Definition:Lebesgue Decomposition/Complex Measure
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Definition
Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a measure on $\struct {X, \Sigma}$.
Let $\nu$ be a complex measure on $\struct {X, \Sigma}$.
Let $\nu_a$ and $\nu_s$ be complex 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: Complex Measure shows the existence and uniqueness of the Lebesgue decomposition