Mat 206 Recap

6. Suppose X is a metric space, let BX be the σ-algebra generated by the open sets of X, we call BX the Borel σ-algebra and its elements Borel sets. 7. A countable intersection of open sets is called a Gδ set ; a countable union of closed sets is called an Fσ set . 8. We call a collection of subsets of X an elementary family if (i) ∅ ∈ E ; (ii) if E,F ∈ E , then E ∩ F ∈ E ; and (iii) if E ∈ E , then E is a finite disjoint union of sets in E . 9. If E is an elementary family, then the collection of finite disjoint unions of E is an algebra. 10. Let X be a nonempty subset and M a σ-algebra of sets of X, we say a function μ : M → [0,∞] is a measure if (i) μ(∅) = 0; (ii) if { Ej }∞ j=1 is a collection of disjoint sets in M , then μ(∪j=1Ej) = ∑∞ j=1 μ(Ej). We call the triple (X,M , μ) a measure space and sets in M measurable sets.