52 CHAPTER 4. THE RIEMANN STIELTJES INTEGRAL

For this partition,∫ b

a( f +g)(x) dF ∈ [L( f +g,P) ,U ( f +g,P)]

⊆ [L( f ,P)+L(g,P) ,U ( f ,P)+U (g,P)]

and ∫ b

af (x) dF +

∫ b

ag(x) dF ∈ [L( f ,P)+L(g,P) ,U ( f ,P)+U (g,P)] .

Therefore, ∣∣∣∣∫ b

a( f +g)(x) dF−

(∫ b

af (x) dF +

∫ b

ag(x) dF

)∣∣∣∣≤U ( f ,P)+U (g,P)− (L( f ,P)+L(g,P))< ε/2+ ε/2 = ε.

This proves 4.4.7 since ε is arbitrary.It remains to show that

α

∫ b

af (x) dF =

∫ b

aα f (x) dF.

Suppose first that α ≥ 0. Then∫ b

aα f (x) dF ≡ sup{L(α f ,P) : P is a partition}=

α sup{L( f ,P) : P is a partition} ≡ α

∫ b

af (x) dF.

If α < 0, then this and Lemma 4.4.2 imply∫ b

aα f (x) dF =

∫ b

a(−α)(− f (x)) dF

= (−α)∫ b

a(− f (x)) dF = α

∫ b

af (x) dF.

This proves the theorem.In the next theorem, suppose F is defined on [a,b]∪ [b,c] .

Theorem 4.4.4 If f ∈ R([a,b]) and f ∈ R([b,c]) , then f ∈ R([a,c]) and∫ c

af (x) dF =

∫ b

af (x) dF +

∫ c

bf (x) dF. (4.4.8)

Proof: Let P1 be a partition of [a,b] and P2 be a partition of [b,c] such that

U ( f ,Pi)−L( f ,Pi)< ε/2, i = 1,2.

Let P≡ P1∪P2. Then P is a partition of [a,c] and

U ( f ,P)−L( f ,P)