This note provides necessary and sufficient conditions for the law of large numbers for partial sum processes indexed by sets of $\lbrack 0, 1\rbrack^d$ and based on i.i.d. integrable random variables.
Evarist Gine. Joel Zinn. "The Law of Large Numbers for Partial Sum Processes Indexed by Sets." Ann. Probab. 15 (1) 154 - 163, January, 1987. https://doi.org/10.1214/aop/1176992261