Probability Theory/The algebra of sets

Boolean algebras Within the subject of algebra, there is a structure called algebra. In order to meet our needs, we need to strongly modify this concept to obtain Boolean algebras. 0 100% #FFFFFF Definition 1.1 (Boolean algebras): A Boolean algebra is a set together with two binary operations and , an unary operation and such that the following axioms hold for all : Associativity of and : , Commutativity of and : , Absorbtion laws: , Distributivity laws: , Neutral elements: , Complementation laws: , Fundamental example 1.2 (logic): If we take and to be the usual operations from logic, we obtain a Boolean algebra. 0 100% #FFFFFF Fundamental example and theorem 1.3: Let be an arbitrary set, and let such that , where denotes the complement of . We set , , , , and for all . Then is a Boolean algebra, called an algebra of subsets of . Proof: Closedness under the operations follows from 1. - 3. We have to verify 1. - 6. from definition 1.1. 1. 2. 3. 4. 5. 6. We thus see that the laws of a Boolean algebra are "elevated" from the Boolean algebra of logic to the Boolean algebra of sets. Exercises Exercise 1.1.1: Let be a Boolean algebra and . Prove that and . Inclusion Infinite numbers of subsets Limits Notation During the remainder of the book, we shall adhere to the following notation conventions (due to Felix Hausdorff). If the sets are pairwise disjoint, we shall write for ; with this notation we already indicate that the are pairwise disjoint. That is, if we encounter an expression such as and the are sets, the are assumed to be pairwise disjoint. If are sets and , we replace by . This means: In any occasion where you find the notation within this book, it means and (note that in this way a set obtains a unique "additive inverse").