CAPT OF ACR MLG ACFT DEV FROM ASSIGNED ALT DURING MISSED APCH IN RESPONSE TO A TCASII RA.

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EXECUTED A MISSED APCH FROM LAX RWY 24R -- TOLD TO CLB TO 2000 FT; HDG 250 DEG. AT 2000 FT LEVEL AT APPROX LA SHORELINE; TCASII ISSUED A 'TFC' FOLLOWED IMMEDIATELY BY A 'CLB; CLB IMMEDIATELY' RA. IT SHOWED TFC 200 FT BELOW US. I WAS CONCERNED THAT THERE WAS A DEP OFF OF 24L OR R; SO I IMMEDIATELY CLBED TO 2200 FT; WHERE TCASII ISSUED A 'CLR OF CONFLICT.' I RETURNED TO LA AND LANDED WITHOUT FURTHER INCIDENCE. AFTER LNDG I WAS REQUESTED BY GND TO CALL LA APCH ON THE TELEPHONE. I DID THIS AND SPOKE WITH A SUPVR AT LA TRACON. HE INFORMED ME THAT WHEN I CLBED FROM 2000 FT I WAS IN CONFLICT WITH AN IFR ACFT AT 3000 FT WITHIN 1 MI OF MY ACFT. LA TRACON GAVE THE 3000 FT TFC AN ABRUPT R TURN AS THEY DIDN'T KNOW HOW HIGH MY RA WOULD TAKE ME. I QUESTIONED LA TRACON AS TO WHAT TFC MY TCASII COULD HAVE PICKED UP. HE SAID HE HAD A DEP OFF RWY 25; BUT THAT ACFT WAS HDG 235 DEG (AWAY FROM ME). WE WERE BOTH CONCERNED THAT THE TCASII DID NOT PICK UP THE TFC IT TRIED TO CLB ME INTO. I NOTIFIED MY AIRLINE AND THEY WILL PULL AND CHK THE TCASII. HOWEVER; MY FAITH IN THE TCASII SYS IS FURTHER DIMINISHED. PS -- I HAVE HAD AT LEAST 12 FALSE RA'S.