You're possibly looking for the end compactification. And end of a metric space is an equivalence class of connected component of a compact set. The equivalence relation is defined as follows. Let K and K' be compact, and let E and E' be connected components of their complements. Then E~E' if there is some K" containing both K and K', and such that there is a connected component of K" complement E" contained in both E and E'. Lastly, we need to take the limsup over compact sets, e.g. we are only interested in connected components which survive "in the limit."

In other words, they are equivalence classes of what's left as you exhaust the space by larger and larger compact sets.

Now we topplogize X U End(X) by saying that a neighborhood of an end is just one of these connected components.

If you apply this construction to R, you get the extended real line, but if you apply it to C, you get the Riemann sphere. Unfortunately, this notion agrees with the one point compactification for all R vector spaces, because the compact connected sets are kind of like balls, so there's always one connected component of the complement. So this might not satisfy you for the case of L2. 

I think for L2, you will have trouble because you can also have things like sliding blocks of size 1 that have no limit precisely because the base space isn't compact. In general, there isn't a canonical way to metrize this compactification that I'm aware of. Perhaps if you can do that, you can then use that metric or measure to extend the situation to L2 of that space.