WestudytheexistenceanduniquenessproblemforthenonhomogeneouspressurelessEulersystemwiththeinitialdensitybeingaRadonmeasure.Ouruniquenessresultisobtainedinthesamespaceastheexistencetheorem.Besides,bycounterexampleweprovethatHuang-Wang'senergyconditionisalsonecessaryforournonhomogeneoussystem.