In　this　paper,　we　study　the　Riemann　problem　with　the　initial　data　containing　the　Dirac　delta　function　for　the　isentropic　relativistic　Chaplygin　Euler　equations.　Under　suitably　generalized　Rankine-Hugoniot　relation　and　entropy　condition,　we　constructively　obtain　the　global　existence　of　generalized　solutions　including　delta　shock　waves　that　explicitly　exhibit　four　kinds　of　different　structures.　Moreover,　it　can　be　found　that　the　solutions　constructed　here　are　stable　for　the　perturbation　of　the　initial　data.