14.3. THE LEBESGUE STIELTJES MEASURES AND BOREL SETS 325

It remains to verify Property 2 so let B ∈ GA. I need to verify that BC ∈ GA. In otherwords, I need to show that A∩BC ∈ G . However, from De Morgan’s laws,

A∩BC =(AC ∪B

)C=(AC ∪ (A∩B)

)CNow AC ∈ G because A ∈K ⊆ G and G is closed with respect to complements. Also,since B∈GA,A∩B∈G and so AC∪(A∩B)∈G because G is closed with respect to disjointunions. Therefore,

(AC ∪ (A∩B)

)C ∈ G because G is closed with respect to complements.Thus BC ∈ GA as hoped. Thus GA satisfies 1 - 3 and this implies, since G is the smallestsuch, that GA ⊇ G . However, GA is constructed as a subset of G . This proves that for everyB ∈ G and A ∈K , A∩B ∈ G . Now pick B ∈ G and consider

GB ≡ {A ∈ G : A∩B ∈ G } .

I just proved K ⊆ GB. The other arguments are identical to show GB satisfies 1 - 3 and istherefore equal to G . This shows that whenever A,B ∈ G it follows A∩B ∈ G .

This implies G is a σ algebra. To show this, all that is left is to verify G is closed undercountable unions because then it follows G is a σ algebra. Let {Ai} ⊆ G . Then let A′1 = A1and

A′n+1 ≡ An+1 \ (∪ni=1Ai) = An+1∩

(∩n

i=1ACi)= ∩n

i=1(An+1∩AC

i)∈ G

because the above showed that finite intersections of sets of G are in G . Since the A′i aredisjoint, it follows ∪∞

i=1Ai = ∪∞i=1A′i ∈ G Therefore, G ⊇ σ (K ).

14.3 The Lebesgue Stieltjes Measures and Borel SetsThe σ algebra of interest here consists of B (R) , the Borel sets of R. B (R) is defined asthe smallest σ algebra which contains the open sets.

Definition 14.3.1 Let B (R) denote σ (O) where O denotes the set of all open setsof R.

Then the following lemma is available.

Lemma 14.3.2 Let I denote the set of open intervals. Then σ (I ) = B (R).

Proof: By Theorem 6.5.9, every open set is a countable or finite union of open intervals.Therefore, each open set is contained in σ (I ). It follows that σ (I )⊇B (R)≡ σ (O)⊇σ (I ).

Let G be those Borel sets E satisfy XE∩[p,q] ∈ R∗ [p,q]. Thus G ⊆B (R) = σ (I ).By Lemma 14.1.6 G contains the open intervals I and is closed with respect to countabledisjoint unions and complements. Hence, from Dynkin’s lemma, G ⊇ σ (I ) and so S ⊇B (R), see Definition 14.1.5. This has proved the following.

Theorem 14.3.3 Every set E in B (R) is measurable which means that the indica-tor function XE is in R∗ [p,q] for any [p,q] .

From this, it is easy to define the Lebesgue Stieltjes measures on the Borel sets. I willgive the definition first and then show that it really is a measure.