No, it's in the City of Los Angeles. However, Westchester is part of Los Angeles, just as Venice is part of LA.
Al baby is correct. I just went and checked a map linked at the City of Los Angeles site. Westchester is now a separate city and does contain the airport in what was once an unincorporated area adjacent to Inglewood. LAX was known as "Inglewood" for many years (one entered from Inglewood on Century Boulevard).