Jordan Decomposition Theorem
Theorem
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 $\mu$.
Then there exists measures $\mu^+$ and $\mu^-$ on $\struct {X, \Sigma}$ such that:
- $\mu = \mu^+ - \mu^-$
Further, at least one of $\mu^+$ and $\mu^-$ is finite.
Proof
From the definition of a Hahn decomposition, the set $P$ is $\mu$-positive, the set $N$ is $\mu$-negative and:
- $X = P \cup N$
with $P$ and $N$ disjoint.
From Sigma-Algebra Closed under Countable Intersection, we have:
- $A \cap P \in \Sigma$
and:
- $A \cap N \in \Sigma$
for each $A \in \Sigma$.
Now, for each $A \in \Sigma$, define:
- $\map {\mu^+} A = \map \mu {A \cap P}$
and:
- $\map {\mu^-} A = -\map \mu {A \cap N}$
We verify that $\mu^+$ and $\mu^-$ are indeed measures by first showing that they are signed measures.
Lemma
- $\mu^+$ and $\mu^-$ are signed measures.
$\Box$
We show that these are in fact measures by showing that $\mu^+ \ge 0$ and $\mu^- \ge 0$.
From Intersection is Subset, we have:
- $A \cap P \subseteq P$
Since $P$ is a $\mu$-positive set, we have:
- $\map \mu {A \cap P} \ge 0$
for each $A \in \Sigma$.
From Intersection is Subset, we have:
- $A \cap N \subseteq N$
Since $N$ is a $\mu$-negative set, we have:
- $\map \mu {A \cap N} \le 0$
for each $A \in \Sigma$.
So:
- $\map {\mu^+} A \ge 0$
and:
- $\map {\mu^-} A \ge 0$
for each $A \in \Sigma$.
So, from Non-Negative Signed Measure is Measure, we have:
- $\mu^+$ and $\mu^-$ are measures.
We now verify that at least one of the measures are finite.
From Intersection with Subset is Subset, we have:
- $\map {\mu^+} X = \map \mu {X \cap P} = \map \mu P$
and:
- $\map {\mu^-} X = -\map \mu {X \cap N} = -\map \mu N$
From the definition of a finite measure, we show that either:
- $\map \mu P < \infty$
or:
- $-\map \mu N < \infty$
Since $\mu$ is countably additive, we have that:
- $\map \mu X = \map \mu P + \map \mu N$
From Set is Subset of Itself, we have:
- $P \subseteq P$
so, since $P$ is $\mu$-positive, we have:
- $\map \mu P \ge 0$
Applying Set is Subset of Itself again we have:
- $N \subseteq N$
so, since $N$ is $\mu$-negative, we have:
- $\map \mu N \le 0$
By the definition of a signed measure, we cannot have both $\map \mu P = +\infty$ and $\map \mu N = -\infty$, so at least one of $\map \mu P$ and $\map \mu N$ is finite.
So, either:
- $\map {\mu^+} X = \map \mu P < \infty$
or:
- $\map {\mu^-} X = -\map \mu N < \infty$
That is, at least one of $\mu^+$ and $\mu^-$ are finite.
Finally, we verify that:
- $\mu = \mu^+ - \mu^-$
Note that $A \cap P$ and $A \cap N$ are disjoint with:
- $A = \paren {A \cap P} \cup \paren {A \cap N}$
so:
\(\ds \map \mu A\) | \(=\) | \(\ds \map \mu {A \cap P} + \map \mu {A \cap N}\) | since $\mu$ is countably additive | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \mu {A \cap P} - \paren {-\map \mu {A \cap N} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {\mu^+} A - \map {\mu^-} A\) |
So:
- $\mu = \mu^+ - \mu^-$
as required.
$\blacksquare$
Source of Name
This entry was named for Marie Ennemond Camille Jordan.
Sources
- 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $4.1$