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.)