  We study non-linear functionals, including quasi-linear functionals, p-conic
quasi-linear functionals, d-functionals, r-functionals, and their relationships
to deficient topological measures and topological measures on locally compact
spaces. We prove representation theorems and show, in particular, that there is
an order-preserving, conic-linear bijection between the class of finite
deficient topological measures and the class of bounded p-conic quasi-linear
functionals. Our results imply known representation theorems for finite
topological measures and deficient topological measures. When the space is
compact we obtain four equivalent definitions of a quasi-linear functional and
four equivalent definitions of functionals corresponding to deficient
topological measures.
