MA40042 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 ∈ 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 = {a1, a2, . . . , an}, and has cardinality ∞ otherwise. The empty set has cardinality 0.) • Probability, finite, and σ-finite measures • Let (X, Σ) be a measurable space, and let x ∈ X. Then the Dirac measure (or δ measure or point measure) at x of a set A ∈ Σ is given by ( 1 if x ∈ A, 3.1 Definition δx(A) = 0 if x∈ / 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, ∞]. (Here, • Let X be a countable set, and let Σ be its powerset. Associate with each [0, ∞] := [0, ∞) ∪ {∞}.) x ∈ X a ‘weight’ wx ∈ [0, ∞]. 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, x∈A x∈X 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) = ∞ 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, ∞] • The 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. ∞ ! ∞ [ 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) < ∞, we have N ! N [ X ∞ ! µ An = µ(An). \ lim µ(An) = µ An . n=1 n=1 n→∞ n=1 (Monotonicity) For A, B ∈ 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 \ An−1 for n ≥ 2. Then ∞ ! ∞ [ X N ∞ ∞ 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 ∈ Σ, we have µ(A) + µ(Ac) = µ(X). N ∞ ∞ ! ∞ ! 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 \ A), where the set difference B \ A was shown to as N → ∞. be measurable in Lecture 1. Note also that the A and B \ A are disjoint. Hence by finite additivity we have µ(B) = µ(A) + µ(B \ A). But µ(B \ A) ≥ 0, as µ:Σ → [0, ∞]. Hence µ(B) ≥ µ(A). 3.4 Probability measures A 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) < ∞, then we call µ a finite measure. n→∞ n→∞ 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 ∈ 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. 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.

∞ ! [ Matthew Aldridge lim µ(An) = µ An . n→∞ [email protected] n=1