  Physical systems are often neither completely closed nor completely open, but
instead they are best described by dynamical systems with partial escape or
absorption. In this paper we introduce classical measures that explain the main
properties of resonance eigenfunctions of chaotic quantum systems with partial
escape. We construct a family of conditionally-invariant measures with varying
decay rates by interpolating between the natural measures of the forward and
backward dynamics. Numerical simulations in a representative system show that
our classical measures correctly describe the main features of the quantum
eigenfunctions: their multi-fractal phase space distribution, their product
structure along stable/unstable directions, and their dependence on the decay
rate. The (Jensen-Shannon) distance between classical and quantum measures goes
to zero in the semiclassical limit for long- and short-lived eigenfunctions,
while it remains finite for intermediate cases.
