A non-singular version of the Oseledec ergodic theorem

Kingman's subadditive ergodic theorem is traditionally proved in the setting of a measure-preserving invertible transformation T of a measure space (X, mu). We use a theorem of Silva and Thieullen to extend the theorem to the setting of a not necessarily invertible transformation, which is non-singular under the assumption that mu and mu omicron T have the same null sets. Using this, we are able to produce versions of the Furstenberg-Kesten theorem and the Oseledec ergodic theorem for products of random matrices without the assumption that the transformation is either invertible or measure-preserving.