I never thought I'd see that language in a theorem

In connection with an earlier post, I started reading a paper titled On the Stability of Geodesics in the Brownian Map(*) On page 2, after their first Definition came "Theorem 1. Almost surely, for all x,y etc."

That doesn't sound much like your old geometry course, does it? "Almost surely" is a phrase I'd never seen before. In context, it makes sense. In good old ordinary Euclidean geometry, if you pick any point and any distance R > 0 (no matter how small), you can, guaranteed, find an infinite number of points closer to the first point than R. In the systems they're writing about, though, it's not quite the same--distances are random. For example, if the Euclidean distance between two points is ρ, a random distance might be ρ times a random scale--which could be bigger than 1 or less; it just can't be 0.

In that case, for some given point, it is possible that the scale factor is big enough that the distance is now bigger than R. It is even possible, though infinitely unlikely, that all new distances are bigger than R.

So "almost surely" means something like "it is infinitely unlikely that this is false, but not a mathematical certainty. There might be a finite number of exceptions compared to the infinite number that are true."

Learn something new every day. There's a lot to learn.

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(*)Sounds impressive, doesn't it? I wish I understood it. Some good advice I ran across once said to read a paper all the way through to get what sense you can, no matter what seems to be missing. Then go back and read it again and identify the parts you need to know and don't, and then look those up, and then read it a third time, trying to work through the proofs. One of these days I should take that advice.