Spitzer's Strong Law of Large Numbers in Nonseparable Banach Spaces

It is well known, that for the sums of i.i.d. random variables we have S _n/n 0 a.s. iff _n=1 1/n P(|S _n| > n) < holds for all > 0 (Spitzer's SLLN). The result is also known in separable Banach spaces. It will be shown, that this also holds in nonseparable (= not necessarily separable) Banach spaces without any measurability assumption. In the theory of empirical processes this gives a characterization of Glivenko-Cantelli classes.