User:Caliburn/s/mt/Lebesgue Decomposition Theorem/Complex Measure
Theorem
Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a measure on $\struct {X, \Sigma}$.
Let $\nu$ be a complex measure on $\struct {X, \Sigma}$.
Then there exists complex 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
- $(3) \quad$ $\nu = \nu_a + \nu_s$.
Proof
Let $\cmod \nu$ be the variation of $\nu$.
From Variation of Complex Measure is Finite Measure, $\cmod \nu$ is a finite measure.
Then from Lebesgue Decomposition Theorem for Finite Measures, there exists finite measures ${\cmod \nu}_a$ and ${\cmod \nu}_s$ on $\struct {X, \Sigma}$ such that:
- $(1) \quad$ ${\cmod \nu}_a$ is absolutely continuous with respect to $\mu$
- $(2) \quad$ ${\cmod \nu}_s$ and $\mu$ are mutually singular
- $(3) \quad$ $\cmod \nu = {\cmod \nu}_a + {\cmod \nu}_s$.
More precisely, the proof of this theorem grants that there exists a $\mu$-null set such that:
- $\map {\cmod \nu_a} A = \map {\cmod \nu} {N^c \cap A}$
and:
- $\map {\cmod \nu_s} A = \map {\cmod \nu} {N \cap A}$
for each $A \in \Sigma$, with ${\cmod \nu}_a$ and ${\cmod \nu}_s$ having the desired properties.
Let $\nu_a$ be the intersection measure of $\nu$ by $N^c$.
Let $\nu_s$ be the intersection measure of $\nu$ by $N$.
We verify that $\nu_a$ and $\nu_s$ are our desired complex measures.
Sources
- 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $4.3$: Singularity