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


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