AN SF34 COMMUTER FLC LANDED ON A CLOSED RWY AT LAX. THE APCH CTLR HAD ASSIGNED RWY 24R; BUT THE ATCT LCL CTLR ASSIGNED RWY 24L. THE PNF THOUGHT THAT THE ATCT CTLR WAS MISTAKEN BECAUSE TFC NORMALLY ARRIVED ON RWY 24R; SO THEY READ BACK RWY 24R AND LANDED ON RWY 24R. DURING THE LNDG; THE FLC NOTICED CONSTRUCTION EQUIP ON THE RWY.

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AFTER BEING CLRED FOR THE VISUAL APCH FOR RWY 24R INTO LAX; WE WERE TOLD TO CONTACT THE TWR AT THE OM. WHEN I CONTACTED THE TWR; THE CTLR TOLD US WE WERE CLRED TO LAND RWY 24L. I ACKNOWLEDGED BY REPLYING 'CLRED TO LAND RWY 24R.' (RWY 24L IS FOR DEPS AND RWY 24R IS FOR ARRS; NORMALLY.) NOT BEING SURE IF I HEARD HIM CORRECTLY; I WAITED FOR HIM TO CORRECT ME IN CASE I WAS WRONG BUT HE DID NOT REPLY AND HE WENT ON TO TALK TO OTHER ACFT. WE CONTINUED TO COMPLETE OUR PRELNDG CHKLIST AND LANDED ON RWY 24R. ON OUR LNDG ROLLOUT WE NOTICED 3 OR 4 VEHICLES CLRING THE RWY AT THE DEP END OF THE RWY ABOUT 1/2 TO 1/4 MI DOWN THE RWY FROM THE POINT WHERE WE TAXIED OFF. THEY WERE NO FACTOR TO SAFETY AND WE TAXIED TO THE TERMINAL WITHOUT FURTHER INCIDENT. AT NO TIME WERE WE TOLD TO SWITCH OR SIDESTEP RWYS; WHICH WOULD HAVE CLARIFIED THE PROPER RWY ASSIGNMENT AND BRING TO OUR ATTN THAT OUR RWY HAD BEEN CHANGED. SUPPLEMENTAL INFO FROM ACN 335607: WE WERE QUITE CONCERNED THAT EITHER WE LANDED ON THE WRONG RWY OR THE CTLR HAD CLRED US TO THE WRONG RWY. I SHOULD HAVE INSISTED THAT MY FO CLR UP ANY AMBIGUITY IN RWY ASSIGNMENT PRIOR TO LNDG.