Generalized solutions of curvature equations

We introduce a notion of “generalized solutions” for the so-called curvature equations. It is an extension of the definition of generalized solutions of the Gauss curvature equation and gives a new framework for the study of more general curvature equations. We prove that if the curvature equation has a convex solution, then the inhomogeneous term must be a Borel measure. We also discuss basic properties of generalized solutions. Among other things, we show that our class of generalized solutions is wider than that of weak solutions.