Radon-Nikodym derivatives add almost everywhere

The sum ${\nu }_1+{\nu }_2$ is finite-valued and countably additive, and vanishes on every $\mu$-null set, so is a finite signed or finite complex measure of the same type and is absolutely continuous. In the signed case [L3] gives finite total variation for both summands and the sum. Any increasing finite-measure exhaustion for $\mu$ is therefore common to these measures. Under AC apply [L1] to choose $h_1,h_2\in L^1(\mu )$ representing ${\nu }_1,{\nu }_2$, and to obtain the almost-everywhere unique density of their sum.

By [L2], $h_1+h_2$ is L1, and for every measurable $E$, $({\nu }_1+{\nu }_2)(E)={\int }_E{h}_{1}d\mu +{\int }_E{h}_{2}d\mu ={\int }_E(h_1+h_2)d\mu \mathrm{.}$ Thus it is a density of the sum. Uniqueness in [L1] gives the asserted equality of RN classes.