Saw this somewhere else. Here's a map of the events from 1984 [1]; or at least many of them.
Two venues were built for the 1984 games; a velodrome at CSU Dominguez Hills, which operated for two decades until it was replaced by a new velodrome will also host (track) cycling and fencing for the 2028 games; the other venue was an olympic swim pool at USC, which was rebuilt 30 years later although it won't host the 2028 games; both venues had corporate sponsorship to help with construction costs.
Only one venue was built for the 1932 games, a swimming stadium that's still in use today, renovated in 2003.
Venues aren't fully confirmed for the 2028 games, but everything so far is either existing or temporary (these tend not to be very expensive). Although softball and canoe slalom are planned to happen in Oklahoma City in order to existing venues, since there apparently aren't appropriate venues in the LA area.
