AN LTT ACR NEARLY JOINED UP ON AN MLG ACR WHILE MAKING A NIGHT VISUAL APCH TO LAX.

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WE WERE BEING VECTORED FOR A VISUAL APCH BY LA APCH CTL. INITIALLY AT 5000 FT OVER SMO VOR AT A HDG OF 070 DEGS FOR THE DOWNWIND LEG. ABOUT 4 MI PAST SMO WE WERE GIVEN 2500 FT AND TOLD TO LOOK FOR AN ACR MLG BASE TO FINAL FOR 24R. BOTH OF US SAW WHAT WE THOUGHT TO BE THE CORRECT ACFT AND RPTED IT IN SIGHT. WE WERE THEN GIVEN A HDG OF 130 DEGS AND TOLD THE TFC WAS AT 12 O'CLOCK. AGAIN; WE BOTH SAW THE SAME ACFT. WE RPTED IT AND WERE CLRED FOR THE VISUAL APCH TO 24R FOLLOWING THE OTHER ACFT. WE SWITCHED TO TWR JUST AS WE WERE ABOUT TO TURN FINAL. I THINK WE WERE AT 2000 FT AT THIS POINT. THE MLG PLT WAS THE ONLY ONE AT THIS TIME TO CALL OUT THE CONFLICT. HE WAS AT OUR 10-9 O'CLOCK POS WHEN HE SAW US AND AT THAT POINT WE SAW HIM ABOUT 3000 FT AWAY FROM US. WE TURNED TOWARD THE ARPT (RWY 24R) AND THE TWR SWITCHED US TO 24L. THE MLG LANDED ON 24R. WHAT DISTURBS ME MOST IS NO CTLR CAUGHT US CUTTING THE MLG OFF UNTIL APPARENTLY HE SPOKE UP. WHEN WE WERE WITHIN 2 MI OF THE RWY AT THE TIME. BUT A MORE SUBTLE FACTOR MIGHT HAVE BEEN OUR FATIGUE LEVEL. WE HAD FLOWN 8 LEGS ALREADY THAT DAY. THE FIRST TWO WERE BOTH RETURNS TO LAX FOR MECH PROBLEMS. WE ONLY HAD SHORT 20 MIN BREAKS IN ALL OF THIS. I FEEL THIS MAY HAVE CONTRIBUTED TO US SEEING THE WRONG ACFT AND NOT CATCHING IT UNTIL LATER. ALSO THE CTLR POINTED OUT TFC AT 12 O'CLOCK WHEN HE CLRLY COULD NOT HAVE BEEN THERE FOR US TO CUT IN FRONT OF HIM.