Given a measurable space X with a probability measure which is invariant by a dynamics T: X --> X, Birkhoff's ergodic theorem states that, for any integrable test function f, its sequence of Cesàro time averages converges almost everywhere, and the integral of the limit is the space mean of f. We will discuss a generalization of this result to a class of non-invariant sigma-finite measures. (This is a joint work with Fernando J. Moreira.)