User:Caliburn/s/mt/Definition:Lebesgue Decomposition/Sigma-Finite 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 $\sigma$-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 $\mu$).