Spitzers 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 (sigma) (infinity) n=1 1/n P(|S n| > n (epsilon) ) < (infinity) holds for all (epsilon) > 0 (Spitzerʹs 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.