Equivalence of Definitions of Variation of Signed Measure

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:

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

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:


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