Infinite invariant measures for non-uniformly expanding transformations of [0, 1]: Weak law of large numbers with anomalous scaling

We consider a class of maps of [0, 1] with an indifferent fixed point at 0 and expanding everywhere else. Using the invariant ergodic probability measure of a suitable, everywhere expanding, induced transformation we are able to study the infinite invariant measure of the original map in some detail. Given a continuous function with compact support in ]0, 1], we prove that its time averages satisfy a 'weak law of large numbers' with anomalous scaling n/log n and give an upper bound for the decay of correlations.