Exercise 2, § 2

a) Let $L$ be a positive linear form on the Riesz space E. Consider an element $a > 0$ of E and the set of positive linear forms $M \leq L$ such that: $1^\circ$ $M(x) = L(x)$ for every $x$ such that $0 \leq x \leq a$; $2^\circ$ $M(x) = 0$ for every $x \geq 0$ alien to $a$. Show that this set of positive linear forms has a largest element $L_a$ and that, for every $x \geq 0$, $L_a(x) = \inf L(y)$, where $y$ runs over the set of all elements such that $0 \leq y \leq x$ and $x - y$ is alien to $a$. Show that $L_{\lambda a} = L_a$ for every scalar $\lambda > 0$, and that if $a$ and $b$ are alien then $L_{a+b} \leq L_a + L_b$. If E is fully lattice-ordered, show that if $a$ and $b$ are alien in E then $L_{a+b} = L_a + L_b$.

b) Take E to be the Riesz space of continuous real-valued functions on the interval $I = [0, 1]$ of $\mathbf{R}$, and $L(x) = x(\frac{1}{2})$; show by means of an example that one can have $L_{a+b} < L_a + L_b$ for two alien elements $a, b$ of E.