Strong law of large numbers (SLLN) says if $X_1, X_2, \dots$ are iid random variables with expectation $\mu$, then $\bar{X}_n \to \mu$ almost surely, or $$P\left(\lim_{n\to \infty} \bar{X}_n = \mu\right)=1.$$
If we use the probability space $(\Omega, \mathcal{F}, P)$ and the concept of the random variable as a function from $\Omega$ to $\mathbb{R}$, we can write $$P\Big( \omega \in \Omega : \lim_{n \to \infty} \bar{X}_n(\omega) = \mu \Big) = 1.$$
Now suppose we have a sample space from tossing a fair six-sided die, the sample space $\Omega=\{1,2,3,4,5,6\}$ and the probability is $P(\{i\})=1/6$. $X_i(\omega)=\omega$ are discrete uniform on $\{1,2,3,4,5,6\}$.
How do I understand $\bar{X}_n(\omega)$? For example, if I get a number $1$ from the die, what is $\bar{X}_n(\{1\})$? Since there is pointwise convergence, $\bar{X}_n(\{1\}) \to 3.5$?
Thank you for the help.