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Questions tagged [signed-measures]

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Radon-Nikodym Derivative of a Total Variation Measure

Let $\nu$ be a signed measure which is absolutely continuous to a sigma-finite measure $\mu$. Show that $\frac{d|\nu|}{d\mu}=|\frac{d\nu}{d\mu}|$, where $|\nu|$ is the total variation measure of $\nu$...
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Equality of finite signed measures by showing that the integrals of every bounded continuous function coincide

In order to show the uniqueness of the Fourier coefficients of a signed measure, I need to show that : For any two finite signed measures $\nu_1, \nu_2$ on $\left([-\pi, \pi], \mathcal{B}_{[-\pi, ...
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1answer
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Continuity of Lebesgue Stieltjes integral

I am trying to prove that Lebesgue-Stieltjes integral defines a cadlag function (i.e. right continuous with left limits) when its integrator is a cadlag function. Assume that $A(s)$, $s\in \mathbb{R}...
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1answer
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An expression for a Total Variation Measure of a Signed Measure

Let $F$ be a Sigma algebra on a set $X$, and let $\mu$ and $\nu$ be probability measures on $(X,F)$, i.e. $\mu(X)=\nu(X)=1$. Finally let $\eta=\mu-\nu$. Show that $$|\eta|(X)=2\sup_{E\in F}|\eta(E)|,...
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Finding a Hahn Decomposition involving a Dirac Measure

Let $(X,F,\mu)$ be a finite measure space, i.e. $\mu(X)<\infty$. And let $x\in X$, and let $\delta_x$ be the Dirac measure with respect to $x$, i.e. $\delta_x(E)=1$ if $x\in E$ and $\delta_x(E)=0$ ...
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1answer
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Prove that $\mu \cup \nu $ is the largest signed measure that is less than or equal to both $\mu,\nu$.

I am learning Signed Measures as a Part of my PhD coursework and I have got this exercise after just learning 1 page. I dont know how to proceed. Let $\mu ,\nu $ ne two finite measures on the ...
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1answer
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Finding possible p's for signed measure

Im new here at StackExchange and hope some of you can help me with a problem I'm dealing with. We are considering a signed measure $\nu$ on $(\mathbb{N},2^{\mathbb{N}})$, which is given by $\nu(\{k\}...
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Signed measure of uncountable set

I have a question and hope some of you can help me :) Consider a signed measure $\nu$ on $(\Omega, \bf{A})$ and let be $P_i \in \bf{A}$ positive sets, such that $ \forall B \subset P_i: \nu(B) \geq0 $...
3
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1answer
82 views

Countably additive finite signed measures form a Banach Space.

I'm currently studying some topics in measure theory and I am not sure how to prove the following: Let $X$ a set, $\mathcal A$ a $\sigma$-algebra on X. Consider the set: $$ca(\mathcal A) = \{\mu:\...
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0answers
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How to introduce a signed finite measure into a Borel $\sigma$-algebra?

This question is a sequence to: How to introduce a probability measure on the space of curves? Given the set $\Omega_{n} = \{\alpha:[0,1]\rightarrow[0,1]^{n}\,|\,\alpha\,\,\text{is smooth}\}$, ...
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Correspondence between dual of the space of continuous maps and signed measures

On the first page of Chapter 4 of Einsiedler an Ward's Ergodic Theory: With a View Towards Number Theory, the following statement is found: Let $(X, d)$ be a compact metric. Recall that the dual ...
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1answer
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Equivalence of the Variation of F$_\mu$(x) and $| \mu |$(($-\infty, x])$

Let $\mu$ be a finite signed Borel measure. I want to see why V$_{F_{\mu}}$($-\infty, x]$ = $| \mu | $ $((-\infty , x])$ for all x in R, where F$_\mu$(x) = $\mu((- \infty, x])$ and V$_{F_{\mu}}$($-\...
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If $\mathcal A$ is an algebra and $\mu$ is a vector measure on $\sigma(\mathcal A)$, can we approximate $\sigma(\mathcal A)$ by $\mathcal A$?

Let $\Omega$ be a set $\mathcal A$ be an algebra on $\Omega$ $E$ be a $\mathbb R$-vector space $\mu:\sigma(\mathcal A)\to E$ be $\sigma$-additive I want to show that for all $A\in\sigma(\mathcal A)...
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Characterizing the dual space of the linear space of the signed measures generated by a given set of measures.

Let $(\Omega,\mathcal{F})$ be a measurable space, $\varphi,\psi_{1},\ldots,\psi_{p}$ be real valued measurable functions on $(\Omega,\mathcal{F})$ and the set $\mathcal{M}$ of all non-negative ...
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1answer
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If $\mu$ is a finite measure and $ν$ is a signed measure with $|\nu|\le C\mu$, are we able to show $\left|\frac{{\rm d}ν}{{\rm d}\mu}\right|\le C$?

Let $(\Omega,\mathcal A,\mu)$ be a finite measure space $\nu$ be a signed measure on $(\Omega,\mathcal A)$ with $$|\nu(A)|\le C\mu(A)\;\;\;\text{for all }A\in\mathcal A\tag1$$ for some $C\ge0$ Note ...
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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?

Let $(\Omega,\mathcal A,\mu)$ be a finite measure space $\mathcal E$ be a semiring with $\sigma(\mathcal E)=\mathcal A$ $\nu$ be a signed measure on $(\Omega,\mathcal A)$ with $$|\nu(A)|\le C\mu(A)\;\...
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Convergence of supremum of the series of a signed measure

Let $(X, \Sigma, \mu)$ be a measure space of finite measure and $f\in L^1 (\mu)$. For every $E\in \Sigma$ we define: $$v(E) = \int _ E f d\mu$$ Suppose that $f\in L^p (\mu)$ for some $p \in [1, + \...
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If $μ$ is a signed measure, $Ω^±$ is a Hahn decomposition and $μ^±$ is the Jordan decomposition, then $\int_{A∩Ω^±}X{\rm d}μ=\pm\int_AX{\rm d}μ^±$

Let $(\Omega,\mathcal A,\mu)$ be a finite signed measure space $(\Omega^+,\Omega^-)$ be a Hahn decomposition of $\Omega$ with respect to $\mu$ and $(\mu^+,\mu^-)$ denot the Jordan decomposition of $\...
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Show that $L^1(\nu)=L^1(|\nu|)$, where $\nu$ is a complex

Suppose that $\nu$ is a complex measure. Show that $L^1(\nu)=L^1(|\nu|)$. This is my attempt to prove this assertion. To begin with, suppose that $\mu=|\nu_r|+|\nu_i|$. Then $\nu_r <<\mu$ and $...