The space of Monge-Ampère functions, introduced by J. H. G. Fu, is a space of rather rough functions in which the map $u\mapsto \operatorname{Det} D^2 u$ is well defined and weakly continuous with respect to a natural notion of weak convergence. We prove a rigidity theorem for Lagrangian integral currents that allows us to extend the original definition of Monge-Ampère functions. We also prove that if a Monge-Ampère function $u$ on a bounded set $\Omega\subset\mathcal{R}^2$ satisfies the equation $\operatorname{Det} D^2 u=0$ in a particular weak sense, then the graph of $u$ is a developable surface, and moreover $u$ enjoys somewhat better regularity properties than an arbitrary Monge-Ampère function of $2$ variables.