AN ACR DC9-50 HAD A FALSE FIRE WARNING ON ITS 'A' LOOP FOR THE APU. THE 'B' LOOP WAS INOP.

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I DEPARTED STL IN A DC9-50 WITH THE APU 'B' FIRE WARNING LOOP LIGHT PLACARDED INOP. JUST PRIOR TO REACHING CRUISE ALT; THE APU 'A' LOOP LIGHT AND THE APU FIRE WARNING LIGHT CAME ON. THE WARNING LIGHTS CONTINUED INTERMITTENTLY TO GO ON AND OFF. WHEN THE LIGHTS CAME ON AND REMAINED ON; I INITIATED THE APU FIRE WARNING CHKLIST WHICH REQUIRES FOR THE FIRE EXTINGUISHER TO BE DISCHARGED IF THE WARNING PERSISTS. FOLLOWING DISCHARGE; THE WARNINGS CONTINUED INTERMITTENTLY AS BEFORE. THE CHKLIST REQUIRES LNDG AT NEAREST SUITABLE ARPT IF WARNING PERSISTS. BECAUSE OF SIMILAR PROBS WITH THE 'B' LOOP LIGHTS THE PREVIOUS LEG AND A LOGBOOK HISTORY OF LOOP MALFUNCTIONS; I BELIEVED THE PROB TO BE A MALFUNCTIONING LOOP SYS GIVING US A FALSE WARNING AND CONTINUED TO MY DEST. MAINT FOUND THE LOOP WIRE SHORTING TO GND. THE CHKLIST SHOULD ALLOW FOR JUDGEMENT; EXPERIENCE; AND COMMON SENSE TO BE A FACTOR IN DETERMINING WHERE TO LAND. MR X; AN FAA INSPECTOR; IN THE JUMP SEAT OBSERVED THE ENTIRE FLT AND CONCURRED WITH MY DECISION. SUPPLEMENTAL INFO FROM ACN 337628: THE APU 'A' LOOP ILLUMINATED ALONG WITH THE APU FIRE WARNING LIGHT AND THE MASTER WARNING LIGHT. THE APU HAD BEEN OFF APPROX 35-40 MINS. THE CHKLIST WAS TERMINATED TO AVOID EXHAUSTING THE CAPABILITY TO PROVIDE FIRE EXTINGUISHING FOR THE ENGS. MAINT PERSONNEL CONFIRMED THAT NO FIRE OR ABNORMAL HEAT HAD OCCURRED.