Jordan Decomposition Theorem/Lemma

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

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

Let $\tuple {P, N}$ be a Hahn decomposition of $\mu$.

For each $A \in \Sigma$, define:


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

and:


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

Then:
 * $\mu^+$ and $\mu^-$ are signed measures.

Proof
We verify both of the conditions given in the definition of a signed measure.

Proof of $(1)$
We have, from Intersection with Empty Set:


 * $\map {\mu^+} \O = \map \mu \O = 0$

verifying $(1)$ for $\mu^+$.

We also have:


 * $\map {\mu^-} \O = -\map \mu \O = 0$

verifying $(2)$ for $\mu^-$.

Proof of $(2)$
Let $\sequence {D_n}_{n \in \N}$ be a sequence of pairwise disjoint $\Sigma$-measurable sets.

We have:

Since:


 * $D_i \cap D_j = \O$ whenever $i \ne j$

we have:


 * $\paren {P \cap D_i} \cap \paren {P \cap D_j} = \O$

from Intersection with Empty Set.

So, since $\mu$ is countably additive, we have:

That is:


 * $\ds \map {\mu^+} {\bigcup_{n \mathop = 1}^\infty D_n} = \sum_{n \mathop = 1}^\infty \map {\mu^+} {D_n}$

for any sequence $\sequence {D_n}_{n \mathop \in \N}$ of pairwise disjoint $\Sigma$-measurable sets, so $(2)$ is satisfied for $\mu^+$.

We also have:

As before, the sets $N \cap D_n$ are pairwise disjoint.

Since $\mu$ is countably additive, we have:

That is:


 * $\ds \map {\mu^-} {\bigcup_{n \mathop = 1}^\infty D_n} = \sum_{n \mathop = 1}^\infty \map {\mu^-} {D_n}$

for any sequence $\sequence {D_n}_{n \mathop \in \N}$ of pairwise disjoint $\Sigma$-measurable sets, so $(2)$ is satisfied for $\mu^-$.

So $\mu^+$ and $\mu^-$ satisfy both conditions $(1)$ and $(2)$, and so are both signed measures.