# 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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### 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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### 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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### 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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### (Signed) measures defined on algebra instead of sigma algebra

So some authors consider (sigma additive) signed measures on algebras (that are closed only under finite unions) instead of on sigma algebras. This happens for instance when considering the subset of ...
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### Looking for books covering signed measures in detail

so I am looking for good text books that treat signed measures, integration thereof and total variation etc. In particular also for finitely additive signed measures. I have found Dunford and Schwartz ...
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### Is this functional jointly continuous in the measure and the integrand?

Let $\Delta[0,1]$ be the space of all signed Borel measures over $[0,1]$, ranging between -1 and 1. $[0,1]^{[0,1]}$ is the set of all functions with domain $[0,1]$ that takes values in $[0,1]$. ...
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### Show that a set function is sigma additive on an algebra, but not extendable to a signed measure on the generated sigma algebra.

I am currently preparing for a measure theory exam and struggling with the following problem: Consider the algebra \begin{align} \mathfrak A = \{ A \subseteq \mathbb R: |A| < \infty ~ \text{or} ...
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### Finiteness of signed measure

Let $\nu$ be a signed measure and $|\nu|$ be its total variation. Then does $\nu \text{ finite } \implies |\nu| \text{ finite}$? If so, why? I can almost see this by appealing to the Jordan ...
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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 ...
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Let $\mathcal{H}$ be a reproducing kernel Hilbert space of functions over $X$, with a bounded kernel $\mathcal{K}: X\times X\to \mathbb{R}$. Let us assume there is a sigma-algebra over $X$ such that \$\...