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Further Notes on the Stieltjes Integral
Published online by Cambridge University Press: 20 January 2009
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The definition here used of the Stieltjes Integral is the same as that of a previous note, viz.:—
Let f (x), φ (x) be two real functions defined in (a, b) a finite interval on the axis of the real variable x. Let Δ1, Δ2, …, Δn, be a finite set of sub-intervals which together make up (a, b). Δrφ denotes the increment of φ (x) in Δr. Let ξr be any point of Δr, and form the sum
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- Copyright © Edinburgh Mathematical Society 1929
References
page 234 note 1 Proc. Edin. Math. Soc., 44 (1926), 79.Google Scholar
page 234 note 2 Proc. Lond. Math. Soc., 13 (1913), 113.Google Scholar
page 235 note 1 Math. Zeitschr., 29 (1928), 217. See also additional note at the end of the present paper.Google Scholar
page 235 note 2 It seems to me that the Theory of the Riemann and Stieltjes Integrals is essentially concerned with intervals rather than with sets, and that it is of interest to develop it so far as possible without recourse to the theory of sets.
page 236 note 1 Hobson, : loc. cit., pp. 295–303, in particular p. 301.Google Scholar
page 237 note l CfYoung, W. H., loc. cit., pp. 133–134.Google Scholar
page 237 note 2 Hobson, : loc. cit., p. 301, 334.Google Scholar
page 237 note 3 Ibid., p. 325.
page 238 note 1 See additional note at the end of this paper.
page 239 note 1 Lot. cit., p. 84.Google Scholar
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