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72 1. Measure theory Remark 1.4.15. From the definitions, it is clear that we have the following principle, somewhat analogous to the principle of mathematical induction: if F is a family of sets in X, and P (E) is a property of sets E ⊂ X which obeys the following axioms: (i) P (∅) is true. (ii) P (E) is true for all E ∈ F. (iii) If P (E) is true for some E ⊂ X, then P (X\E) is true also. (iv) If E1,E2,... ⊂ X are such that P (En) is true for all n, then P ( ∞ n=1 En) is true also. Then one can conclude that P (E) is true for all E ∈ F. Indeed, the set of all E for which P (E) holds is a σ-algebra that contains F, whence the claim. This principle is particularly useful for establishing properties of Borel measurable sets (see below). We now turn to an important example of a σ-algebra: Definition 1.4.16 (Borel σ-algebra). Let X be a metric space, or more generally a topological space. The Borel σ-algebra B[X] of X is defined to be the σ-algebra generated by the open subsets of X. Elements of B[X] will be called Borel measurable. Thus, for instance, the Borel σ-algebra contains the open sets, the closed sets (which are complements of open sets), the countable unions of closed sets (known as Fσ sets), the countable intersections of open sets (known as Gδ sets), the countable intersections of Fσ sets, and so forth. In Rd, every open set is Lebesgue measurable, and so we see that the Borel σ-algebra is coarser than the Lebesgue σ-algebra. We will shortly see, though, that the two σ-algebras are not equal. We defined the Borel σ-algebra to be generated by the open sets. How- ever, they are also generated by several other sets: Exercise 1.4.14. Show that the Borel σ-algebra B[Rd] of a Euclidean set is generated by any of the following collections of sets: (i) The open subsets of Rd. (ii) The closed subsets of Rd. (iii) The compact subsets of Rd. (iv) The open balls of Rd. (v) The boxes in Rd. (vi) The elementary sets in Rd.