250 CHAPTER 9. MEASURES AND MEASURABLE FUNCTIONS

9.7 An Outer Measure on P (R)A measure on R is like length. I will present something more general than length becauseit is no trouble to do so and the generalization is useful in many areas of mathematics suchas probability.

Definition 9.7.1 The following definition is important.

F (x+)≡ limy→x+

F (y) , F (x−) = limy→x−

F (y)

Thus one of these is the limit from the left and the other is the limit from the right.

In probability, one often has F (x)≥ 0, F is increasing, and F (x+) = F (x). This is thecase where F is a probability distribution function. In this case, F (x) ≡ P(X ≤ x) whereX is a random variable. In this case, limx→∞ F (x) = 1 but we are considering more generalfunctions than this including the simple example where F (x) = x. This last example willend up giving Lebesgue measure on R. Recall the following definition.

Definition 9.7.2 P (S) denotes the set of all subsets of S.

Also recall

Definition 9.7.3 For two sets, A,B in a metric space,

dist(A,B)≡ inf{d (x,y) : x ∈ A,y ∈ B} .

Theorem 9.7.4 Let F be an increasing function defined on R. This will be called anintegrator function. There exists a function µ : P (R)→ [0,∞] which satisfies the followingproperties.

1. If A⊆ B, then 0≤ µ (A)≤ µ (B) ,µ ( /0) = 0.

2. µ(∪∞

k=1Ai)≤ ∑

∞i=1 µ (Ai)

3. µ ([a,b]) = F (b+)−F (a−) ,

4. µ ((a,b)) = F (b−)−F (a+)

5. µ ((a,b]) = F (b+)−F (a+)

6. µ ([a,b)) = F (b−)−F (a−).

7. If dist(A,B) = δ > 0, then µ (A∪B) = µ (A)+µ (B) .

Then the σ algebra of µ measurable sets F contains the Borel sets. This measure iscalled Lebesgue Stieltjes measure.

Proof: First it is necessary to define the function µ . This is contained in the followingdefinition.

Definition 9.7.5 For A⊆ R,

µ (A) = inf

{∞

∑i=1

(F (bi−)−F (ai+)) : A⊆ ∪∞i=1 (ai,bi)

}.