1602 CHAPTER 50. RIEMANN STIELTJES INTEGRALS

Theorem 50.0.3 Let φ and γ be as just described. Then assuming that∫γ

f dγ

exists, so does ∫γ◦φ

f d (γ ◦φ)

and ∫γ

f dγ =∫

γ◦φf d (γ ◦φ) . (50.0.1)

Proof: There exists δ > 0 such that if P is a partition of [a,b] such that ||P|| < δ ,then ∣∣∣∣∫

γ

f dγ−S (P)

∣∣∣∣< ε.

By continuity of φ , there exists σ > 0 such that if Q is a partition of [c,d] with ||Q|| <σ ,Q = {s0, · · · ,sn} , then

∣∣φ (s j)−φ(s j−1

)∣∣< δ . Thus letting P denote the points in [a,b]given by φ (s j) for s j ∈Q, it follows that ||P||< δ and so∣∣∣∣∣

∫γ

f dγ−n

∑j=1

f (γ (φ (τ j)))(γ (φ (s j))− γ

(φ(s j−1

)))∣∣∣∣∣< ε

where τ j ∈[s j−1,s j

]. Therefore, from the definition 50.0.1 holds and∫

γ◦φf d (γ ◦φ)

exists.This theorem shows that

∫γ

f dγ is independent of the particular γ used in its computa-tion to the extent that if φ is any nondecreasing continuous function from another interval,[c,d] , mapping to [a,b] , then the same value is obtained by replacing γ with γ ◦φ .

The fundamental result in this subject is the following theorem. We have in mindfunctions which have values in C but there is no change if the functions have values in anycomplete normed vector space.

Theorem 50.0.4 Let f : γ∗ → X be continuous and let γ : [a,b]→ C be continuous andof bounded variation. Then

∫γ

f dγ exists. Also letting δ m > 0 be such that |t− s| < δ m

implies || f (γ (t))− f (γ (s))||< 1m ,∣∣∣∣∫

γ

f dγ−S (P)

∣∣∣∣≤ 2V (γ, [a,b])m

whenever ||P||< δ m.