In the presence of a (quantum or classical) statistical ensemble of metrics one can consider averages of distances between points/events, as long as a prescription is assigned for identifying such points.
These average distances, in general, are not geodesic distances of any metric because they are not additive, in a sense that I will specify. Deviations from additivity can be measured by a quantity that, in any coordinate expansion, starts only at forth order. In Euclidean signature average distances are always subadditive. In Lorentzian signature it proves convenient to identify the events by anchoring them to a set of free falling observers. This prescription, by no means unique, naturally conveys the point of view of these observers, whose causal relations are inevitably affected by the fluctuations of the metric field. Average Lorentzian distances portray an interesting ``average causal structure” that has no classical analogue.
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