Equivalence of Definitions of Variation of Signed Measure

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Theorem

The following definitions of the concept of Variation of Signed Measure are equivalent:

Definition 1

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

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

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


We define the variation $\size \mu$ of $\mu$ by:

$\size \mu = \mu^+ + \mu^-$


Definition 2

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

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

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


We define the variation $\cmod \mu : \Sigma \to \overline \R$ of $\mu$ by:

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

for each $A \in \Sigma$, where the supremum is taken in the set of extended real numbers $\overline \R$.


Proof

We aim to prove that for each $A \in \Sigma$, we have:

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

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

Then $\set {P \cap A, N \cap A}$ is a finite partition of $A$ into $\Sigma$-measurable sets.

Then, we have from the definition of Jordan decomposition:

$\map \mu {P \cap A} = \map {\mu^+} A \ge 0$

and:

$\map \mu {N \cap A} = -\map {\mu^-} A \le 0$

Then:

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

and:

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

so:

\(\ds \map {\mu^+} A + \map {\mu^-} A\) \(=\) \(\ds \size {\map \mu {P \cap A} } + \size {\map \mu {N \cap A} }\)
\(\ds \) \(\le\) \(\ds \sup \set {\sum_{j \mathop = 1}^n \cmod {\map \mu {A_j} } : \set {A_1, A_2, \ldots, A_n} \in \map P A}\) Definition of Supremum of Subset of Extended Real Numbers

Conversely, for each $\set {A_1, A_2, \ldots, A_n} \in \map P A$, we have:

\(\ds \sum_{j \mathop = 1}^n \size {\map \mu {A_j} }\) \(=\) \(\ds \sum_{j \mathop = 1}^n \size {\map {\mu^+} {A_j} - \map {\mu^-} {A_j} }\) Definition of Jordan Decomposition
\(\ds \) \(\le\) \(\ds \sum_{j \mathop = 1}^n \paren {\size {\map {\mu^+} {A_j} } + \size {\map {\mu^-} {A_j} } }\) Triangle Inequality
\(\ds \) \(=\) \(\ds \sum_{j \mathop = 1}^n \paren {\map {\mu^+} {A_j} + \map {\mu^-} {A_j} }\) since $\mu^+ \ge 0$ and $\mu^- \ge 0$
\(\ds \) \(=\) \(\ds \sum_{j \mathop = 1}^n \map {\mu^+} {A_j} + \sum_{j \mathop = 1}^n \map {\mu^-} {A_j}\)
\(\ds \) \(=\) \(\ds \map {\mu^+} A + \map {\mu^-} A\) using finite additivity of $\mu^+$ and $\mu^-$

so:

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

so:

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

$\blacksquare$

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