Lemma 13.1.13. Let $(E, \le)$ be a vector lattice, $C = \bracs{x \in E|x \ge 0}$ and $\phi: C \to [0, \infty)$ such that:
For any $x \in C$ and $\lambda \in \real$ with $\lambda \ge 0$, $\phi(\lambda x) = \lambda \phi(x)$.
For any $x, y \in C$, $\phi(x + y) = \phi(x) + \phi(y)$.
then the mapping
\[\Phi: E \to \real \quad x \mapsto \phi(x^{+}) - \phi(x^{-})\]
is a positive linear functional on $E$.
Proof. For any $\lambda \in \real$ with $\lambda \ge 0$,
\begin{align*}\Phi(\lambda x)&= \phi((\lambda x)^{+}) - \phi((\lambda x)^{-}) \\&= \lambda\phi(x^{+}) - \lambda\phi(x^{-}) = \lambda\Phi(x)\end{align*}
Likewise, if $\lambda < 0$, then
\begin{align*}\Phi(\lambda x)&= \phi((\lambda x)^{+}) - \phi((\lambda x)^{-}) \\&= -\lambda\phi(x^{-}) + \lambda\phi(x^{+}) = \lambda\Phi(x)\end{align*}
For any $x, y \in E$, let $z = x + y$, then $z = z^{+} - z^{-} = x^{+} + y^{+} - x^{-} - y^{-}$. Thus
\begin{align*}z^{+} + x^{-} + y^{-}&= z^{-} + x^{+} + y^{+} \\ \phi(z^{+}) + \phi(x^{-}) + \phi(y^{-})&= \phi(z^{-}) + \phi(x^{+}) + \phi(y^{+}) \\ \phi(z^{+}) - \phi(z^{-})&= \phi(x^{+}) - \phi(x^{-}) + \phi(y^{+}) - \phi(y^{-}) \\ \Phi(z)&= \Phi(x) + \Phi(y)\end{align*}
$\square$