The paper extends the ergodic theorems of information theory (Shannon-MacMillan-Breiman theorems) to spaces with an infinite invariant measure. An L1 difference theorem and a pointwisa ratio theorem are proved, for the information of spreading partitions. For the validity of the theorems it is assumed that the supremum f* of the conditional information given the increasing “past” is integrable. Simple necessary and sufficient conditions for the integrability of f* are obtained in special cases: If the initial partition is composed of one state of a null-recurrent Markov chain, then f* is integrable if and only if the partition of this state according to the first return times has finite entropy.