Radon-Nikodym derivative can depend on the $\sigma$-algebra Suppose $X$ is a set and $\mathcal{E} \subset \mathcal{F}$ are two $\sigma$-algebras of subsets of $X$. Let $\mu,\nu$ be two finite positive measures on $(X,\mathcal{F})$ and suppose $\nu \ll \mu$. Let $\bar{\mu}$ be the restriction of $\mu$ to $(X,\mathcal{E})$ and $\bar{\nu}$ the restriction of $\nu$ to $\mathcal{E}$. Find an example of the above framework where $\frac{d\bar{\nu}}{d\bar{\mu}} \ne \frac{d\nu}{d\mu}$, i.e, where the Radon-Nikodym derivative of $\bar{\nu}$ with respect to $\bar{\mu}(\text{in terms of $\mathcal{E}$)} $ is not the same as the Radon-Nikodym derivative of $\nu$ with respect to $\mu$ (in terms of $\mathcal{F}$). 
For this I took $X=\{0,1,2,3\},\mathcal{E}=\{\{0,1\},X,\phi,\{3,4\}\}, \mathcal{F}=\mathcal{P}(X)$. I chose $\mu$ as counting measure and $\nu(E)=\mu(E \cap\{1\})$. Then $\nu \ll \mu$. Then $\frac{d\nu}{d\mu}=\chi_{\{1\}}$ where as $\frac{d\bar{\nu}}{d\bar{\mu}}=\chi_{\{0,1\}}$. 
Is this alright? I was wondering if there are examples which look a little more beautiful. 
Thanks for the help!!
 A: Let $(X,\mathcal{F},m)$ be a finite measure space. Let $\mathcal{G}$ be a $\sigma$-algebra on $X$ contained in $\mathcal{F}$. Let $f \in L^1((X,\mathcal{F},\mu))$ and introduce the (in general signed) measure $\mu(A)=\int_A f dm$. Define $\overline{\mu}$ and $\overline{m}$ to be the restrictions to $\mathcal{G}$ of of $\mu$ and $m$ respectively.
Let $g=\frac{d \overline{\mu}}{d \overline{m}}$. We have $\frac{d \mu}{dm}=f$ and in general $g \neq f$. The reason is that $g$ is by definition $\mathcal{G}$-measurable (otherwise it makes no sense to integrate it with respect to $\overline{m}$). If $f$ isn't, then the two can't be the same. In this case $g$ is the best $\mathcal{G}$-measurable approximation to $f$, in a certain sense.
This construction originates in probability theory. In probability measurable functions are called random variables, and we say the random variable $g$ is the conditional expectation of the random variable $f$ with respect to the $\sigma$-algebra $\mathcal{G}$. In the case $\mathcal{G}=\{ X,\emptyset \}$, this is actually the ordinary expectation (with which you are probably familiar even without studying measure-theoretic probability theory). Radon-Nikodym is one way to prove that a $g$ with the desired properties exists and is unique up to a null set. Another is to introduce $f_n \in L^2$ such that $f_n \to f$ in $L^1$, and take $g$ to be the $L^1$-limit of the $L^2$-projections of the $f_n$ onto the space of $\mathcal{G}$-measurable functions. Cf. Approximation of conditional expectation
