that $f,g \in L_{10}(\mu)$. Prove that their product $fg$ belongs to $L_5(\mu)$.

Let $g_n(x)=\frac{1}{x} \chi_{(1,n)} (x) : \mathbb{R} \rightarrow \mathbb{R}$. Show $\forall$ $n>1$, $\exists$ a m>1 such that $||g_n - g_m||_1>1$.

Riemann Integrable implies Lebesgue Integrable

Calculate $\lim_{n\to\infty}\int_{[0,1]}\frac{nx}{1+nx^2}$.

Show that $\lim_{n\to\infty}\int_{[0,1]}\frac{nx}{1+n^2x^2}=0$.

Explain the Equality

Example of decreasing sequence of non-negative measurable functions such that $\int f_n \ d\mu= +\infty \ \text{ and } \ \int f \ d\mu=0.$

If a function is Riemann integrable and Lebesgue integrable, the two integrals are the same?

Prove that if f $\in L^1$, then $\forall t \in \mathbb{R}: \int_{\mathbb{R}} f(x)dx = \int_{\mathbb{R}}f(x+t)dx$

Inequality of Schmidt: $\int_S |x-y|^{-s} dy \leq \omega_n (n-s)^{-1} R^{n-s},$ where $|S| = e_n R^n = | B_R(x) |$ and $e_n = \frac{\omega_n}{n}$

Lebesgue theory and equivalence integral-series

Measurability of continuous function in R

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