The answer is no.
Consider any event $\omega$ such that $\xi_i(\omega) = 1$ for infinitely many $i$. Then for a given finite set $G$ one can find a finite superset $F$ such that $\frac{1}{\lvert F \rvert} \sum_{i \in F} \xi_i(\omega)$ is close to $1$ (to obtain $F$, simply add a lot of indices $i$ to $G$ for which $\xi_i(\omega) = 1$.
Similarly, if $\xi_i(\omega) = -1$ for infinitely many $i$, then for any given finite set $G$ we can find a finite superset $F$ such that $\frac{1}{\lvert F \rvert} \sum_{i \in F} \xi_i(\omega)$ is close to $-1$.
Hence, the net does not converge unless only the sequence $(\xi_i(\omega))$ is eventually constant. But the set of $\omega$ for which this happens has probability $0$.