I have trouble understanding Radon-Nikodym theorem. Here's an example.
Consider the measurable space $(\Omega,F)$, where $\Omega = R$, $F~$ is the $\sigma$-algebra of Borel sets. Let $P[dw] = \frac{1}{\sqrt(2\pi)}e^{-w^2/2}dw$ and $\tilde{P}[dw] = e^{-w}1_{[0,\infty)(w)}dw$ be two probability measures on $\omega$. How to test whether $\tilde{P} << P$ and $P << \tilde{P}$. How do we compute Radon-Nikodym derivative $\frac{d\tilde{P}}{dP}$?