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.