282 CHAPTER 12. THE CONSTRUCTION OF MEASURES

It will be assumed in what follows that (Ω,τ) is a locally compact Hausdorff space.This means it is Hausdorff: If p,q ∈ Ω such that p ̸= q, there exist open sets, Up and Uqcontaining p and q respectively such that Up ∩Uq = /0 and Locally compact: There existsa basis of open sets for the topology, B such that for each U ∈B, U is compact. RecallB is a basis for the topology if ∪B = Ω and if every open set in τ is the union of sets ofB. Also recall a Hausdorff space is normal if whenever H and C are two closed sets, thereexist disjoint open sets, UH and UC containing H and C respectively. A regular space is onewhich has the property that if p is a point not in H, a closed set, then there exist disjointopen sets, Up and UH containing p and H respectively.

12.2 Urysohn’s lemmaUrysohn’s lemma which characterizes normal spaces is a very important result which isuseful in general topology and in the construction of measures. Because it is somewhattechnical a proof is given for the part which is needed.

Theorem 12.2.1 (Urysohn) Let (X ,τ) be normal and let H ⊆ U where H is closed andU is open. Then there exists g : X → [0,1] such that g is continuous, g(x) = 1 on H andg(x) = 0 if x /∈U.

Proof: Let D≡ {rn}∞n=1 be the rational numbers in (0,1]. Choose Vr1 an open set such

thatH ⊆Vr1 ⊆V r1 ⊆U.

This can be done by applying the assumption that X is normal to the disjoint closed sets, Hand UC, to obtain open sets V and W with

H ⊆V, UC ⊆W,and V ∩W = /0.

ThenH ⊆V ⊆V , V ∩UC = /0

and so let Vr1 =V .Suppose Vr1 , · · · ,Vrk have been chosen and list the rational numbers r1, · · · ,rk in order,

rl1 < rl2 < · · ·< rlk for {l1, · · · , lk}= {1, · · · ,k}.

If rk+1 > rlk then letting p = rlk , let Vrk+1 satisfy

V p ⊆Vrk+1 ⊆V rk+1 ⊆U.

If rk+1 ∈ (rli ,rli+1), let p = rli and let q = rli+1 . Then let Vrk+1 satisfy

V p ⊆Vrk+1 ⊆V rk+1 ⊆Vq.

If rk+1 < rl1 , let p = rl1 and let Vrk+1 satisfy

H ⊆Vrk+1 ⊆V rk+1 ⊆Vp.