Questions tagged [signed-measures]
A signed measure is a countably additive set function on a sigma-algebra and taking values in the extended reals, but not permitted to assign negative infinity to a set.
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Prove: $F:M(X,\mathscr{A},\mathbb{R})\to\mathbb{R}$ defined by $F(\mu)=\int fd\mu$ is a linear functional. [closed]
I need to prove the following result:
Remark 4.30$\quad$ Let $M(X,\mathscr{A},\mathbb{R})$ be the collection of all finite signed measures on $(X,\mathscr{A})$. Let $B(X,\mathscr{A},\mathbb{R})$ be ...
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Signed measure and bounded total variation on algebras
Let $X$ be a compact Hausdorff second countable topological space. Let $\mathcal{B}$ a countable basis of the topology of $X$, which we can assume to be closed under finite unions and intersections, ...
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Why does $\mu\mapsto\int fd\mu$ defines a linear functional on $M(X,\mathscr{A},\mathbb{R})$?
I am self-studying signed measure.I got stuck on the following remark:
Remark$\quad$ Let $M(X,\mathscr{A},\mathbb{R})$ be the collection of all finite signed measures on $(X,\mathscr{A})$. Then the ...
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Question About Signed Measures
I am self-studying signed measure, and I come across the following construction:
Let $\mu$ be a signed measure on the measurable space $(X,\mathscr{A})$, and let $A$ be a subset of $X$ that belongs ...
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Prove: Define $\nu$ on $\mathscr{A}$ by $\nu(A) = \int_Afd\mu$. Then $\nu$ is a signed measure on $(X,\mathscr{A})$.
I am self-studying measure theory and I come across the following question:
Prove: Let $(X,\mathscr{A},\mu)$ be a measure space, let $f$ belong to $\mathscr{L}^1(X,\mathscr{A},\mu,\mathbb{R})$, and ...
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Relation between total variation measure and total variation of a function
It is well known that if $f$ defined on a compact interval $[0,T]$ is a continuous function with finite variation, then $f$ induces a signed measure $\mu$ on $[0,T]$. Let $|\mu|$ be the total ...
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Sum of two signed measures is well defined iff both do not take the same sign of $\pm \infty$
Let $(X,\mathcal{A})$ be a measurable space and $\mu$ and $\nu$ be a signed measures on $(X,\mathcal{A})$. We say that $\lambda =\mu+\nu$ is well-defined as a function
$\lambda \: \mathcal{A} \to \...
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Uniqueness of extension of signed measures
By Caratheordory's extension theorem, if the measure space is $\sigma-$ finite and we have two measures that are equal on an algebra generating the $\sigma-$ algbera, then they are equal. Will the ...
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Sigma additivity of signed measure
Suppose we have $(X, M, v)$ as a measure space. $v$ is a signed measure, then it also satisfies sigma additivity, i.e.
If {$E_j$} is a sequence of disjoint sets in $M$ , then $v(\cup_1^\infty E_j)=\...
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Why are complex measures not allowed to attain $\infty$ while signed measures are?
I saw another question similar to this one but I'm not satisfied by answers. Here I changed the question to clearify the point I am interested in. I study Measure Theory for Real & Complex ...
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Characterization of absolute continuity for finite additive functions
Let $\nu:\mathbb{X}\longrightarrow \mathbb{R}$ and additive function, where $(X,\mathbb{X},\mu)$ is a measure space. That is, $\nu(\emptyset)=0$, and for any finite disjoint family of measurable sets $...
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Clarification on the definition of signed measure: Is the rearrangement issue assumed to be impossible by the definition?
Let $(\Omega,M,\mu)$ be a $\underline{\text{signed}}$ measure space. Is it true that the definition of signed measure implicitly implies that
Either for any sequence of disjoint measurable sets with ...
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Folland Theorem 3.22 Proof Explanation
At the first step of Theorem 3.22 of Folland, he mentioned that $d\nu = d\lambda + f\,dm$ implies $d|\nu| = d|\lambda| + |f| dm$:
I am aware of the similar questions has been asked before on this ...
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Won't signed measures upset Riemann?
I can't seem to wrap my head around signed measures.
If $\mu$ is a signed measure on a measurable space $(X, \Sigma)$, then there'll be sets of both positive and negative measure. Let $E_1, E_2, \...
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boundedness of signed measures
Let us consider signed charges, these are finitely additive signed measures (I suppose this would also work with sigma additive signed measures). We work on a measure space $(\Omega, \mathcal{A})$, ...
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