Exercise on Radon–Nikodym derivative

Question

Consider the measure space $(\mathbb{N}, P(\mathbb{N}))$ and two measures $\mu$ and $\nu$, where $\mu$ is the counting measure and $\nu$ is defined as $$ \nu(E) = \sum_{n \in E}(n+1)$$

Compute $d\mu/d\nu$ if its possible.

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How would you proceed?

Thanks in advice!

Julian

Answer 1

By definition the Radon-Nikodym derivative is a function $f$ such that $$\nu (E)=\int _Efd\mu,$$ in the counting measure this means $$\nu (E)=\int _Efd\mu =\sum _{n\in E}f(n),$$ so is not $f(n)=n+1.$
Edit: Taking the inverse problem, $d\mu /d\nu =\frac{1}{n+1}.$