252 CHAPTER 9. MEASURES AND MEASURABLE FUNCTIONS

the last inequality from the definition. Now letting δ decrease to 0 it follows F (b−)−F (a+)≤ µ ((a,b))≤ F (b−)−F (a+) . This shows 4.)

Consider 5.). From 3.) and 4.), for small δ > 0,

F (b+)−F ((a+δ ))≤ F (b+)−F ((a+δ )−)= µ ([a+δ ,b])≤ µ ((a,b])≤ µ ((a,b+δ ))

= F ((b+δ )−)−F (a+)≤ F (b+δ )−F (a+) .

Now let δ converge to 0 from above to obtain F (b+)−F (a+) = µ ((a,b]) . This estab-lishes 5.) and 6.) is entirely similar to 5.).

Finally, consider 7.). Let

V ≡ ∪{

B(

x,δ

10

): x ∈ A∪B

}.

Let A∪B⊆ ∪∞i=1 (ai,bi) where

µ (A∪B)+ ε > ∑i

F (bi−)−F (ai+)

Then, taking the intersection of each of these intervals with V, it can be assumed that all ofthe intervals are contained in V since such an intersection will only strengthen the aboveinequality. Now refer to V as the union of these intervals, none of which can intersectboth A and B. Thus V consists of disjoint open sets, one containing A consisting of theintervals which intersect A,UA and the other consisting of those which intersect B,UB. LetIA denote the intervals which intersect A and let IB denote the remaining intervals. Alsolet ∆((ai,bi))≡ F (bi−)−F (ai+) . Then from the above,

µ (A∪B)+ ε > ∑I∈IA

∆(I)+ ∑I∈IB

∆(I)≥ µ (A)+µ (B)≥ µ (A∪B)

Since ε > 0 is arbitrary, this shows 7.). That F contains the Borel sets follows from 7.)also. ■

We have just shown that µ is an outer measure on P (R). Unlike what was presentedearlier, this outer measure did not begin with a measure.

9.8 Measures and RegularityIt is often the case that Ω is not just a set. In particular, it is often the case that Ω is some sortof topological space, often a metric space. In this case, it is usually if not always the casethat the open sets will be in the σ algebra of measurable sets. This leads to the followingdefinition.

Definition 9.8.1 A Polish space is a complete separable metric space. For a Polishspace E or more generally a metric space or even a general topological space, B (E)denotes the Borel sets of E. This is defined to be the smallest σ algebra which contains theopen sets. Thus it contains all open sets and closed sets and compact sets and many others.

For example, R is a Polish space as is any separable Banach space. Amazing thingscan be said about finite measures on the Borel sets of a Polish space. First the case of afinite measure on a metric space will be considered.