  Can every measure-valued solution to the compressible Euler equations be
approximated by a sequence of weak solutions? We prove that the answer is
negative: Generalizing a well-known rigidity result of Ball and James to a more
general situation, we construct an explicit 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 M\"uller. This difference between weak and measure-valued solutions
in the compressible case is in contrast with the incompressible situation,
where every measure-valued solution can be approximated by weak solutions, as
shown by Sz\'ekelyhidi and Wiedemann.
