I am reading some lecture notes that have the following definition of a left invariant mean on a group:
$G$ is called amenable if there exists a bounded linear functional $m:L^{\infty}(G) \to \mathbb{C}$ such that:
$1)$ $m(f)\geq0$ for all $f\geq 0$
$2)$ $m(1_{G})=1$
$3)$ $m$ is left-invariant $(m(f)=m(\lambda_{g}f)$ for all $f\in L^{\infty}(G)$ and $g \in G$)
What I don't understand is, if $m$ is a function to $\mathbb{C}$ then what does $m(f)\geq 0$ mean? Or should $m$ actually be a function onto $\mathbb{R}$?
Thanks!