Consider Ln=n11incnig(Xn:i) for order statistics Xn:i and let cni=n(i1)/ni/nJdλ for some (Lebesgue) λ-summable over (0,1) function J. Sufficient as well as necessary conditions for limnLn=01Jgdλ to hold almost surely and in probability are given. Superposition (or Nemytskii) operators have been used to derive the laws of large numbers for L-statistics from the laws of large numbers in quasi-Banach function spaces for the empirical distribution functions based on X1,,Xn.