Likelihood ratio as a Radon–Nikodym derivative

Let $P_0$ and $P_1$ be probability measures on the same measurable space. Assume AC. Their Lebesgue decomposition relative to $P_0$ is the unique decomposition $P_1=P_{1,\mathrm{a}\mathrm{c}}+P_{1,\mathrm{s}}$ with $P_{1,\mathrm{a}\mathrm{c}}\ll P_0$ and $P_{1,\mathrm{s}}\perp P_0$ (Every sigma-finite signed measure admits a Lebesgue decomposition relative to a sigma-finite positive measure, The Lebesgue decomposition of a sigma-finite signed measure is unique). Both components are positive by Lebesgue components of a positive finite measure are positive, and $P_{1,\mathrm{a}\mathrm{c}}$ is finite because their sum has mass one. Define the likelihood-ratio density of $P_1$ against $P_0$ to be the $P_0$-almost-everywhere class

$$\frac{d{P}_{1,\mathrm{a}\mathrm{c}}}{d{P}_0}$$

of The Radon-Nikodym derivative as an almost-everywhere equivalence class. Under AC this class has a nonnegative measurable representative by A sigma-finite positive measure has a nonnegative Radon–Nikodym density version.

This is a full density of $P_1$ with respect to $P_0$ only in the absolutely continuous case $P_1\ll P_0$. If that condition fails, the singular component remains separate. A displayed representative is not a canonical pointwise function: its values are determined only $P_0$-almost everywhere.