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

Given: AC, a sigma-finite positive measure $\mu$ and a finite complex measure $\nu$ with $\nu \ll \mu$.

The real and imaginary parts of $\nu$ are finite signed measures. (The real and imaginary parts of a complex measure are finite signed measures, and nu = Re nu + i Im nu)

Absolute continuity means vanishing on every measurable reference-null set. Thus $\nu \ll \mu$ implies both real and imaginary parts vanish there too. (Absolute continuity of a signed or complex measure with respect to a positive measure)

A signed measure satisfying a common finite exhaustion with $\mu$ and absolutely continuous with respect to $\mu$ has an almost-everywhere unique density; if its total variation is finite, the density is integrable (A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density).

A complex function is integrable exactly when its real and imaginary parts are integrable, using $\mathrm{\mid }u\mathrm{\mid },\mathrm{\mid }v\mathrm{\mid }\le \mathrm{\mid }u+iv\mathrm{\mid }\le \mathrm{\mid }u\mathrm{\mid }+\mathrm{\mid }v\mathrm{\mid }$. (Integrable real and complex functions, and their integrals)

A complex measure has finite total variation (Every complex measure has finite total variation).

AC supplies the hypotheses of the signed RN existence theorem and its underlying Hahn/Jordan selections. (The Axiom of Choice)