Question

Why are complex measures not allowed to attain $\infty$ while signed measures are?

score 152 · Answer 1 · 2024-01-11 13:13:03

152

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Why are complex measures not allowed to attain $\infty$ while signed measures are?

Published on 11 Jan 2024 - 13:13

score 27 · Answer 2 · 2024-01-22 00:31:56

27

Views

Sigma additivity of signed measure

Published on 22 Jan 2024 - 0:31

score 42 · Answer 3 · 2024-02-08 13:48:30

42

Views

Uniqueness of extension of signed measures

Published on 08 Feb 2024 - 13:48

score 47 · Answer 4 · 2024-02-13 09:19:01

47

Views

Sum of two signed measures is well defined iff both do not take the same sign of $\pm \infty$

Published on 13 Feb 2024 - 9:19

score 15 · Answer 5 · 2024-03-22 04:39:55

15

Views

Relation between total variation measure and total variation of a function

Published on 22 Mar 2024 - 4:39

score 126 · Answer 6 · 2017-08-14 02:20:10

126

Views

Show that $L^1(\nu)=L^1(|\nu|)$, where $\nu$ is a complex

Published on 14 Aug 2017 - 2:20

score 166 · Answer 7 · 2017-11-07 01:54:02

166

Views

If $μ$ is a signed measure, $Ω^±$ is a Hahn decomposition and $μ^±$ is the Jordan decomposition, then $\int_{A∩Ω^±}X{\rm d}μ=\pm\int_AX{\rm d}μ^±$

Published on 07 Nov 2017 - 1:54

score 120 · Answer 8 · 2017-11-08 01:05:34

120

Views

Convergence of supremum of the series of a signed measure

Published on 08 Nov 2017 - 1:05

score 298 · Answer 9 · 2018-01-09 22:50:27

298

Views

If $\mu$ is a finite measure and $\nu$ is a signed measure, can we extend the inequality $|\nu|\le C\mu$ from a generator to the whole σ-algebra?

Published on 09 Jan 2018 - 22:50

score 238 · Answer 10 · 2018-01-09 23:21:46

238

Views

If $\mu$ is a finite measure and $ν$ is a signed measure with $|\nu|\le C\mu$, are we able to show $\left|\frac{{\rm d}ν}{{\rm d}\mu}\right|\le C$?

Published on 09 Jan 2018 - 23:21

score 59 · Answer 11 · 2020-08-22 09:05:14

59

Views

Show that $\{h d \mu: h \in \mathcal{L}^1(\mu)\}$ is a closed subset of $M(\mathcal{S})$

Published on 22 Aug 2020 - 9:05

score 22 · Answer 12 · 2020-10-10 11:29:59

22

Views

existence of a signed measure on $(\mathbb{R},\mathcal{B})$

Published on 10 Oct 2020 - 11:29

score 223 · Answer 13 · 2026-02-24 01:54:40

223

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Decomposition of the variation of a signed measure as $|\mu|(A) = \int_A |\frac{d\mu_{1a}}{d\mu_2}-1|d\mu_2 + \mu_{1s}(A)$, where $\mu=\mu_1-\mu_2$

Published on 24 Feb 2026 - 1:54

score 40 · Answer 14 · 2020-10-27 18:16:40

For given $σ, σ_1, σ_2, . . . ∈ L^d,$ ($L^d$ set of all signed measures) we should show that $\lim_{n}σ_n = σ ⇔ ∀A ∈ F, \; \lim_{n} σ_n(A) = σ(A).$

score 71 · Answer 15 · 2026-03-24 21:42:58

71

Views

<< notation on measures

Published on 24 Mar 2026 - 21:42

List Question

Why are complex measures not allowed to attain $\infty$ while signed measures are?

Sigma additivity of signed measure

Uniqueness of extension of signed measures

Sum of two signed measures is well defined iff both do not take the same sign of $\pm \infty$

Relation between total variation measure and total variation of a function

Show that $L^1(\nu)=L^1(|\nu|)$, where $\nu$ is a complex

If $μ$ is a signed measure, $Ω^±$ is a Hahn decomposition and $μ^±$ is the Jordan decomposition, then $\int_{A∩Ω^±}X{\rm d}μ=\pm\int_AX{\rm d}μ^±$

Convergence of supremum of the series of a signed measure

If $\mu$ is a finite measure and $\nu$ is a signed measure, can we extend the inequality $|\nu|\le C\mu$ from a generator to the whole σ-algebra?

If $\mu$ is a finite measure and $ν$ is a signed measure with $|\nu|\le C\mu$, are we able to show $\left|\frac{{\rm d}ν}{{\rm d}\mu}\right|\le C$?

Show that $\{h d \mu: h \in \mathcal{L}^1(\mu)\}$ is a closed subset of $M(\mathcal{S})$

existence of a signed measure on $(\mathbb{R},\mathcal{B})$

Decomposition of the variation of a signed measure as $|\mu|(A) = \int_A |\frac{d\mu_{1a}}{d\mu_2}-1|d\mu_2 + \mu_{1s}(A)$, where $\mu=\mu_1-\mu_2$

For given $σ, σ_1, σ_2, . . . ∈ L^d,$ ($L^d$ set of all signed measures) we should show that $\lim_{n}σ_n = σ ⇔ ∀A ∈ F, \; \lim_{n} σ_n(A) = σ(A).$

<< notation on measures

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