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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.

The idea of signed measure is an extension of the measure of . A signed measure also is a function with domain of definition a given sigma-algebra of sets, but is more general than ordinary measure in that the value assigned to a set may be a negative real number.

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Show that the total variation distance of probability measures $\mu,\nu$ is equal to $\frac{1}{2}\sup_f\left|\int f\:{\rm d}(\nu-\mu)\right|$

Let $(E,\mathcal E)$ be a measurable space, $\mu$ and $\nu$ be probability measures on $(E,\mathcal E)$ and $$|\nu-\mu|:=\sup_{B\in\mathcal E}|\nu(B)-\mu(B)|$$ denote the total variation distance of $\mu$ and $\nu$. How can we show that…
0xbadf00d
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Outer signed measure

I would like to ask whether there is some kind of analogue of outer measure when dealing with signed measures. I would like to assign measure to all the subsets, not just some $\sigma$-field. I'm using Evans-Gariepy's book "Measure theory and fine…
Jkbb
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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:\mathcal A \to \mathbb R|\; \mu \; \text{is a …
Victor Ronchim
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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 Analysis and I wonder why in the following…
Alileo
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How to prove that $\mu_n \rightharpoonup \mu$ IFF $\mu_n \overset{*}{\rightharpoonup} \mu$ and $\mu_n (X) \to \mu (X)$?

Let $X$ be a metric space, $\mathcal M(X)$ the space of all finite signed Borel measures on $X$, $\mathcal C_b(X)$ be the space of real-valued bounded continuous functions, $\mathcal C_0(X)$ be the space of real-valued continuous functions that…
Analyst
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If $\mu_n \overset{\ast}{\rightharpoonup}\mu$, then $\mu^+_n \overset{\ast}{\rightharpoonup} \mu^+$ and $\mu^-_n \overset{\ast}{\rightharpoonup}\mu^-$

Let $X$ be a Polish space and $\mu, \mu_n$ finite signed Borel measures on $X$. Assume that $\mu_n \overset{\ast}{\rightharpoonup} \mu$, i.e., $$ \int_X f \mathrm d \mu_n \to \int_X f \mathrm d \mu $$ for all bounded continuous functions $f:X \to…
Akira
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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 $\mu=\mu_1-\mu_2$

Let $\mu_1$ and $\mu_2$ be two finite measures on $(\Omega, \mathcal{F})$. Let $\mu_1 = \mu_{1a}+\mu_{1s}$ be the Lebesgue decomposition of $\mu_1$ w.r.t. $\mu_2$, that is, $\mu_{1a} \ll \mu_2$ and $\mu_{1s}\perp \mu_2$. Let $\mu = \mu_1 - \mu_2$.…
Protosartorium
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Integration with respect to total variation norm measure

Let $\mu$ be a signed measure. Then it has the Jordan decomposition $\mu=\mu^{+}-\mu{-}$. The total variation is defined as $|\mu|=\mu^{+}+\mu{-}$. I am now wondering how to compute an integral $\int f d|\mu|$? I have seen in some other questions…
guest1
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Portmanteau theorem for finite signed Borel measures

Let $X$ be a metric space, $\mathcal M(X)$ the space of all finite signed Borel measures on $X$, $\mathcal M_+(X)$ the space of all finite nonnegative Borel measures on $X$, and $\mathcal C_b(X)$ be the space of real-valued bounded continuous…
Analyst
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Let $\mu_n,\mu \in \mathcal M(X)$ such that $\mu_n \rightharpoonup \mu$ and $[\mu_n] \to [\mu]$, then $|\mu_n| \rightharpoonup |\mu|$

Let $X$ be a metric space, $\mathcal M(X)$ the space of all finite signed Borel measures on $X$, $\mathcal M_+(X)$ the space of all finite nonnegative Borel measures on $X$, and $\mathcal C_b(X)$ be the space of real-valued bounded continuous…
Analyst
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$3$ versions of Riesz–Markov–Kakutani theorem

I'm reading RKM theorem from this lecture note by professor Tomasz Kochanek. I have no question here. This thread is to summarize $3$ versions of the theorem (in an increasing order of generality). I try to include related definitions to remove any…
Analyst
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Is it possible to determine the sign of the determinant of a matrix without knowing/using the formula for the determinant?

I'm trying to build intuition for the orientation of a set of vectors independent of the well-known definition of the determinant. My intuition wants to go something like this: any set of vectors can be transformed into the standard basis by a…
Jagerber48
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Signed Borel Measures and Functions of Bounded Variation

Let $\nu$ be a finite signed Borel measure on the closed interval $[a,b]$. We can define a function $F_\nu : [a,b]$ by $$F_\nu(x) = \nu([a,x]).$$It can be shown that $F_\nu$ has bounded variation and is right-continuous. An exercise in my class…
Nick A.
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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 $A(s)$, $s\in \mathbb{R}_+$ is a right-continuous function, has left…
Mushtandoid
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Signed Measures and Radon-Nikodym Theorem with Total Variation Measure

This is a question from Richard Bass, Real Analysis for Graduate Students. If $\mu$ is a signed measure on $(X,A)$ and $|\mu|$ is the total variation measure, prove that there exists a real-valued function $f$ that is measurable with respect to $A$…
Tyler88
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