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ANALYSIS II
Introduction to Riemann-Stieltjes Integration

Defn. A collection of n+1 distinct points of the interval [a,b]

is called a partition of the interval. In this case, we define the norm of the partition by

where Dx_i : = x_i - x_i-1 is the length of the i-th subinterval [x_i-1,x_i].
For a non-decreasing function a on [a,b], define

Defn.  Suppose that f is a bounded function on [a,b] and a is nondecreasing. For a given partition P, we define the Riemann-Stieltjes upper sum of a function f with respect to a by

where M_i denotes the supremum of f over each of the subintervals [x_i-1, x_i]. Similarly, we define the Riemann-Stieltjes lower sum by

where here m_i denotes the infimum of f over each of the subintervals [x_i-1, x_i]. Since m_i £ M_i and Da_i is nonnegative, we observe that

for any partition P.

Defn.  Suppose P₁, P₂ are both partitions of [a,b], then P₂ is called a refinement of P₁ (denoted by P₁ £ P₂) if as sets P₁ Í P₂.

Note. If P₁ £ P₂, it follows that ||P₂|| £ || P₁|| since each of the subintervals formed by P₂ is contained in a subinterval which arises from P₁.

Lemma. If P₁ £ P₂, then

and

Pf. Suppose first that P₁ is a partition of [a,b] and that P₂ is the partition obtained from P₁ by adding an additional point z. The general case follows by induction, adding one point at a time. In particular, we let

and

for some fixed i. We focus on the upper sum for these two partitions, noting that the inequality for the lower sums follows similarly. Observe that

and

where  M : = sup { f(x) | x_i-1 £ x £ z } and  M^~ : = sup {f(x) | z £ x £ x_i }. It then follows that U(P₂;f,a) £ U(P₁;f,a) since

Defn.  If P₁ and P₂ are arbitrary partitions of [a,b], then the common refinement of P₁ and P₂ is the formal union of the two.

Corollary. Suppose P₁ and P₂ are arbitrary partitions of [a,b], then

Pf.  Let P be the common refinement of P₁ and P₂, then

Defn.  The lower Riemann-Stieltjes integral of f with respect to a over [a,b] is defined to be

Similarly, the upper Riemann-Stieltjes integral of f with respect to a over [a,b] is defined to be

By the definitions of least upper bound and greatest lower bound, it is evident that for any function f there holds

Defn.  A function f is Riemann-Stieltjes integrable over [a,b] if the upper and lower Riemann-Stieltjes integrals coincide. We denote this common value by   ò_a^b f(x) d a(x).

Examples: 

1. Obviously, if a(x) : = x, then the Riemann-Stieltjes integral reduces to the Riemann integral of f.
2. ò_a^b f(x)  d a(x) = f(x₀), if f is continuous at x₀ and a is defined to be the step function which is one for x larger than x₀ and zero otherwise.