The Radon-Nikodym derivative as an almost-everywhere equivalence class

Let $\mu$ be a sigma-finite positive measure.

1. If $\nu$ is a signed measure satisfying the hypothesis of A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density and $\nu \ll \mu$, the Radon-Nikodym derivative $d\nu \mathrm{/}d\mu$ is the $\mu$-almost-everywhere equivalence class of measurable real-valued functions $f$ satisfying the measurable-set integral formula in that theorem.
2. Assuming the Axiom of Choice (The Axiom of Choice), if $\nu$ is a finite complex measure and $\nu \ll \mu$, the Radon-Nikodym derivative $d\nu \mathrm{/}d\mu$ is the $\mu$-almost-everywhere equivalence class of complex $L^1(\mu )$ functions $h$ satisfying $\nu (E)={\int }_{E}hd\mu (E\in \mathcal{A}),$ as given by A finite complex measure absolutely continuous with respect to a sigma-finite positive measure has an integrable complex density.

In both cases the derivative is not a distinguished pointwise function: it is determined only up to $\mu$-almost-everywhere equality in the sense of Measure-null sets and almost-everywhere statements relative to a measure.

A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density

The Axiom of Choice

A finite complex measure absolutely continuous with respect to a sigma-finite positive measure has an integrable complex density

Measure-null sets and almost-everywhere statements relative to a measure