Lemma 21.3.1 (Jordan Decomposition).label Let $X$ be a topological space and $I \in C_{0}(X; \real)^{*}$, then there exists positive linear functionals $I^{+}, I^{-} \in C_{0}(X; \real)^{*}$ such that:
- (1)
$I = I^{+} - I^{-}$.
- (2)
$I^{+} \perp I^{-}$.
- (3)
$\norm{I^+}_{C_0(X; \real)^*}, \norm{I^-}_{C_0(X; \real)^*}\le \norm{I}_{C_0(X; \real)^*}$.
Proof. (1), (2): Since $C_{0}(X; \real)$ is a Banach lattice, $C_{0}(X; \real)^{*}$ is also a Banach lattice by Proposition 15.2.3. Therefore there exists positive linear functionals $I^{+}, I^{-} \in C(X; \real)^{*}$ such that $I = I^{+} - I^{-}$ and $I^{+} \perp I^{-}$.
(3): By Proposition 15.2.2.$\square$