Definition 6.1: Riemann Integral • Partition ○ A partition P of a closed interval [a,b] is a finite set of points ○ {x_0,x_1,…,x_n } where a=x_0≤x_1≤…≤x_(n−1)≤x_n=b • Let f be a bounded real function on [a,b], for each partition P of [a,b] ○ Define M_i and m_i to be § M_i=sup┬(x∈[x_(i−1),x_i ] )f(x) § m_i=inf┬(x∈[x_(i−1),x_i ] )f(x) ○ Define the upper sum and lower sum to be § U(P,f)=∑_(i=1)^n▒〖M_i Δx_i 〗 § L(P,f)=∑_(i=1)^n▒〖m_i Δx_i 〗 § where Δx_i=x_i−x_(i−1) ○ Define the upper and lower Reimann integral to be § (∫_a^b▒ ) ̅fdx=inf┬(All P)U(P,f) § ▁(∫_a^b▒ ) fdx=sup┬(All P)L(P,f) • If (∫_a^b▒ ) ̅fdx=▁(∫_a^b▒ ) fdx, then ○ We say that f is Riemann\integrable on [a,b], and write f∈R ○ Their common value is denoted by∫_a^b▒fdx or ∫_a^b▒f(x)dx • Well-definedness of upper and lower Riemann integral ○ Since f is bounded, ∃m,M∈R s.t. § m≤f(x)≤M (a≤x≤b) ○ Therefore for every partition P of [a,b] § m(b−a)≤L(P,f)≤U(P,f)≤M(b−a) ○ So (∫_a^b▒ ) ̅fdx and ▁(∫_a^b▒ ) fdx are always defined Definition 6.2: Riemann-Stieltjes Integral • Let α be a monotonically increasing function on [a,b] • Let f be a real-valued function bouned on [a,b] • For each partition P of [a,b], define ○ M_i=sup┬(x∈[x_(i−1),x_i ] )f(x) ○ m_i=inf┬(x∈[x_(i−1),x_i ] )f(x) ○ Δα_i=α(x_i )−α(x_(i−1) ) ○ U(P,f,α)=∑_(i=1)^n▒〖M_i Δα_i 〗 ○ L(P,f,α)=∑_(i=1)^n▒〖m_i Δα_i 〗 ○ (∫_a^b▒ ) ̅fdx=inf┬(All P)U(P,f,α) ○ ▁(∫_a^b▒ ) fdx=sup┬(All P)L(P,f,α) • If(∫_a^b▒ ) ̅fdx=▁(∫_a^b▒ ) fdx ○ We denote the common value by∫_a^b▒fdα or ∫_a^b▒f(x)dα(x) ○ This is the Riemann-Stieltjes integral of f with respect to α over [a,b] ○ We say f is integrable with respect to α with on [a,b], and write f∈R(α) • Note ○ When α(x)=x, this is just Riemann integral Definition 6.3: Refinement and Common Refinement • We say that the partition P^∗ is a refinement of P if P^∗⊃P • Given two partitions P_1 and P_2, their common refinement is P_1∪P_2 Theorem 6.4: Properties of Refinement • If P^∗ is a refinement of P, then ○ L(P,f,α)≤L(P^∗,f,α) ○ U(P^∗,f,α)≤U(P,f,α) Theorem 6.5: Properties of Common Refinement • Statement ○ (∫_a^b▒ ) ̅fdx≤▁(∫_a^b▒ ) fdx • Proof Outline ○ Given 2 partitions P_1 and P_2 ○ Let P^∗ be the common refinement ○ Then L(P_1,f,α)≤L(P∗,f,α)≤U(P∗,f,α)≤U(P_2,f,α) Theorem 6.6 • Statement ○ f∈R(α) on [a,b] if and only if ○ ∀ε 0, there exists a partition P s.t. U(P,f,α)−L(P,f,α) ε • Proof Outline ○ ∀P,L(P,f,α)≤▁(∫_a^b▒ ) fdx≤(∫_a^b▒ ) ̅fdx≤U(P,f,α) ○ (⟸) If U(P,f,α)−L(P,f,α) ε § Then 0≤(∫_a^b▒ ) ̅fdx−▁(∫_a^b▒ ) fdx ε ○ (⟹) If f∈R(α) § Then ∃P_1,P_2 s.t. □ U(P_1,f,α)−∫_a^b▒fdα ε/2 □ ∫_a^b▒fdα−L(P_1,f,α) ε/2 § Consider their common refinement P § By Theorem 6.4, U(P,f,α)−L(P,f,α) ε Theorem 6.8 • If f is continuous on [a,b], then f∈R(α) on [a,b] Theorem 6.9 • If f is monotonic on [a,b], and α is continuous on [a,b] • Then f∈R(α) on [a,b] Theorem 6.10 • If f is bounded on [a,b] with finitely many points of discontiunity • And α is continuous on these points, then f∈R(α)
