The law of large numbers for non-additive capacities without continuity

We consider a capacity on a general measurable space associated with a sequence of identically distributed random variables. With a weak assumption concerned with the independence and very few restrictions on the capacity, we show that, on a set with full lower-capacity, any cluster point of empirical averages lies in the mean interval with the lower and upper Choquet integrals of the random variables being its bounds, and, under a Fubini-independent product capacity introduced by Ghirardato (1997), cluster points fill the whole mean interval with full upper-capacity. The results give answers to two open questions of Terán (2014).