Question

Textbooks about integration w.r.t. signed measures

score 33 · Answer 1 · 2022-05-03 09:17:29

33

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Textbooks about integration w.r.t. signed measures

Published on 03 May 2022 - 9:17

score 115 · Answer 2 · 2022-05-03 10:05:56

115

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If $\lambda = \mu-\nu$, is it true that $\int f \mathrm d \lambda^+- \int f \mathrm d \lambda^- = \int f \mathrm d \mu -\int f \mathrm d \nu$?

Published on 03 May 2022 - 10:05

score 54 · Answer 3 · 2022-06-29 09:10:12

54

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What is the equivalent operation of measures to the multiplication of (right-continuous, bounded variation) function

Published on 29 Jun 2022 - 9:10

score 24 · Answer 4 · 2022-07-07 07:51:20

24

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Let $L = \{f:X \to [0, 1] \mid f \text{ is measurable}\}$. Then $\mu_+(X) = \sup_{f \in L} \int_X f \mathrm d \mu$

Published on 07 Jul 2022 - 7:51

score 301 · Answer 5 · 2022-07-08 18:35:39

301

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How to find total variation of a signed measure??

Published on 08 Jul 2022 - 18:35

score 104 · Answer 6 · 2022-07-09 04:24:38

104

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If $\mu_n \overset{\ast}{\rightharpoonup}\mu$, then $\mu^+_n \overset{\ast}{\rightharpoonup} \mu^+$ and $\mu^-_n \overset{\ast}{\rightharpoonup}\mu^-$

Published on 09 Jul 2022 - 4:24

score 275 · Answer 7 · 2022-07-26 03:45:47

275

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What does it mean for a signed measure to be "regular" in Riesz-Markov-Kakutani theorem?

Published on 26 Jul 2022 - 3:45

score 362 · Answer 8 · 2022-07-26 08:07:17

362

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$3$ versions of Riesz–Markov–Kakutani theorem

Published on 26 Jul 2022 - 8:07

score 51 · Answer 9 · 2022-10-15 17:50:11

51

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Why signed measures should respect countable additivity and not sub-addivity.

Published on 15 Oct 2022 - 17:50

score 91 · Answer 10 · 2022-10-16 17:47:25

91

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Representation of Functional in the Dual of the Space of Signed Measures on a Measurable Space

Published on 16 Oct 2022 - 17:47

score 337 · Answer 11 · 2022-11-01 20:15:40

337

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How to prove that $\mu_n \rightharpoonup \mu$ IFF $\mu_n \overset{*}{\rightharpoonup} \mu$ and $\mu_n (X) \to \mu (X)$?

Published on 01 Nov 2022 - 20:15

score 113 · Answer 12 · 2022-11-03 10:49:08

113

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Does $\mu_n \overset{*}{\rightharpoonup} \mu$ (or $\mu_n \rightharpoonup \mu$) imply $\{[\mu_n] \mid n \in \mathbb N\}$ is bounded?

Published on 03 Nov 2022 - 10:49

score 70 · Answer 13 · 2022-11-03 14:50:43

70

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Does $\mu_n \overset{1}{\rightharpoonup} \mu$ necessarily imply $\mu_n \overset{2}{\rightharpoonup} \mu$?

Published on 03 Nov 2022 - 14:50

score 81 · Answer 14 · 2022-11-06 03:20:35

81

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Let $\mu_n,\mu \in \mathcal M(X)$ such that $\mu_n \rightharpoonup \mu$ and $[\mu_n] \to [\mu]$, then $|\mu_n| \rightharpoonup |\mu|$

Published on 06 Nov 2022 - 3:20

score 87 · Answer 15 · 2022-11-06 03:55:34

87

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Portmanteau theorem for finite signed Borel measures

Published on 06 Nov 2022 - 3:55

List Question

Textbooks about integration w.r.t. signed measures

If $\lambda = \mu-\nu$, is it true that $\int f \mathrm d \lambda^+- \int f \mathrm d \lambda^- = \int f \mathrm d \mu -\int f \mathrm d \nu$?

What is the equivalent operation of measures to the multiplication of (right-continuous, bounded variation) function

Let $L = \{f:X \to [0, 1] \mid f \text{ is measurable}\}$. Then $\mu_+(X) = \sup_{f \in L} \int_X f \mathrm d \mu$

How to find total variation of a signed measure??

If $\mu_n \overset{\ast}{\rightharpoonup}\mu$, then $\mu^+_n \overset{\ast}{\rightharpoonup} \mu^+$ and $\mu^-_n \overset{\ast}{\rightharpoonup}\mu^-$

What does it mean for a signed measure to be "regular" in Riesz-Markov-Kakutani theorem?

$3$ versions of Riesz–Markov–Kakutani theorem

Why signed measures should respect countable additivity and not sub-addivity.

Representation of Functional in the Dual of the Space of Signed Measures on a Measurable Space

How to prove that $\mu_n \rightharpoonup \mu$ IFF $\mu_n \overset{*}{\rightharpoonup} \mu$ and $\mu_n (X) \to \mu (X)$?

Does $\mu_n \overset{*}{\rightharpoonup} \mu$ (or $\mu_n \rightharpoonup \mu$) imply $\{[\mu_n] \mid n \in \mathbb N\}$ is bounded?

Does $\mu_n \overset{1}{\rightharpoonup} \mu$ necessarily imply $\mu_n \overset{2}{\rightharpoonup} \mu$?

Let $\mu_n,\mu \in \mathcal M(X)$ such that $\mu_n \rightharpoonup \mu$ and $[\mu_n] \to [\mu]$, then $|\mu_n| \rightharpoonup |\mu|$

Portmanteau theorem for finite signed Borel measures

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