# Questions tagged [signed-measures]

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29 questions
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### Signed Measure equivalent definition

Let $\mu$ be a signed measure, define $$|\mu|(E) = \mu^+(E) +\mu^-(E)$$ Why is it that $$|{\mu}|(E) = \text{sup}\sum_{k=1}^n|\mu(E_k)|,$$ where the supremum is taken over all finite disjoint ...
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### Can anyone prove this proposition? (Frame Functions)

Can anyone prove this proposition? It is proposition 3.2.2. out of A. Dvurecenskij's book on the applications of Gleason's Theorem. The proof in the book just says that it is obvious and doesn't ...
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### Prove that $m$ is a positive measure.

Let $X$ be a compact metrizable space, and $m$ be a finite signed, Borel, regular measure on $X$. Assume that for every continuous function $f:X\to [0,\infty)$ one has $\int_Xfdm\ge0$. How to show $m$ ...
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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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### 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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