We first show with proofs the basic and fundamental concepts and theorems from abstract and geometric measure theory. These include, in particular, the three classical covering theorems: 4r, Besicovitch, and Vitali type. We also include a short section on probability theory: conditional expectations and Martingale Theorems. We devote quite a significant amount of space to treating Hausdorff and packing measures. In particular, we formulate and prove Frostman Converse Lemmas, which form an indispensable tool for proving that a Hausdorff or packing measure is finite, positive, or infinite. Some of these are frequently called, in particular in the fractal geometry literature, the mass redistribution principle, but these lemmas involve no mass redistribution. We then deal with Hausdorff, packing, box counting, and dimensions of sets and measures, and provide tools to calculate and estimate them.

We prove in this chapter the Bogolyubov-Krylov Theorem about the existence of Borel probability invariant measures for continuous dynamical systems acting on compact metrizable topological spaces. We also establish in this chapter the basic properties of invariant and ergodic measures and provide several examples of such measures.

This chapter is devoted to the stochastic laws for measurable endomorphisms preserving a probability measure that are finer than the mere Birkhoff Ergodic Theorem. Under appropriate hypotheses, we prove the Law of the Iterated Logarithm. We then describe another powerful method of ergodic theory, namely Young towers, which are also frequently called Kakutani towers. With appropriate assumptions imposed on the first return time, Young's construction yields the exponential decay of correlations, the Central Limit Theorem, and the Law of the Iterated Logarithm follows too.

We start with quasi-invariant measures and early on, in the second section of this chapter, we introduce the powerful concept of the first return map. This concept, along with the concept of nice sets, forms our most fundamental tool in Part IV of our book, which is devoted to presenting a refined ergodic theory of elliptic functions. We introduce, in this chapter, the notions of ergodicity and conservativity (always satisfied for finite invariant measures), and prove the Poincaré Recurrence Theorem, Birkhoff Ergodic Theorem, and Hopf Ergodic Theorem, the last pertaining to infinite measures. We also provide a powerful, though perhaps somewhat neglected by the ergodic community, tool for proving the existence of invariant s-finite measures absolutely continuous with respect to given quasi-invariant measures.