in the definition of recurrent
we say that the probability of returning to i (starting at i) is 1. How is this different than saying that the statement itself is true (i.e. for a recurrent
state i we always come back to i starting at i) ?
As far as I understand, saying "the probability of returning to i is 1" is the same as saying "we almost always return to i". Almost always (or almost surely) is a technical term here.
Saying "we always return to i" is wrong or imprecise because there can be events where we don't return to i (these events happen with a probability of 0).
