Positive and Negative Parts of Signed Measure are Mutually Singular

Theorem

Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.

Let $\tuple {\mu^+, \mu^-}$ be the Jordan decomposition of $\mu$.

Then $\mu^+$ and $\mu^-$ are mutually singular.

Proof

From the Hahn Decomposition Theorem, there exists $\mu$-positive set and a $\mu$-negative set such that:

$X = P \cup N$

and:

$P \cap N = \O$

From the Jordan Decomposition Theorem, we have:

$\map {\mu^+} A = \map \mu {A \cap P}$

and:

$\map {\mu^-} A = -\map \mu {A \cap N}$

for each $A \in \Sigma$.

From the definition of a signed measure we have $\map \mu \O = 0$.

Hence $\map {\mu^+} N = 0$ and $\map {\mu^-} {X \setminus N} = 0$.

So $\mu^+$ is concentrated on $N^c$ and $\mu%-$ is concentrated on $N$.

So $\mu^+$ and $\mu^-$ are mutually singular.

$\blacksquare$