I will compare (classical) Ergodic Theory and Infinite Ergodic Theory, i.e. when the space has infinite measure. In particular I will describe how to modify the Birkhoff Ergodic Theorem in the infinite setting. As examples, I will discuss the (classical) Euclidean continued fractions and the (less classical) continued fractions with even partial quotients. Time permitting, I will show the connection of such continued fraction expansions with theta sums.