How could we know that the universe is homogeneous and isotropic at a moment? Cosmologists say that the universe is homogeneous and isotropic within our Hubble volume  based upon the astronomical observations. But how can we argue that the universe is homogeneous and isotropic at any particular moment while we can't picture the entire universe simultaneously due to the finite speed of light? i.e. when we picture the universe we picture the past. The further we go the further in time we observe.
 A: I think there are two ways to approach the question.
If you are coming from the point of view of a theoretician trying to come up with a working model of the universe, you would definitely like to make the assumption of isotropy and homogeneity. This is usually what we do, as least to first order approximation. One of the reasons for that is that the symmetry allows us to write useful and relatively simple solutions to Einstein's equations, like the FRW metric. Motivated by the fact that the universe is definitely not homogeneous at very small scales (read, scale of several galaxies), Cosmologists then add perturbations to this homogeneous and isotropic background, and see what they can get.
The second, more pragmatic approach is to look at the stars and see if the universe is homogeneous and isotropic. In other words, are the assumptions I discussed above justified. It turns out that they are. You were asking more specifically about the Hubble Radius, but it is not very much different from the observable universe. To sum up, we don't know for sure that the universe doesn't have a big "lump" just out of reach of what we can see today, but given what we can see and the fact that we have extremely accurate and predictive models of the universe that crucially rely on the basic assumptions of isotropy and homogeneity, it's a very justifiable thing to say.
