Radon--Nikod\'{y}m Theorem

Let $(\Omega, \mathcal{F})$ be a measurable space (Definition~Sigma-Algebra) and let $\mu, \nu$ be $\sigma$-finite measures (Definition~Measure Space) with $\nu \ll \mu$ (i.e.\ $\mu(A) = 0 \Rightarrow \nu(A) = 0$). Then there exists a non-negative measurable function $f$, called the Radon--Nikod\'{ym derivative} $\frac{d\nu}{d\mu}$, such that \[ \nu(A) = \int_A f\, d\mu \quad for all A \in \mathcal{F}. \] The function $f$ is unique $\mu$-a.e.