Definition:Lebesgue Decomposition/Finite Signed Measure

Definition

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

Let $\mu$ be a measure on $\struct {X, \Sigma}$.

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

Let $\nu_a$ and $\nu_s$ be finite signed measures on $\struct {X, \Sigma}$.

We say that $\struct {\nu_a, \nu_s}$ is the Lebesgue decomposition of $\nu$ if and only if:

$(1) \quad$ $\nu_a$ is absolutely continuous with respect to $\mu$

$(2) \quad$ $\nu_s$ and $\mu$ are mutually singular

$(3) \quad$ $\nu = \nu_a + \nu_s$.

Also see

• Lebesgue Decomposition Theorem: Finite Signed Measure shows the existence and uniqueness of the Lebesgue decomposition