Several notions of weak or 'very weak' solutions have been suggested for the incompressible and compressible Euler systems, motivated by the lack of a satisfactory well-posedness theory for these equations in turbulent regimes. Surprisingly, the speaker and L. Székelyhidi showed in 2012 that dis- tributional and measure-valued solutions are in a sense the same, although the latter had been expected to be a much weaker notion. In this talk, we turn to the isentropic compressible Euler system, where the situation is fundamentally different.