User:Caliburn/s/mt/Definition:Lebesgue Decomposition/Sigma-Finite Measure

Definition
Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu$ be a measure on $\struct {X, \Sigma}$. Let $\nu$ be a $\sigma$-finite measure on $\struct {X, \Sigma}$.

Then from the Lebesgue Decomposition Theorem for Sigma-Finite Measures there exists unique $\sigma$-finite measures $\nu_a$ and $\nu_s$ on $\struct {X, \Sigma}$ such that:


 * $(1) \quad$ $\nu_a$ is absolutely continuous with respect to $\mu$
 * $(2) \quad$ $\nu_s$ and $\mu$ are mutually singular
 * $(3) \quad$ $\nu = \nu_a + \nu_s$.

We say that $\struct {\nu_a, \nu_s}$ is the Lebesgue decomposition of $\nu$ (with respect to $\mu$).