Convergence of random variables

When talking about almost sure convergence, do we have to clarify the underlying sample space beforehand?

Say for example, let X_n be iid such that
P(X_n)=1=P(X_n)=0=1/2,
then what's lim X_n?

It looks to me that the answer is X_1. However, by the Borel-Cantelli lemma (second), we conclude that X_n=1 io and X_n=0 io. And lim X_n is a tail function, so P(limsup X_n = 1) is either 0 or 1. But all these two things are not true if we clarify the probability space, say ([0,1], B[0,1], Lebesgue measure), then if we set X_n(x) = 1 if 0<=x<1/2, X_n(x) = 0 if 1/2<=x<=1, we have P(x: limsup X_n(x) = 1) = 1/2.

What's wrong there? I'm really confused.
.

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