In this chapter we consider so-called measure preserving maps. These are measurable maps from the state space into the state space where the pre-image of any measurable set has the same probability as the set itself. If the map is such that any set coinciding with its pre-image has either probability one or zero we call the map ergodic. The Birkhoff-Khinchin ergodic theorem is proven based on the maximal ergodic theorem, and a generalized form of the Strong Law of Large Numbers is deduced.