Question

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

score 153 · Answer 1 · 2026-03-25 11:10:20

153

Views

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

Published on 25 Mar 2026 - 11:10

score 29 · Answer 2 · 2026-04-05 21:52:57

29

Views

Sigma additivity of signed measure

Published on 05 Apr 2026 - 21:52

score 43 · Answer 3 · 2026-03-25 11:16:08

43

Views

Uniqueness of extension of signed measures

Published on 25 Mar 2026 - 11:16

score 48 · Answer 4 · 2026-03-25 11:15:04

48

Views

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

Published on 25 Mar 2026 - 11:15

score 16 · Answer 5 · 2026-03-25 12:46:09

16

Views

Relation between total variation measure and total variation of a function

Published on 25 Mar 2026 - 12:46

score 127 · Answer 6 · 2026-03-25 12:54:17

127

Views

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

Published on 25 Mar 2026 - 12:54

score 167 · Answer 7 · 2026-03-25 12:49:39

167

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 25 Mar 2026 - 12:49

score 121 · Answer 8 · 2026-03-25 12:50:05

121

Views

Convergence of supremum of the series of a signed measure

Published on 25 Mar 2026 - 12:50

score 299 · Answer 9 · 2026-03-25 12:50:52

299

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 25 Mar 2026 - 12:50

score 239 · Answer 10 · 2026-03-25 12:50:49

239

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 25 Mar 2026 - 12:50

score 60 · Answer 11 · 2026-03-25 11:16:39

60

Views

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

Published on 25 Mar 2026 - 11:16

score 23 · Answer 12 · 2026-03-25 12:53:59

23

Views

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

Published on 25 Mar 2026 - 12:53

score 224 · Answer 13 · 2026-04-12 00:09:17

224

Views

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 12 Apr 2026 - 0:09

score 41 · Answer 14 · 2026-03-25 12:50:59

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 93 · Answer 15 · 2026-04-30 06:50:57

93

Views

<< notation on measures

Published on 30 Apr 2026 - 6:50

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

Trending Questions

Popular # Hahtags

Popular Questions