# Tagged Questions

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### diffeomorphisms preserve zero measure

Suppose $\Omega\subset \mathbb R^N$ is an open set and $f:\Omega\rightarrow f(\Omega)$ is a $C^1$ diffeomorphism. Show that if $F \subset \Omega$ has zero measure then $f(F)$ has zero measure. I ...
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### Signed Measure Decomposition

Disclaimer: This thread is meant as record and written in Q&A style. Additional answers are heartly welcome, too! A complex measure can be decomposed by exploiting the Radon-Nikodym derivative: ...
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### Complex Measure Integration

Disclaimer: This thread is meant as record and written in Q&A style. Additional answers are heartly welcome, too! Integration w.r.t. complex measure usually is defined via the Radon-Nikodym ...
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### Spectral Measure Integration: Unbounded Functions?

Given a Hilbert space $\mathcal{H}$ and spectral a measure $E:\Sigma(\Omega)\to\mathcal{B}(\mathcal{H})$. How to define the integral for unbounded measurable functions: ...
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### Lesbegue integration w.r.t scaled measure

I was wondering if we have a measure $\mu$ and $a \in \mathbb{R}$ with $\lambda = a\mu$ if we get : \begin{align} \int fd\lambda = a\int fd\mu \end{align} It seems to work for the basic definition of ...
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### What types of integrals cannot be solved using improper Riemann-Stieltjes Integration?

I came across the wikipedia discussion of the Riemann-Stieltjes integral. The first sentence in the "Generalization" section gave me pause: An important generalization is the Lebesgue–Stieltjes ...
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### Open problems in Federer's Geometric Measure Theory

I wanted to know if the problems mentionned in this book are solved. More specifically, at some places, the author says that he doesn't know the answer, for example :"I do not know whether this ...