Radon–Nikodym Derivative

If measures $\nu$ and $\mu$ satisfy $\nu\ll\mu$ (absolute continuity), the Radon–Nikodym derivative $\frac{d\nu}{d\mu}$ is the (a.e. unique) function $X$ with $\nu(S)=\int_S X\,d\mu$ for all measurable $S$.

Key property (paper use):

• Shale writes $X(T)=\frac{dn(T)}{dn}$ for the Gaussian pushforward (the “Jacobian”).

Example: For $\nu=f\mu$ with $f\ge0$, one has $d\nu/d\mu=f$.