What is the probability measure in strong law of large number?

Question

The statement is that $P(\bar{X}_n \to \mu)=1$. $\{\bar{X}_n\}$ is a sequence of random variables, which is essentially a sequence of functions on $\Omega$, the domain of all outcomes. Mathematically, Strong LLN states that this sequence of functions converges to the constant function $\mu$ pointwise almost everywhere. "Almost everywhere" here is with respect to probability measure $P$. But how exactly is $P$ defined here. It seems that distribution of $\{\bar{X}_n\}$ changes for each $n$. So where does this $P$ come from.