If $Y_n=\frac{X_1+X_2+...X_n}{n}$ and $X_i's$ are i.i.d and $E(X_i)=0$, when we want to show that $Y_n$ converges to zero almost surely we need to prove that
$$\mathbb{P}\{w:Y_n(w)\rightarrow0\}=1.$$
Here we choose a $w$ for all $Y_n's$ and hence one $w$ for all $X_i$'s. However, in real world we expect that if we realize every random variable $X_i$ independently then the sum would converge to zero. In other words, in realization of random variables we do not choose same $w$ for all $X_i$. $X_1$ can choose $w_1$ and $X_2$ can choose $w_2$ and so on. But $Y_n$ would converge to zero.
I am a little bit confused with the definition of almost surely convergence that why we choose same $w$ for all random variables while this is not the case when we want ,for example, to verify the law of large numbers using simulations.
To me, the "almost surely convergence" simply means something will happen with a probability, and this probability is 1. The "almost surely convergence " is different than something will surely happen, it is a slightly weaker certainty.
Every possibility we have Sample Space, Borel set, and Probability space. w represents outcomes, which are from the sample space. So, for almost all outcomes(w) from Sample space, we have the probability equals to 1. For the same outcome w, if its associated Y_n converges to its associated X, then Y_n converges almost surely to x.