Signed Measure $\nu$ mapping $A \mapsto \int_A f d\mu$ and the Radon-Nikodym derivative

Question

I am supposed to take an exam in August and so I am trying to prepare (this is not homework). So far I am pretty good at most topics, but anything related to Radon-Nikodym I don't quite understand. Here is a question that is stumping me for example:

Suppose $(X,\mathscr{F},\mu)$ is a finite measure space and there exists a $E \in \mathscr{F}$ such that $0<\mu(E)<\mu(X)$. In addition, suppose $\mathscr{G}$ is the $\sigma$-algebra generated by $\{E\}$.

(1) Describe $\mathscr{G}$ and $\mathscr{G}$-measurable functions, $f: X \rightarrow \mathbb{R}$

(2) Let $f:X \rightarrow \mathbb{R}$ be $\mathscr{F}$-measurable and $\mu$-integrable. Let $\nu$ be the signed measure on $(X, \mathscr{G})$ given by $A \mapsto \int_A f d \mu$ for $A \in \mathscr{G}$. On $(X, \mathscr{G})$ determine the Radon-Nikodym derivative of $\nu$ with respect to $\mu \restriction _\mathscr{G}$

any hints or suggestions would be greatly appreciated.

Answer 1

• $\mathscr{G}=\{ \emptyset , X , E , X\setminus E \}$.
• $f$ is $\mathscr{G}$-measurable means that $f=a\chi_{\{E\}} + b\chi_{\{X\setminus E\}}$ where $a$, $b\in \mathbb{R}$.
• I don't get the last part, $f$ is the R.-N. derivative by definition. Is there maybe something more to that question?