Question

Is there an analogue of Lebesgue’s Dominated Convergence Theorem for a net $ (f_{\alpha})_{\alpha \in A} $ of measurable functions?

score 1.3k · Answer 1 · 2014-06-24 19:49:26

1.3k

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Is there an analogue of Lebesgue’s Dominated Convergence Theorem for a net $ (f_{\alpha})_{\alpha \in A} $ of measurable functions?

Published on 24 Jun 2014 - 19:49

score 643 · Answer 2 · 2014-06-24 21:11:20

643

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Why is $fg$ integrable w.r.t. a probability measure if $f,g$ are Lebesgue integrable?

Published on 24 Jun 2014 - 21:11

score 104 · Answer 3 · 2014-06-25 15:58:11

104

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Suppose there are constants $K > 0$ and $p > 1$ such that $μ\{x∈X : |f(x)|>M\}< \frac{K}{M^p}$ for all $M >0.$

Published on 25 Jun 2014 - 15:58

score 512 · Answer 4 · 2014-06-26 08:10:00

512

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an argument that strengthen Lusin's theorem

Published on 26 Jun 2014 - 8:10

score 304 · Answer 5 · 2014-06-26 17:47:17

304

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How to show convergence in distribution

Published on 26 Jun 2014 - 17:47

score 320 · Answer 6 · 2014-06-27 02:41:18

320

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example concerning Lusin's theorem

Published on 27 Jun 2014 - 2:41

score 1.4k · Answer 7 · 2014-06-27 10:06:51

1.4k

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Proof the that compact support is a vector space

Published on 27 Jun 2014 - 10:06

score 156 · Answer 8 · 2014-06-27 16:37:47

156

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Show $\lim_{n_\rightarrow \infty}\int_0^\infty ne^{-\frac{2n^2x^2}{x + 1}}dx = \infty$

Published on 27 Jun 2014 - 16:37

score 107 · Answer 9 · 2014-06-27 23:23:42

107

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Let $\{f_n\}^∞_{n=1} ⊂ L_2(μ)$ be a sequence of functions such that $∥f_n∥_2 ≤ 1.$

Published on 27 Jun 2014 - 23:23

score 134 · Answer 10 · 2014-06-28 18:05:41

134

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Prove that $\lim_{x\to y} \frac{d(x, F )}{|x−y|} = 0$ for a.e. $y \in F$.

Published on 28 Jun 2014 - 18:05

score 772 · Answer 11 · 2014-06-29 01:21:18

772

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Limit of an integral, as the measure of the region of integration approaches zero

Published on 29 Jun 2014 - 1:21

score 648 · Answer 12 · 2014-06-29 16:18:38

648

Views

Partial integration for lebesgue integrable functions

Published on 29 Jun 2014 - 16:18

score 106 · Answer 13 · 2014-06-30 11:18:54

106

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Approximation of integration by simple functions.

Published on 30 Jun 2014 - 11:18

score 333 · Answer 14 · 2014-06-30 11:39:11

333

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Integral change that I don't understand

Published on 30 Jun 2014 - 11:39

score 137 · Answer 15 · 2014-06-30 17:31:29

137

Views

Show that $\lim_{x→0+ } x^{−1/2}f(x)$ exists and determine the value of this limit.

Published on 30 Jun 2014 - 17:31

List Question

Is there an analogue of Lebesgue’s Dominated Convergence Theorem for a net $ (f_{\alpha})_{\alpha \in A} $ of measurable functions?

Why is $fg$ integrable w.r.t. a probability measure if $f,g$ are Lebesgue integrable?

Suppose there are constants $K > 0$ and $p > 1$ such that $μ\{x∈X : |f(x)|>M\}< \frac{K}{M^p}$ for all $M >0.$

an argument that strengthen Lusin's theorem

How to show convergence in distribution

example concerning Lusin's theorem

Proof the that compact support is a vector space

Show $\lim_{n_\rightarrow \infty}\int_0^\infty ne^{-\frac{2n^2x^2}{x + 1}}dx = \infty$

Let $\{f_n\}^∞_{n=1} ⊂ L_2(μ)$ be a sequence of functions such that $∥f_n∥_2 ≤ 1.$

Prove that $\lim_{x\to y} \frac{d(x, F )}{|x−y|} = 0$ for a.e. $y \in F$.

Limit of an integral, as the measure of the region of integration approaches zero

Partial integration for lebesgue integrable functions

Approximation of integration by simple functions.

Integral change that I don't understand

Show that $\lim_{x→0+ } x^{−1/2}f(x)$ exists and determine the value of this limit.

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