Outer Measure R ∈ R to Be Ρ(R)

MA40042 Measure Theory and Integration 4.2 Set-up Let X be a nonempty set. We start with

Lecture 4 • A collection R of subsets of X. We shall assume throughout that ∅ ∈ R. Constructing measures I: • A function ρ: R → [0, ∞]. We shall assume throughout that ρ(∅) = 0. The idea is that R is a collection of ‘simple’ sets, and we want the measure of Outer measure R ∈ R to be ρ(R). We then need to ‘grow’ ρ into a measure, by deﬁning a measure on some σ-algebra on X that has been built up from R and ρ. An important case is that of the Lebesgue measure. On X = R, we take R to • How to construct measures be the collection of intervals I, and deﬁne the length ρ by • The Lebesgue measure ρ(∅) = 0 ρ [a, b) = b − a ρ [−∞, b) = ρ [a, ∞) = ρ(R) = ∞.

on intervals and boxes d On X = R , we take R to be the collection of interval boxes Id and ρd to be the • Constructing an outer measure (hyper)volume ρd(I1 × I2 × · · · × Id) = ρ(I1)ρ(I2) ··· ρ(Id). • Properties of outer measures (Remember our convention that 0 × ∞ = 0.) In particular, a Cartesian product of semi-open intervals has volume

d ! d Y Y 4.1 How to construct measures: a roadmap ρd [ai, bi) = (bi − ai). So far we have only seen simple examples of measures, such as the counting measure i=1 i=1 or some discrete measures. To create more powerful measures, such as the Lebesgue (As ever, we suppress the dimension d when it’s clear by context.) measure of length/area/volume, we have to build them up from basic pieces. Suppose we are working on a set X. 4.3 Constructing µ∗ 0. We begin with a collection R of subsets of X, and a function ρ on R. The Given X, R, and ρ, we will construct a function µ∗ : P(X) → [0, ∞], which gives idea is that R is the collection of sets we ‘know’, and, for R ∈ R, we ‘want’ a concept of size to all subsets of X. We shall see later that µ∗ has many of the the measure of R to be ρ(R). properties of a measure. 1. The first step is to construct an outer measure µ∗ from R and ρ. This gives The idea is that, for any set A ⊂ X, we approximate A from the outside by a a ‘size’ to all subsets of X, and has many of the properties of a measure. countable union of sets from R, and we know what we want their sizes to be. 2. The outer measure µ∗ is (in general) not a measure. But if we restrict ourselves Definition 4.1. Let X be a nonempty set, R a collection of subsets containing to the σ-algebra of measurable sets satisfying a certain ‘splitting condition’, the empty set, and ρ a function from R to [0, ∞] satisfying ρ(∅) = 0. this does give a measure µ on those measurable sets. Given a countable subcollection C ⊂ R, we write X 3. The measure µ does not necessarily ‘extend’ (that is, agree with) our original ρ(C) = ρ(R). set-up: we can’t guarantee that every R ∈ R is a measurable set, and even if R R∈C is measurable, there’s no guarantee that µ(R) = ρ(R). However, Carathéo- We call C a covering of a set A ⊂ X if A ⊂ S R. dory’s extension theorem tells us that if R and ρ have certain properties, R∈C We define a function µ∗ : P(X) → [0, ∞] by then µ is a proper extension of ρ. In certain other conditions, the µ is the unique extension of ρ. µ∗(A) := inf {ρ(C): C is a covering of A} . (By convention, inf ∅ = ∞, meaning we take µ∗(A) = ∞ if there is no covering of Proof of Theorem 4.2. For the first point, note that one covering of ∅ is C = {∅}, A.) and ρ(C) = 0. No covering has a negative size, so µ∗(∅) = 0. Monotonicity is immediate, since any covering of B is also a covering of A. The infimum in the definition means we want to take the tightest outer approx- ∗ Now for countable subadditivity. First note that if for some An we have µ (An) = imation – or at least the limit of increasingly tight approximations – of the set ∗ ∞, then there’s nothing to prove. So assume µ (An) is finite for all n. A. Since the definition of µ∗ includes an infimum, we will want to show that, for arbitrary > 0, ∞ ! ∗ [ X ∗ µ An ≤ µ (An) + , n=1 n which would prove the theorem. Let’s pause for a moment, and look at a failed attempt to prove the statement. By definition, for each An there exists a covering Cn of An with

∗ ρ(Cn) ≤ µ (An) + . S∞ S∞ Now the union C = n=1 Cn is countable and is a covering of n=1 An. Hence

∞ ! ∞ ∞ ∞ ∗ [ X X ∗ X ∗ µ An ≤ ρ(C) ≤ ρ(Cn) ≤ µ (An) + = µ (An) + ∞ × . When R = I and ρ is the length/volume as above, we write λ∗ for the function n=1 n=1 n=1 n=1 obtained from this covering construction, and call it the Lebesgue outer measure. Had we been left with ‘5’ at the end, we go back to the proof and replace all our ‘’s with ‘/5’s. But we can’t go back and divide through by ∞. But, we can 4.4 Properties of outer measures use ‘the /2n trick’.

The function µ∗ has many of the properties of a measure. Proof continued. By deﬁnition, for each An there exists a covering Cn of An with

∗ ∗ Theorem 4.2. Let X, R, ρ, and µ be as above. Then ρ(Cn) ≤ µ (An) + . 2n 1. µ∗( ) = 0 S∞ S∞ ∅ Now the union C = n=1 Cn is countable and is a covering of n=1 An. Hence 2. µ∗ is monotone, in that for A ⊂ B ⊂ X we have µ∗(A) ≤ µ∗(B). ∞ ! ∞ ∞ ∞ ∗ [ X X ∗ X ∗ µ An ≤ ρ(C) ≤ ρ(Cn) ≤ µ (An) + n = µ (An) + , ∗ 2 3. µ is countably subadditive, in that for a countably infinite sequence A1, n=1 n=1 n=1 n=1 A ,... of subsets of X, we have 2 since ∞ X 1 1 1 ∗ [ X ∗ = + + + ··· = × 1 = . µ An ≤ µ (An). 2n 2 4 8 n=1 n n The statement is proved. Definition 4.3. Any function that satisfies points 1, 2, and 3 from the previous theorem is called an outer measure on X. Matthew Aldridge Note that ‘an outer measure’ is any function satisfying these properties, and the [email protected] construction of µ∗ above is one way of forming an outer measure.