abstract: Székelyhidi and Wiedemann showed that any measure-valued solution to the incompressible Euler equations in several space dimensions can be generated by a sequence of exact solutions. This means that measure-valued solutions and weak solutions are substantially the same for incompressible Euler, thus leading to a very large set of weak solutions. In this talk we address the corresponding problem for the compressible Euler system: can every measure-valued solution to the compressible Euler equations be approximated by a sequence of weak solutions? We show that the answer is negative: generalizing a well-known rigidity result of Ball and James, we give an explicit example of a measure valued solution for the compressible Euler equations which can not be generated by a sequence of distributional solutions. We also give an abstract necessary condition for measure-valued solutions to be generated by weak solutions, relying on work of Fonseca and Müller. The dichotomy between weak and measure-valued solutions in the compressible case is in contrast with the incompressible situation. The results presented are joint work with E. Feireisl, O. Kreml and E. Wiedemann.