$\textbf{Rudin's Real and Complex Analysis}$
$\textbf{6.10 The Theorem of Lebesgue-Radon-Nikodym}$
Let $\mu$ be a positive $\sigma$-finite measure on a $\sigma$-algebra $\mathfrak{M}$ in a set $X$, and let $\lambda$ be a complex measure on $\mathfrak{M}$.
(a) There is then a unique pair of complex measures $\lambda_a$ and $\lambda_s$ on $\mathfrak{M}$ such that
$\lambda=\lambda_a+\lambda_s$, $\lambda_a\ll \mu$, $\lambda_s\bot \mu$.
If $\lambda$ is positive and finite, then so are $\lambda_a$ and $\lambda_s$.
(b) There is a unique $h\in L^1(\mu)$ such that
$\lambda_a(E) = \int_Eh\,d\mu$
for every set $E\in\mathfrak{M}$.