Question

Question about a theorem concerning the continuity of integral functions

score 432 · Answer 1 · 2014-06-15 23:03:29

432

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Question about a theorem concerning the continuity of integral functions

Published on 15 Jun 2014 - 23:03

score 284 · Answer 2 · 2014-06-16 09:21:59

284

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Finding the integral of $\frac{x}{e^x + 1}$

Published on 16 Jun 2014 - 9:21

score 432 · Answer 3 · 2014-06-16 15:43:08

432

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Suppose that all functions ${f_n},f$ are integrable. Is $\lim_{n \rightarrow \infty} \int f_n(x)dx = \int f(x)dx?$

Published on 16 Jun 2014 - 15:43

score 179 · Answer 4 · 2014-06-19 19:38:01

179

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Prove g is Lebesgue intergrable

Published on 19 Jun 2014 - 19:38

score 1.6k · Answer 5 · 2014-06-20 03:43:47

1.6k

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Square Integrable and Continuous

Published on 20 Jun 2014 - 3:43

score 125 · Answer 6 · 2014-06-20 15:27:58

125

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Change of variable for Lebesegue Integral

Published on 20 Jun 2014 - 15:27

score 125 · Answer 7 · 2014-06-20 16:04:58

125

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Prove that $\int|f − g| = \int_{-\infty}^{\infty} μ(F_t △ G_t) dt.$

Published on 20 Jun 2014 - 16:04

score 515 · Answer 8 · 2014-06-20 16:21:59

515

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$m(E∩I)≥αm(I)$ for all intervals $I⊂[0,1].$ Prove that $m(E) = 1.$

Published on 20 Jun 2014 - 16:21

score 393 · Answer 9 · 2014-06-21 17:29:18

393

Views

One dimensional integrals in Green's theorem

Published on 21 Jun 2014 - 17:29

score 116 · Answer 10 · 2014-06-21 18:15:10

116

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Let $f ∈ L_1([0,1])$ be a function such that $\int_E f(x)dx = 0$ for any measurable set $E ⊂ [0,1]$ of Lebesgue measure $0.99.$

Published on 21 Jun 2014 - 18:15

score 1.7k · Answer 11 · 2014-06-21 20:20:46

1.7k

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Show that $\lim _{r \to 0} \|T_rf−f\|_{L_p} =0.$

Published on 21 Jun 2014 - 20:20

score 156 · Answer 12 · 2014-06-22 15:21:44

156

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Construct a function $F : X → \mathbb{R}$ such that $F ∈ L_p(μ)$ for all $p > 1,$ but $F \notin L_1(μ).$

Published on 22 Jun 2014 - 15:21

score 68 · Answer 13 · 2014-06-23 13:44:49

68

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Prove that $\sup_n |T_n f(x)|<\infty,$ for a.a. $x$ for every $f \in L^1(R)$.

Published on 23 Jun 2014 - 13:44

score 61 · Answer 14 · 2014-06-23 16:00:34

61

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Let $E⊂\{(x,y)|0≤x≤1, 0≤y≤x\}, E_x =\{y|(x,y)∈E\}, E^y =\{x|(x,y)∈E\}$ and assume that $m(E_x) ≥ x^3$ for any $x ∈ [0, 1].$

Published on 23 Jun 2014 - 16:00

score 227 · Answer 15 · 2014-06-24 17:12:36

227

Views

Prove that the set $A$ is measurable and find its Lebesgue measure.

Published on 24 Jun 2014 - 17:12

List Question

Question about a theorem concerning the continuity of integral functions

Finding the integral of $\frac{x}{e^x + 1}$

Suppose that all functions ${f_n},f$ are integrable. Is $\lim_{n \rightarrow \infty} \int f_n(x)dx = \int f(x)dx?$

Prove g is Lebesgue intergrable

Square Integrable and Continuous

Change of variable for Lebesegue Integral

Prove that $\int|f − g| = \int_{-\infty}^{\infty} μ(F_t △ G_t) dt.$

$m(E∩I)≥αm(I)$ for all intervals $I⊂[0,1].$ Prove that $m(E) = 1.$

One dimensional integrals in Green's theorem

Let $f ∈ L_1([0,1])$ be a function such that $\int_E f(x)dx = 0$ for any measurable set $E ⊂ [0,1]$ of Lebesgue measure $0.99.$

Show that $\lim _{r \to 0} \|T_rf−f\|_{L_p} =0.$

Construct a function $F : X → \mathbb{R}$ such that $F ∈ L_p(μ)$ for all $p > 1,$ but $F \notin L_1(μ).$

Prove that $\sup_n |T_n f(x)|<\infty,$ for a.a. $x$ for every $f \in L^1(R)$.

Let $E⊂\{(x,y)|0≤x≤1, 0≤y≤x\}, E_x =\{y|(x,y)∈E\}, E^y =\{x|(x,y)∈E\}$ and assume that $m(E_x) ≥ x^3$ for any $x ∈ [0, 1].$

Prove that the set $A$ is measurable and find its Lebesgue measure.

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