Several strong limit theorems are proved for sums of logarithms ofmth order spacings from general distributions. In all given results, the ordermof the spacings is allowed to increase to infinity with the sample size. These results provide a nonparametric strongly consistent estimator of entropy as well as a characterization of the uniform distribution on [0,1]. Furthermore, it is shown that Cressie's (1976) goodness of fit test is strongly consistent against all continuous alternatives.