In this preliminary chapter we collect the basic notions of measure theory that will be used throughout the book. We work on a nonempty set \(\mathbb {X}\) , denote by \(\mathcal {P}(\mathbb {X})\) its power set, and let \(\mathcal {T}\) denote a distinguished family of subsets of \(\mathbb {X}\) , typically a \(\sigma \) –algebra. We will recall the specific constructions that form the foundation of geometric measure theory and fractal analysis. We begin with the abstract notion of outer measure, describe the Carathéodory criterion for measurability, and recall how premeasures defined on an algebra extend to genuine measures through the Hahn–Kolmogorov theorem. In the metric setting, these ideas lead to two fundamental examples: the Hausdorff measure, constructed by coverings with sets of small diameter, and the packing measure, based on disjoint families of balls. Both constructions rely in an essential way on classical covering theorems, such as the basic covering lemma, the Vitali covering theorem, and their variants for doubling or Radon measures. These theorems provide the geometric mechanism needed to refine arbitrary covers into almost disjoint families, a key step in the analysis of Hausdorff and packing dimensions. The material presented in this chapter is classical and may be found in any standard reference on measure theory. We include it here for completeness and in order to establish a unified notation for the developments of the subsequent chapters [90–94].