MA40042 Measure Theory and Integration The triple (X; Σ; µ) is called a measure space. Recall that a sequence (An) of sets is disjoint (or `pairwise disjoint') if Ai \ Aj Lecture 3 for i 6= j. Measures 3.2 Examples The following are measures. (Checking they really are measures is a homework • Definition of measure problem.) • Examples of measures • Let X be a nonempty set, and let Σ be its powerset. For A 2 Σ, the count- ing measure # is given by setting #(A) to be the cardinality of A. (Re- • Basic properties and call that a set A has cardinality n if its elements can be enumerated as continuity of measure A = fa1; a2; : : : ; ang, and has cardinality 1 otherwise. The empty set has cardinality 0.) • Probability, finite, and σ-finite measures • Let (X; Σ) be a measurable space, and let x 2 X. Then the Dirac measure (or δ measure or point measure) at x of a set A 2 Σ is given by ( 1 if x 2 A, 3.1 Definition δx(A) = 0 if x2 = A. Recall from Lecture 0 the properties we wanted a measure to have: • A measure is a function µ from measurable subsets of X to [0; 1]. (Here, • Let X be a countable set, and let Σ be its powerset. Associate with each [0; 1] := [0; 1) [ f1g.) x 2 X a `weight' wx 2 [0; 1]. Then the associated discrete measure is • µ( ) = 0. X X ? µ(A) = wx = wxδx(A): • For a finite or countably infinite sequence of disjoint measurable sets, x2A x2X S P µ ( An) = µ(An). n n • Let X be a nonempty set, and let Σ be the σ-algebra of countable and co- S • For a finite or countably infinite sequence of measurable sets, µ ( n An) ≤ countable sets. Then we can set P µ(A ). n n ( 0 if A is countable, • µ(A) + µ(Ac) = µ(X). µ(A) = 1 otherwise We take some of these as a definition, and prove the rest as properties later. Definition 3.1. Let (X; Σ) be a measurable space. Then a function µ:Σ ! [0; 1] • The Lebesgue measure on R is the unique measure λ on the measurable space is a measure on (X; Σ) if (R; B) such that λ [a; b) = b − a. Showing that this measure exists and is unique is highly nontrivial (and will take about four lectures' work). 1. µ(?) = 0; 2. µ is countably additive, in that if A1;A2;:::; is a countably infinite sequence 3.3 Properties of disjoint sets in Σ, then Let's start with some simple properties. 1 ! 1 [ X µ An = µ(An): Theorem 3.2. Let (X; Σ; µ) be a measure space. Then we have the following n=1 n=1 properties: (Finite additivity) For A1;A2;:::;AN a finite sequence of disjoint sets in Σ, we (Downward continuity) For A1 ⊃ A2 ⊃ · · · a countably infinite `contracting' se- have quence of sets in Σ with µ(A1) < 1, we have N ! N [ X 1 ! µ An = µ(An): \ lim µ(An) = µ An : n=1 n=1 n!1 n=1 (Monotonicity) For A; B 2 Σ with A ⊂ B, we have µ(A) ≤ µ(B). Proof. We'll prove upward continuity, and leave downward continuity for home- (Countable subadditivity) For A1;A2;:::; a countably infinite sequence of sets in work. Σ, we have Write B1 = A1, and Bn = An n An−1 for n ≥ 2. Then 1 ! 1 [ X N 1 1 µ An ≤ µ(An): [ [ [ Bn = AN and Bn = An; n=1 n=1 n=1 n=1 n=1 (Finite subadditivity) For A1;A2;:::;AN a finite sequence of sets in Σ, we have with the unions being disjoint. So by finite additivity we have N ! N N ! N [ X [ X µ An ≤ µ(An): µ(AN ) = µ Bn = µ(Bn): n=1 n=1 n=1 n=1 Hence, by countable additivity, Also, for A 2 Σ, we have µ(A) + µ(Ac) = µ(X). N 1 1 ! 1 ! Proof. Most of these are left for homework, but let's do monotonicity (assuming X X [ [ µ(AN ) = µ(Bn) ! µ(Bn) = µ Bn = µ An finite additivity) for an example. n=1 n=1 n=1 n=1 We can write B = A [ (B n A), where the set difference B n A was shown to as N ! 1. be measurable in Lecture 1. Note also that the A and B n A are disjoint. Hence by finite additivity we have µ(B) = µ(A) + µ(B n A). But µ(B n A) ≥ 0, as µ:Σ ! [0; 1]. Hence µ(B) ≥ µ(A). 3.4 Probability measures A probability measure fulfils the definition of a measure, but also `adds up to 1'. This proof illustrates a general rule: to prove a statement about measures, turn it into a statement about disjoint unions. Definition 3.4. Let (X; Σ; µ) be a measure space. The following properties are known as `continuity of measure', by analogy with • If µ(X) = 1, then we call µ a probability measure, and (X; Σ; µ) a probability the usual definition of continuity, that space. lim f(xn) = f lim xn : • If µ(X) < 1, then we call µ a finite measure. n!1 n!1 For the `limit' of a sequence of sets, we can think of the union of a sequence of • If X can be written as a countable union of sets in Σ with finite measure, then expanding sets (an `upward limit'), or the intersection of a sequence of contracting we call µ a σ-finite measure. sets (a `downward limit'). A probability space is usually written as (Ω; F; P). Note that it follows from Theorem 3.2 that (A) ≤ 1 and (Ac) = 1 − (A) for all A 2 F. Theorem 3.3. Let (X; Σ; µ) be a measure space. Then we have the following P P P Probability measures are well-behaved, and your intuition is useful in dealing properties: with them. Finite measure behave essentially the same as probability measures, (Upward continuity) For A1 ⊂ A2 ⊂ · · · a countably infinite `expanding' sequence and σ-finite measures are fairly well-behaved too. Measures that are not σ-finite of sets in Σ, we have can be pretty weird. 1 ! [ Matthew Aldridge lim µ(An) = µ An : n!1 [email protected] n=1.