Signed Measure may not be Monotone
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Theorem
Let $\struct {X, \Sigma}$ be measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Then $\mu$ may not be monotone.
Proof
Let:
- $\struct {X, \Sigma} = \struct {\R, \map \BB \R}$
where $\map \BB \R$ is the Borel $\sigma$-algebra on $\R$.
Define:
- $\mu = \delta_1 - 2 \delta_2$
where $\delta_1$ and $\delta_2$ are the Dirac measures at $1$ and $2$ respectively.
Since $\delta_1$ and $\delta_2$ are both finite measures, we have:
- $\mu$ is a signed measure
from Linear Combination of Signed Measures is Signed Measure.
Then, we have:
- $\closedint 0 1 \subseteq \closedint 0 2$
with:
\(\ds \map \mu {\closedint 0 1}\) | \(=\) | \(\ds \map {\delta_1} {\closedint 0 1} - 2 \map {\delta_2} {\closedint 0 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 - 0\) | Definition of Dirac Measure | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
and:
\(\ds \map \mu {\closedint 0 2}\) | \(=\) | \(\ds \map {\delta_1} {\closedint 0 2} - 2 \map {\delta_2} {\closedint 0 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 - 2\) | Definition of Dirac Measure | |||||||||||
\(\ds \) | \(=\) | \(\ds -1\) |
So:
- $\closedint 0 1 \subseteq \closedint 0 2$ and $\map \mu {\closedint 0 2} \le \map \mu {\closedint 0 1}$
So:
- $\mu$ is not monotone.
$\blacksquare$