User:Caliburn/s/mt/Lebesgue Decomposition Theorem/Finite Signed Measure
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Theorem
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 $\struct {X, \Sigma}$.
Then there exists finite signed 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
- $(3) \quad$ $\nu = \nu_a + \nu_s$.
Sources
- 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $4.3$: Singularity