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.