Question

how prove that $\lim_{n\to{}\infty} {\sum_{(i, j)\in{K_n}}^\infty{a_{ij}}}=\sum_{i=1}^\infty\sum_{j=1}^\infty{a_ {ij}}$

score 91 · Answer 1 · 2026-04-15 19:31:59

91

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how prove that $\lim_{n\to{}\infty} {\sum_{(i, j)\in{K_n}}^\infty{a_{ij}}}=\sum_{i=1}^\infty\sum_{j=1}^\infty{a_ {ij}}$

Published on 15 Apr 2026 - 19:31

score 197 · Answer 2 · 2026-04-12 04:36:24

197

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Equivalent definitions of Lebesgue measure

Published on 12 Apr 2026 - 4:36

score 1.6k · Answer 3 · 2026-04-15 20:19:52

1.6k

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the $\sigma$ algeba generated by the class of open intervals with rational end points coincide with the borel $\sigma$ algebra on the real line.

Published on 15 Apr 2026 - 20:19

score 572 · Answer 4 · 2026-04-17 17:49:11

572

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"Intermediate value theorem" for Lebesgue integrals and subsets

Published on 17 Apr 2026 - 17:49

score 67 · Answer 5 · 2026-04-17 18:06:43

67

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Measurable functions such that $\int\underline{\text{lim}}f_n=0 $ and $\underline{\text{lim}}\int f_n=+\infty$.

Published on 17 Apr 2026 - 18:06

score 586 · Answer 6 · 2026-04-09 22:19:16

586

Views

Lebesgue outer measure is not finitely additive

Published on 09 Apr 2026 - 22:19

score 113 · Answer 7 · 2026-04-12 08:22:04

113

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how to find such a sequence of measurable functions?

Published on 12 Apr 2026 - 8:22

score 1.2k · Answer 8 · 2026-04-15 12:18:23

1.2k

Views

Product of outer measure equals outer measure of product

Published on 15 Apr 2026 - 12:18

score 157 · Answer 9 · 2026-04-17 18:06:46

157

Views

Showing $\left(\sum_1^\infty 2^{-n} \mathbf{1}_{x\in(0,1)}(x-r_n)^{-1/2}\right)^2$ is not integrable

Published on 17 Apr 2026 - 18:06

score 91 · Answer 10 · 2026-04-17 18:06:43

91

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$f$ be Lebesgue integrable and $F = m\{f > \alpha\}$, then $F$ is right continuous

Published on 17 Apr 2026 - 18:06

score 120 · Answer 11 · 2026-04-17 18:07:42

120

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Exist $g:\mathbb{R}\to \mathbb{R}$ Lebesgue measurable and $h:[0, 1]\to \mathbb{R}$ Borel measurable such that $f = g \circ{}h$.

Published on 17 Apr 2026 - 18:07

score 373 · Answer 12 · 2026-04-11 21:46:11

373

Views

Rotation send measure zero set to measure zero

Published on 11 Apr 2026 - 21:46

score 95 · Answer 13 · 2026-04-14 00:24:37

95

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Integral of a Lebsgue Integrable Function over Small Sets

Published on 14 Apr 2026 - 0:24

score 468 · Answer 14 · 2026-04-13 23:15:34

468

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Find $E\subseteq\mathbb{R}$ such that $\liminf_{\delta\to 0}\frac{m(E\cap(-\delta,\delta))}{2\delta}=\alpha$

Published on 13 Apr 2026 - 23:15

score 1.2k · Answer 15 · 2026-04-09 18:03:22

1.2k

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Is the Carathéodory definition of Measurable set the most general?

Published on 09 Apr 2026 - 18:03

List Question

how prove that $\lim_{n\to{}\infty} {\sum_{(i, j)\in{K_n}}^\infty{a_{ij}}}=\sum_{i=1}^\infty\sum_{j=1}^\infty{a_ {ij}}$

Equivalent definitions of Lebesgue measure

the $\sigma$ algeba generated by the class of open intervals with rational end points coincide with the borel $\sigma$ algebra on the real line.

"Intermediate value theorem" for Lebesgue integrals and subsets

Measurable functions such that $\int\underline{\text{lim}}f_n=0 $ and $\underline{\text{lim}}\int f_n=+\infty$.

Lebesgue outer measure is not finitely additive

how to find such a sequence of measurable functions?

Product of outer measure equals outer measure of product

Showing $\left(\sum_1^\infty 2^{-n} \mathbf{1}_{x\in(0,1)}(x-r_n)^{-1/2}\right)^2$ is not integrable

$f$ be Lebesgue integrable and $F = m\{f > \alpha\}$, then $F$ is right continuous

Exist $g:\mathbb{R}\to \mathbb{R}$ Lebesgue measurable and $h:[0, 1]\to \mathbb{R}$ Borel measurable such that $f = g \circ{}h$.

Rotation send measure zero set to measure zero

Integral of a Lebsgue Integrable Function over Small Sets

Find $E\subseteq\mathbb{R}$ such that $\liminf_{\delta\to 0}\frac{m(E\cap(-\delta,\delta))}{2\delta}=\alpha$

Is the Carathéodory definition of Measurable set the most general?

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