Consider a functional $G^+:L_p \to \mathbb{R}$. Here $L_p = L_p (X,\textbf{X}, \mu)$ is the collection of all integrable fns (f s.t. $\int \vert f \vert^p d \mu < \infty$ on the measure space $(X,\textbf{X}, \mu)$.
$G^+ (f)$ has already been defined for nonnegative fns f, and has been shown to satisfy $$G^+ (af+g)=a \cdot G^+ (f) + G^+(g) \tag1$$ $\forall$ fns $f,g \geq 0$, $a \in \mathbb{R}_{ \geq 0}$.
Extend the defn of $G^+$ to arbitrary $f \in L_p$ by $$G^+(f):=G^+(f^+)+G^+(f^-)$$where $f^+,f^-$ are the positive & negative parts of f.
Issue: I'm stuck trying to show (1) holds for arbitrary $f,g \in L_p, a \in \mathbb{R}$. I think I can already handle the multiplicative constant, so it's left to show $G^+ (f+g)=G^+ (f) + G^+(g)$. What I have so far is $f+g=f^+ + g^+ - (f^- + g^-)$, but it sadly isn't the case that $(f+g)^+ = f^+ + g^+$, for instance, if $f(x)=2, g(x)=-1$. I'd preferably like to use that (1) already holds for nonnegatives.
By the way, $G^+(f):=\sup \{ \vert G(g) \vert: g \in L_p, 0 \leq g \leq f \}$ for nonnegative f, where $G:L_p \to \mathbb{R}$ is another functional known to be linear and bounded. I'm stating this defn for completeness but hoping to avoid using it again.