The density line and its positive cone

Fix a basis $e$ (the empty basis when $n=0$). Every basis is uniquely $eA$ for $A\in GL(n,\mathbb{R})$. A density is determined by $c=\delta (e)$ since $\delta (eA)=c\mathrm{\mid }\det A\mathrm{\mid }$. Conversely this formula, with zero on dependent tuples for $n\ge 1$, satisfies the required transformation law by multiplicativity of determinants.

The formula is linear in $c$, so evaluation and its displayed inverse are linear bijections. Since $\mathrm{\mid }\det A\mathrm{\mid }>0$, positivity is equivalent to $c>0$ and nonnegativity to $c\ge 0$, independently of the chosen basis. Thus the nonzero nonnegative densities are exactly one ray. For $n=0$ this says precisely that scalars form $\mathbb{R}$.