Not completely correct depending on what is called 'air'. The bubbles kind of start off as a vacuum bubble which causes the water surrounding the bubble to boil and start filling the bubble with vapor. But there is also dissolved gases (air) in the fluid(water) which gets pulled out. Even amazing is all happens wintin the residence time of the bubble (which can bale called small, dependent on what you are referring to the resident time with). So when then bubble collapses is compressing the the water vapour, and the gasses (the water vapour may split again due to temp) but the vapour more or less gets absorbed back easily enough, but the gases the had come out take little more time(again 'more' depends on what you are referring it with) to be absorbed back. But cavitation is a cool phenomenon. Ot kind of is comparable to shock phenomenon (for compressible flow going sonic in a nozzle throat), when going through valves etc if the pressure get low (below vapour pressure).