I have been trying to pin down a precise definition of large-scale homogeneity, in the context of saying, per the Cosmological Principle, that all constant-time hypersurfaces (CTHs) of a foliation are large-scale homogeneous. Here is my attempt: Let M represent any coordinate-independent, physical quantity that can be measured for any sphere of given centre and given finite radius in a CTH. Let MH(P,x) be the value of M obtained from measuring M on a sphere of radius x, centred on P, in H. Let LMH(P) = limit as x-> infinity of MH(P,x), if the limit exists. If, for given M, H and P and any e>0, there exists Ne such that x>Ne => MH(P,x)>1/e then we say that LMH(P) = infinity. Then we say that a CTH H is “large-scale homogeneous” iff, (1) for any M and P, LMH(P) exists (including that it may be infinity) and (2) for any M and any points P1 and P2 in H, LMH(P1)=LMH(P2). This definition still isn’t exactly what I’d like because it doesn’t rule out trivial, local-focused definitions of M such as MH(P,x) = 1 if there is a proton within one micron of x, otherwise zero. But I couldn’t think of an easy, concise way of ruling out such definitions (suggestions would be much appreciated!), so I’ll live with that bit of ambiguity for the time being. The sorts of M I imagine wanting to use are related to average density of mass-energy in the sphere. It could be mass-energy generally, or a particular form such as just protons. Alternatively it could be something like average pressure, average entropy or average Ricci scalar. I would appreciate comments on this definition, especially: - is it well-defined (unambiguous)? - is it too narrow? (does it rule out CTHs that we would wish to regard as large-scale homogeneous?) - is it too broad? (does it allow CTHs that we would not wish to regard as large-scale homogeneous) - is there a reasonably concise way of ruling out annoying locally-focused measurements M like the above example with the proton?