The Stieltjes Integral
In this chapter we discuss a generalization of the Riemann integral that is often used in both theoretical and applied mathematics. Stieltjes originally introduced this concept to deal with infinite continued fractions, but it was soon apparent that the c
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The Stieltjes Integral
In this chapter we discuss a generalization of the Riemann integral that is often used in both theoretical and applied mathematics. Stieltjes1 originally introduced this concept to deal with infinite continued fractions,2 but it was soon apparent that the concept is useful in other areas of mathematics—and thus in mathematical physics, probability, and number theory, independently of its role in continued fractions. We illustrate the usefulness of the concept with two simple examples. Example 18.1. Consider a planar curve parameterized by γ (t) = x(t), y(t) (t ∈ [a, b]), where the x-coordinate function is strictly monotone increasing and continuous, and the y-coordinate function is nonnegative on [a, b]. The problem is to find the area under the region bounded by the curve. If a = t0 < t1 < · · · < tn = b is a partition of the interval [a, b] and ci ∈ [ti−1 ,ti ] for all i, then the area can be approximated by the sum n ∑ y(ci ) x(ti ) − x(ti−1 ) . i=1
We can expect the area to be the limit—in a suitable sense—of these sums. Example 18.2. Consider a metal rod of negligible thickness but not negligible mass M > 0. Suppose that the rod lies on the interval [a, b], and let the mass of the rod over the subinterval [a, x] be m(x) for all x ∈ [a, b]. Our task is to find the center of mass of the rod. We know that if we place weights m1 , . . . , mn at the points x1 , . . . , xn , then the center of mass of this system of points {x1 , . . . , xn } is m1 x1 + · · · + mn xn . m1 + · · · + mn 1 2
Thomas Joannes Stieltjes (1856–1894), Dutch mathematician. For more on continued fractions, see [5].
© Springer New York 2015 M. Laczkovich, V.T. S´os, Real Analysis, Undergraduate Texts in Mathematics, DOI 10.1007/978-1-4939-2766-1 18
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18 The Stieltjes Integral
Consider a partition a = t0 < t1 < · · · < tn = b and choose points ci ∈ [ti−1 ,ti ] for all i. If we suppose that the mass distribution of the rod is continuous (meaning that the mass of the rod at every single point is zero), then the mass of the rod over the interval [ti−1 ,ti ] is m(ti ) − m(ti−1 ). Concentrating this weight at the point ci , the center of mass of the system of points {c1 , . . . , cn } is c1 m(t1 ) − m(t0 ) + · · · + cn m(tn ) − m(tn−1 ) . M This approximates the center of mass of the rod itself, and once again, we expect that the limit of these numbers in a suitable sense will be the center of mass. We can see that in both examples, a sum appears that depends on two functions. In these sums, we multiply the value of the first function (which, in Example 18.2, was the function x) at the inner points by the increments of the second function. We use the following notation and naming conventions. Let f , g : [a, b] → R be given functions, let F : a = x0 < x1 < · · · < xn = b be a partition of the interval [a, b], and let ci ∈ [xi−1 , xi ] (i = 1, . . . , n) be arbitrary inner points. Then the sum n
∑ f (ci ) ·
is denoted by σF respect to g.
g(xi ) − g(xi−1 )
i=1
f , g; (ci ) , and is call
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