Chapter 2 Measure spaces 2.1 Sets and σ-algebras 2.1.1 Definitions Let X be a (non-empty) set; at this stage it is only a set, i.e., not yet endowed with any additional structure. Our eventual goal is to define a measure on subsets of X. As we have seen, we may have to restrict the subsets to which a measure is assigned. There are, however, certain requirements that the collection of sets to which a measure is assigned—measurable sets (;&$*$/ ;&7&"8)—should satisfy. If two sets are measurable, we would like their union, intersection and comple- ments to be measurable as well. This leads to the following definition: Definition 2.1 Let X be a non-empty set. A (boolean) algebra of subsets (%9"#-! ;&7&"8 -:) of X, is a non-empty collection of subsets of X, closed under finite unions (*5&2 $&(*!) and complementation. A (boolean) σ-algebra (%9"#-! %/#*2) ⌃ of subsets of X is a non-empty collection of subsets of X, closed under countable unions (%**1/ 0" $&(*!) and complementation. Comment: In the context of probability theory, the elements of a σ-algebra are called events (;&39&!/). Proposition 2.2 Let be an algebra of subsets of X. Then, contains X, the empty set, and it is closed under finite intersections. A A 8 Chapter 2 Proof : Since is not empty, it contains at least one set A . Then, A X A Ac and Xc ∈ A. Let A1,...,An . By= de∪ Morgan’s∈ A laws, = ∈ A n n c ⊂ A c Ak Ak . k 1 k 1 = = = ∈ A n Given in 2018 . Exercise 2.1 Let X be a set and let be a collection of subsets containing X and closed under set subtraction (i.e, if A, B then A B . Show that is an algebra. A The following useful proposition∈ A states ∈ thatA for an algebraA to be a σ-algebra, it suffices to require closure under countable disjoint union: Proposition 2.3 Let X be a non-empty set, and let P X be an algebra of subsets of X, closed under countable disjoint unions. Then, is a σ-algebra. A ⊂ ( ) A Proof : We need to show that is closed under countable unions. Let An , and define recursively the sequence of disjoint sets Bn , A ( ) ⊂ A ( n )1 B A and Bn An Ak . 1 1 − k 1 = ∈ A = ∈ A Then, = An Bn . ∞ ∞ n 1 n 1 (The inclusion n 1 Bn n 1 An= is obvious.= = Let∈ Ax n 1 An. Then, there exists a minimal n such that∞ x An.∞ By definition of the Bn’s, ∞x Bn, hence x Bn, = = = n 1 which proves the reverse⊂ inclusion.) ∈ ∞ n = ∈ ∈ ∈ Trick #1: Turn a sequence of sets into a sequence of disjoint sets having the same —1h(2018)— union. Definition 2.4 A non-empty set endowed with a σ-algebra of subsets is called a measurable space ($*$/ "(9/). Measure spaces 9 To conclude, a σ-algebra is a collection of sets closed under countably-many set- theoretic operations. Examples: 1. Let X be a non-empty set. Then, , X is its smallest σ-algebra (the trivial{σ-algebra).} 2. Let X be a non-empty set. Then, its power set (%8'(% ;7&"8) P X is its maximal σ-algebra. ( ) 3. Let X be a non-empty set. Then, ⌃ A X A is countable or Ac is countable is a σ-algebra (the= { countable⊂ ∶ or co-countable sets). Clearly,} ⌃ is non- empty and is closed under complementation. Let An ⌃. If all An are countable, then their union is also countable, hence in ⌃. Conversely, if one of the An is co-countable, then their union is also co-countable,( ) ⊂ hence in ⌃. The following proposition states that every intersection of σ-algebras is a σ- algebra. This property will be used right after to prove that every collection of subsets defines a unique σ-algebra, which is the smallest σ-algebra containing that collection. Proposition 2.5 Let ⌃↵ P X ↵ J be a (not necessarily countable) collection of σ-algebras. Then, their intersection is a σ-algebra. { ⊂ ( ) ∶ ∈ } Proof : Set ⌃ ⌃↵. ↵ J Since for every ↵ J, , X ⌃↵, it follows= ∈ that ⌃ contains and X. Let A ⌃; by definition, ∈ ∈ A ⌃↵ ↵ J. ∈ ∈ ∀ ∈ 10 Chapter 2 Since each of the ⌃↵ is a σ-algebra, c A ⌃↵ ↵ J, hence Ac ⌃, proving that ⌃ is∈ closed under∀ ∈ complementation. Likewise, let An ⌃. Then, ∈ An ⌃↵ n N and ↵ J. ( ) ⊂ It follows that ∈ ∀ ∈ ∀ ∈ An ⌃↵ ↵ J, ∞ n 1 hence = ∈ ∀ ∈ An ⌃, ∞ n 1 proving that ⌃ is closed under countable= unions.∈ n . Exercise 2.2 Let X be a set. (a) Show that the collection of subsets that are either finite or co-finite is an algebra. (b) Show that it is a σ-algebra if and only if X is a finite set. Given in 2018 . Exercise 2.3 (a) Let n be an increasing sequence of algebras of subsets of X, i.e., n n 1. Show that A A ⊂ + n ∞ A n 1 is an algebra. (b) Give an example in whichA =n=isA an increasing sequence of σ-algebras, and its union is not a σ-algebra. A Let E P X be a collection of sets and consider the family of all σ-algebras of subsets of X containing E . This family is not empty since is includes at least the power⊂ set P( )X . By Proposition 2.5, ( ) σ ⌃ ⌃ is a σ-algebra containing is a σ-algebra; it is( theE) smallest= { ∶σ-algebra containing ; it is calledE} the σ-algebra generated by (*$* -3 ;97&1) . E A comment about nomenclature: we will repeatedly deal with sets of sets and sets E of sets of sets; to avoid confusion, we will use the terms collection (42&!), class (%8-(/) and family (%(5:/) for sets whose elements are sets. The following proposition comes up useful: Measure spaces 11 Proposition 2.6 Let P X and let ⌃ P X be a σ-algebra. If E ⊂ ( ) ⌃⊂, ( ) then E ⊂ σ ⌃. (E) ⊂ Proof : Since ⌃ is a σ-algebra containing , then, by definition, it contains σ . n E (E) Corollary 2.7 Let , P X . If E F ⊂ σ( ) and σ , then E ⊂ (F) F ⊂ (E) σ σ . (E) = (F) . Exercise 2.4 Let X be a non-countable set. Let x x X . What is the σ-algebra generated by ? E = {{ } ∶ ∈ } . Exercise 2.5 Let A1,...,An beE a finite number of subsets of X. (a) Prove that if the A1,...,An are disjoint and their union is X, then n σ A1,...,An 2 . (b) Prove that for arbitrary A1,...,An, ({ }) = σ A1,...,An contains a finite number of elements. ({ }) . Exercise 2.6 Let X, ⌃ be a measurable space. Let be a collection of sets. Let P be a property of sets in X (i.e., a function P P X true, false ), such that ( ) C (a) P true. ∶ ( ) → { } () = 12 Chapter 2 (b) P C true for all C . (c) P A true implies that P Ac true. ( ) = ∈ C (d) if P A true for all n, then P A true. ( ) =n ( ) = n 1 n ∞ = Show that P( A) = true for all A σ (. ) = Given in 2018 ( ) = ∈ (C) . Exercise 2.7 Prove that every member of a σ-algebra is countably-generated: i.e., let be a collection of subsets of X and let A σ . Prove that there exists a countable sub-collection A , such that A σ A . C Given in 2018 ∈ (C) C ⊂ C ∈ (C ) . Exercise 2.8 Let f X1 X2 be a function between two sets. Prove that if ⌃2 is a σ- algebra on X2, then ∶ → 1 ⌃1 f A A ⌃2 − is a σ-algebra on X1. = { ( ) ∶ ∈ } Given in 2018 . Exercise 2.9 Let X, ⌃ be a measurable space and let An ⌃ be a sequence of measurable set. The superior limit of this sequence is the set of points x which belong to An for infinitely- many n’s, i.e., ( ) ∈ lim sup An x X n x An . n = { ∈ ∶ { ∶ ∈ } = ∞} The inferior limit of this sequence→∞ is the set of points x which belong to An for all but finitely- many n’s, i.e., lim inf An x X n x An . n →∞ = { ∈ ∶ { ∶ ∉ } < ∞} (a) Prove that lim supn An n 1 k n Ak. (b) Prove that lim inf A ∞ ∞ A . n →∞ n = n =1 k =n k c ∞ ∞ c (c) Prove that lim sup→∞ A lim inf A . n =n = = n n c c (d) Prove that lim inf A lim sup →∞ A . ( n →∞ n ) = n n (e) Denote by χ the →∞ of a measurable set A. Prove that ( A indicator) function= →∞ χ χ χ χ . lim supn An lim sup An and lim infn An lim inf An n n →∞ →∞ = →∞ = →∞ 2.1.2 The Borel σ-algebra on R A σ-algebra is a structure on a set. Another set structure you are familiar with is a topology. Open sets and measurable sets are generally distinct entities, however, we often want to assign a measurable structure to a topological space. For that, there is a natural construction: Measure spaces 13 Definition 2.8 Let X,⌧X be a topological space (*#&-&5&) "(9/). The σ-algebra generated by the open sets is called the Borel σ-algebra and is denoted ( ) B X σ ⌧X . Its elements are called Borel sets (-9&"( ) ;&7&"8= ( )) (roughly speaking, it is the collec- tion of all sets obtained by countably-many set-theoretic operations on the open sets). Since we have special interest in the topological spaces Rn (endowed with the topology induced by the Euclidean metric), we want to get acquainted with its Borel sets.