The space of Monge–Ampère functions, introduced by J. H. G. Fu, is a space of rather rough functions in which the map
$u\,\mapsto \,\text{Det}\,{{\text{D}}^{2}}u$ 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
$\Omega \,\subset \,{{\mathbb{R}}^{2}}$ satisfies the equation
$\text{Det}\,{{D}^{2}}u\,=\,0$ 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.