BELOW GS; AN ATC LOW ALT ALERT; FOLLOWED BY A GAR; STARTED AND THEN ABANDONED; FINALLY ENDING IN AN 'UNEVENTFUL LNDG'?

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IN A DSCNT FROM 6000 FT TO 2000 FT ON VECTOR FOR VISUAL TO RWY 12R. ASSIGNED HEADING 050 DEGS FROM BIZEL INTXN. THEN FINAL SAYS L TO 030 DEGS FOR VECTOR ILS FINAL APCH COURSE. (WX CLR; 7 SM VISIBILITY). I SAID SPLIT THE DIFFERENCE AND WE'LL LIKELY PROCEED VISUALLY (1ST ERROR). FO ASKING FOR FINAL CONFIGN FOR LNDG. VECTOR WAS THROUGH THE LOC FINAL AND A TURN TO 150 DEGS TO REACQUIRE WAS GIVEN. FINAL CONFIGN WAS ACCOMPLISHED. CAPT ON VOR; FO ON ILS; CAPT ON VOR FOR AZIMUTH. CAPT ACQUIRED THE FIELD; VISUAL GIVEN. FO 'THOUGHT' HE HAD THE RWY. RAPCON SAYS 'STAY AT 2000 FT UNTIL 1000 FT.' I ASKED IF THE FO WAS ON GS AND LOOKED OVER -- NOT SAME TIME LOW ALT ALERT GIVEN BY RAPCON AND GO AROUND INSTRUCTED. DSCNT STOPPED; CLB INITIATED AND THEN FO ACQUIRES THE RWY. STABILIZED APCH RESUMED; TWR SWITCH; AND UNEVENTFUL LNDG MADE. TWR AND GND HAD SOME QUESTIONS ABOUT UNRELIABLE DME ON THE ILS. WE SAID NO. MY ERROR IN NOT MONITORING FO APCH. MY ERROR IN NOT SWITCHING THE ILS ONCE RWY WAS ACQUIRED. MY JUDGEMENT ERROR IN BEING COMPLACENT WITH FO PERFORMANCE. MY ERROR IN ASSUMING FO REALLY HAD RWY. ATTABOY TO FINAL CTLR; ALTHOUGH I DON'T THINK HE SAVED ANY 'BACON.' THIS WILL NOT HAPPEN AGAIN TO ME. CONSTANT VIGILANCE IN APCH PHASE AND ADHERENCE TO PROC IS THE ONLY ANSWER.