  Random Overlap Structures (ROSt's) are random elements on the space of
probability measures on the unit ball of a Hilbert space, where two measures
are identified if they differ by an isometry. In spin glasses, they arise as
natural limits of Gibbs measures under the appropriate algebra of functions. We
prove that the so called `cavity mapping' on the space of ROSt's is continuous,
leading to a proof of the stochastic stability conjecture for the limiting
Gibbs measures of a large class of spin glass models. Similar arguments yield
the proofs of a number of other properties of ROSt's that may be useful in
future attempts at proving the ultrametricity conjecture. Lastly, assuming that
the ultrametricity conjecture holds, the setup yields a constructive proof of
the Parisi formula for the free energy of the Sherrington-Kirkpatrick model by
making rigorous a heuristic of Aizenman, Sims and Starr.
