GAP Measures and Wave Function Collapse

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 $\rho$ there is a unique measure GAP$_\rho$. We describe and prove a property of these measures that was not recognized so far: If a wave function $\Psi$ is GAP$_\rho$ distributed and a collapse occurs, then the collapsed wave function $\Psi'$ is again GAP distributed (relative to the appropriate $\rho'$). 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 $\Psi'$, given the measurement outcome (respectively, the noise in CSL or the collapse history in GRW), that is GAP$_{\rho'}$.