Abstract
GAP measures (also known as Scrooge measures) are a natural class of probability distributions on the unit sphere of a Hilbert space that come up in quantum statistical mechanics; for each density matrix ρ there is a unique measure GAP_ρ. We describe and prove a property of these measures that was not recognized so far: If a wave function Ψ is GAP_ρ distributed and a collapse occurs, then the collapsed wave function Ψ' is again GAP distributed (relative to the appropriate ρ'). This fact applies to collapses due to a quantum measurement carried out by an observer, as well as to spontaneous collapse theories such as CSL or GRW. More precisely, it is the conditional distribution of Ψ', given the measurement outcome (respectively, the noise in CSL or the collapse history in GRW), that is GAP_ρ'.