Bound for Variation of Complex Measure in terms of Jordan Decomposition

Theorem

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

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

Let $\cmod \mu$ be the variation of $\mu$.

Let $\tuple {\mu_1, \mu_2, \mu_3, \mu_4}$ be the Jordan decomposition of $\mu$.

Then:

$\map {\cmod \mu} A \le \map {\mu_1} A + \map {\mu_2} A + \map {\mu_3} A + \map {\mu_4} A$

for all $A \in \Sigma$.

Proof

Let $A \in \Sigma$.

Let $\map P A$ be the set of finite partitions of $A$ into $\Sigma$-measurable sets.

From the definition of variation, we have:

$\ds \map {\cmod \mu} A = \sup \set {\sum_{j \mathop = 1}^n \cmod {\map \mu {A_j} } : \set {A_1, A_2, \ldots, A_n} \in \map P A}$

Let:

$\set {A_1, A_2, \ldots, A_n} \in \map P A$

Consider:

$\ds \sum_{i \mathop = 1}^n \cmod {\map \mu {A_i} }$

We have, from the definition of Jordan decomposition:

$\mu = \mu_1 - \mu_2 + i \paren {\mu_3 - \mu_4}$

so that:

\(\ds \sum_{i \mathop = 1}^n \cmod {\map \mu {A_i} }\)

\(=\)

\(\ds \sum_{i \mathop = 1}^n \cmod {\map {\mu_1} {A_i} - \map {\mu_2} {A_i} + i \paren {\map {\mu_3} {A_i} - \map {\mu_4} {A_i} } }\)

\(\ds \)

\(=\)

\(\ds \sum_{i \mathop = 1}^n \cmod {\map {\mu_1} {A_i} } + \sum_{i \mathop = 1}^n \cmod {\map {\mu_2} {A_i} } + \sum_{i \mathop = 1}^n \cmod {\map {\mu_3} {A_i} } + \sum_{i \mathop = 1}^n \cmod {\map {\mu_4} {A_i} }\)

Triangle Inequality for Complex Numbers

\(\ds \)

\(=\)

\(\ds \sum_{i \mathop = 1}^n \map {\mu_1} {A_i} + \sum_{i \mathop = 1}^n \map {\mu_2} {A_i} + \sum_{i \mathop = 1}^n \map {\mu_3} {A_i} + \sum_{i \mathop = 1}^n \map {\mu_4} {A_i}\)

since $\mu_1$, $\mu_2$, $\mu_3$ and $\mu_4$ are measures

Since: $\set {A_1, A_2, \ldots, A_n}$ is a partition of $A$, we have:

$\set {A_1, A_2, \ldots, A_n}$ are pairwise disjoint with:

$\ds A = \bigcup_{i \mathop = 1}^n A_i$

So:

\(\ds \sum_{i \mathop = 1}^n \map {\mu_1} {A_i} + \sum_{i \mathop = 1}^n \map {\mu_2} {A_i} + \sum_{i \mathop = 1}^n \map {\mu_3} {A_i} + \sum_{i \mathop = 1}^n \map {\mu_4} {A_i}\)

\(=\)

\(\ds \map {\mu_1} {\bigcup_{i \mathop = 1}^n A_i} + \map {\mu_2} {\bigcup_{i \mathop = 1}^n A_i} + \map {\mu_3} {\bigcup_{i \mathop = 1}^n A_i} + \map {\mu_4} {\bigcup_{i \mathop = 1}^n A_i}\)

Measure is Finitely Additive Function

\(\ds \)

\(=\)

\(\ds \map {\mu_1} A + \map {\mu_2} A + \map {\mu_3} A + \map {\mu_4} A\)

$\blacksquare$