* The main exception occurs where the formal reciprocity between integrand and integrator peculiar to the one-dimensional theory is involved, namely in the “theorem of integration by parts” (§ 2, III). I may point out that, once this reciprocity is lost, it is far more natural to start with a function of intervals or of sets as integrator.

† “Riemann” integration with respect to an additive function of sets is discussed in my paper in the Proc. Lond. Math. Soc. (2), 29, 479–489 (1928).

‡ Cp. my paper on “Functions of ∑ defined by addition, or functions of intervals in n-dimensional formulation,” Math. Ztschr. 29, 171–216 (1928). Integration with respect to a function of ∑ unchanged by subdivision and bounded is considered in my paper “On Riemann integration with respect to a continuous increment,” Ibid. 216–233. (With regard to the latter paper, it may be well to point out here that the assumption of continuity for the integrating increment corresponds, in the language of integration with respect to a function of a single variable g (t), to taking only points of division at which g (t) is continuous, i.e. involves a restriction on the choice of subdivision only. Cp. p. 207 of the former paper, where the nomenclature is explained.)

§ The latter condition is not satisfied by the type of subdivision used for the most part by Burkill, “Functions of intervals,” Proc. Lond. Math. Soc. (2), 22, 275–310 (1924), and generally in the theory of areas and volumes, and in the topology of surfaces. In all these theories, a unit of a subdivision may not be subdivided independently of “neighbouring” units, to form a subdivision consecutive to the given one.

∥ The crucial point in this method corresponds indeed precisely to, or rather involves a particular case of, the property of functions of ∑ whose S is unchanged by subdivision, given on p. 190 of my first Ztschr. paper quoted above; and the simple proof of this property may be seen at a glance to involve the conditions in question.

¶ The method under its more general form is outlined in my Dissertation for the Cambridge Ph.D. (1929) on “Foundations for the generalisation of the theory of Stieltjes integration and of the theory of length, area and volume.” See the Abstracts of Dissertations published for the year 1928–29 by the University Registry, Cambridge.

* For this reason all methods of summation, mean derivation, etc., as well as all considerations of upper and lower limits, are directed explicitly to the deduction of a single number from the given data.

* This is the only legitimate definition of upper and lower integrals in connection with the usual definition of a unique R-S integral indicated above, where unrestricted integrators are admitted. The definition by means of upper and lower Darboux sums discussed, e.g. by S. Pollard, Quart. Journ. 49 (1920–21), 73—138 (see also E. W. Hobson, Real Variable, Vol. i, 2nd ed. p. 509, or 3rd ed. p. 546), is concerned with a monotone increasing integrator, and its adoption requires the corresponding modification of the idea of a unique R-S integral, which then becomes what S. Pollard has called the “modified” R-S integral.

† See my paper on “The algebra of many-valued quantities,” Math. Ann. 104, 260–290 (1931), with particular reference to the last paragraph. I note that the adoption of the word “limit” to designate eventually an aggregate of values fits in with many habitual phrases (for instance, E. W. Hobson, Real Variable, i, 2nd ed. p. 339, line 2) which, although contrary to classical usage, only serve the more readily to illustrate its inconvenience.

‡ Cp. my paper just quoted, p. 276.

§ The proper treatment of the more general case requires the theory of infinities which I have outlined in my Dissertation and in a short paper read at the British Association Meeting in Bristol, Sept. 1930. I shall indicate as far as possible in footnotes what the relative extensions are.

* When there are (i.e. when in the ordinary sense the upper integral is + ∞, or the lower − ∞, or both), each unbounded monotone sequence of “Riemann” sums for subdivisions of norms tending to 0 defines as well a “simple infinity” of ∫ f (t)dg. The aggregate of these simple infinities defines the “complete infinity” in ∫ f (t)dg.

* When g (t) is monotone, this definition reduces to the usual one, and in that case the upper “Darboux” sum for any given subdivision is not less than (and the lower one not greater than) the corresponding sum for any subdivision consecutive to the given one.

* Loc. cit. Math. Ann. 104, p. 261.

† Ibid. p. 281.

‡ When the integrals are one-valued, the restriction a < b < c in this formula is removed by definition, i.e. by writing

The corresponding procedure here would lead to the supplementary formula

as a conclusion a fortiori from (1) when a < b < c. Cp. footnote †, p. 332.

§ When infinities are allowed, IV is of course maintained unrestrictedly, whether the ordinary or refined sense of “infinity” is adopted. The addition theorems I−III may, however, lose all meaning in terms of ordinary infinity (namely when one upper integral on the addition side is + ∞, and the other lower integral − ∞). This inconvenience is entirely removed by the analysis into distinct “simple infinities” and the relations all remain true without restriction.

* For we have always in the many-valued sense, when either is one-valued, complete equality between their sum and the This also follows at once from the relation of inclusion valid in general, and the property quoted in the next footnote. The conclusion is not invalidated if we admit infinities in the strict sense.

† For any many-valued quantities a, b and c, the relation a + b ⊂ c implies a fortiori a ⊂ c − b, and conversely if b is one-valued. Cp. loc. cit. Math. Ann. p. 268.

‡ Examples showing that it cannot always do so may readily be constructed, e.g. as indicated below, p. 342, footnote.

§ If infinities were being allowed, in the integrals, we should still get equality, more precisely arithmetic equality, described for two infinities defined by sequences of numbers α_i and β_i by saying that the term-by-term difference α_i − β_i between the two sequences tends to zero, and distinct from the more stringent (“geometric”) type of equality for which the term-by-term ratio α_i/β_i must tend to 1. Thus in particular the two theorems that follow remain true for non-finite integrals with the arithmetic meaning of the equality sign.

* The precise condition involved is g (b + 0) = g (b − 0) one-valued, irrespective of the actual value g (b).

† This property does not seem to be mentioned in accounts of R-S integration, although used in ordinary Riemann integration (g (t) = t). In more dimensions the complete continuity of the integrating function g (t) is not necessary, even with the unessential modification mentioned in the preceding footnote; the condition involved, namely

being in fact only in one dimension no more general than the condition

‡ See footnote* above.

§ As f (t) is assumed bounded in the neighbourhood of b, these limits are complete. If we omit the assumption of boundedness of f (t), the ordinary equality of infinite limits would be quite insufficient to secure the vanishing of the difference given above and the consequent validity of the addition theorem. But if we enlist the proper definition of “infinities” and demand arithmetic equality, the theorem remains true without any restriction of boundedness on f (t).

* Rendiconti Lincei (5), 17 (1), 582—587 (1908).

† See also W. H. Young, “La symétrie de structure des functions de variables reélles,” Bull. Soc. Math. (2), 52, 265—280 (1928), where also the extension to functions of several variables is indicated. The proof there given applies to finite values of the above limits, but may easily be adapted for the “infinite values,” if existent, provided that we are content with the ordinary notion of equality of infinities (i.e. regard any two positive infinities as “equal").

* In the second part of the argument, the actual continuity at τ is involved, and not merely g (τ + 0) = g (τ − 0).

† See p. 330, footnote.

* The German use of the term “iteriertes Integral” to describe what we call a repeated integral is of course quite different from that made here of the expression “iterated integral” to designate an integral with respect to another integral.

* This proof is adapted from that of J. Hyslop quoted below, p. 339. It is valid also when non-finite integrals are allowed, the equality obtained between the two integrals being of the arithmetic type, for the analysed notion of infinity. Indeed for this theorem the more refined notion is not necessary in the first instance, and we may be content with the ordinary meaning of the “equality” of two “infinite values,” i.e. with the bare fact that the two upper integrals, and similarly the two lower integrals, if not finite, are infinities of the same sign.

* Hyslop, J., “A theorem on the integral of Stieltjes,” Proc. Edinburgh Math. Soc. 44, 79–84 (1926).CrossRefGoogle Scholar

† The many-valued form of the less obvious equalities of limits usually involves an additional term only vanishing when the limits are one-valued.

* From this point of view, the position of the theorem considered in the theory of integration of Riemann type is quite different from that which it holds in the Young-Lebesgue theory. Cp. for instance Young, L. C., “The Theory of Integration,” Cambridge Tract, 1927, p. 40Google Scholar, Theorem I, where the application of the method of monotone sequences is precisely on a footing with that in the proof of the four elementary properties, p. 32.

† Whittaker, J. M., “On integrals with a nucleus,” Proc. Lond. Math. Soc. (2), 25, 213–218 (1926)CrossRefGoogle Scholar, where also h (t) is assumed non-negative.

‡ And in particular in the proof of W. H. Young's condition for the uniqueness of the R-S integral, the analogue of Riemann's condition of integrability in its modern form in terms of measure (Riemann's condition, in this form, is used in the paper of J. M. Whittaker just quoted). Cp. p. 216 of J. M. Whittaker's paper, first footnote, where it is remarked that the reasoning in the text is “similar” to that used in the proof of W. H. Young's condition. Effectually, as I pointed out in a short note “On Riemann-Stieltjes integrals with respect to a Riemann integral in space of n dimensions” in the Journ. Lond. Math. Soc. 3, 117–19 (1928)Google Scholar, the whole theorem (for the case g (t) = t) could be deduced trivially (without the restriction h (t) ≥ 0) and in n-dimensional form, from W. H. Young's condition, noting that “existence” of the two integrals implies their equality by the known properties of Lebesgue-Stieltjes integrals. (In J. M. Whittaker's paper, this part was deduced, at a suggestion of S. Pollard, from monotony and addition properties of upper and lower Riemann integrals not usually valid for the upper and lower R-S integrals which would have had to be considered if he had not confined himself to the case g (t) = t.)

§ See my companion paper in these Proceedings on integration with respect to functions of bounded variation, where examples of this type of result are given.

∥ It has also the advantage over the purely “elementary theorems” of having a definite meaning in terms of current notions of infinity when no restriction of boundedness is made. Cp. footnotes, pp. 331 and 337.

* t ⇒ − 0 means t tends to 0 in any manner through values ≤ 0, and similarly t ⇒ + 0 through values ≥ 0, and t ⇒ 0 combines both modes of approach to 0.

* To obtain from this a ready example of the “non-additiveness” of many-valued integrals, take for instance

ø (t) = sgn t, i.e. + 1 when t is positive,

− 1 when t is negative,

0 when t = 0.

The sum (3) of the two integrals for (− 1, 0) and (0, + 1) has then the values + 1, − 1 and 0, while the integral (4) for the whole (− 1, + 1) has, besides, the values + 2 and − 2.

Conversely, the above formulae also show that the cases of “additiveness” given in § 3 are not the only ones, since here g (t) is discontinuous (in every sense) at t = 0, and, without having

we can easily arrange that the mean of these two limits should include

and hence (3) be identical with (4). For instance, if

with ø (0) = 1,

then has all values between 0 and 1 inclusive, has all those between 0 and ½ inclusive, and the value 1, their mean has all values between 0 and ¾ and all values between ½ and 1, i.e. all values between 0 and 1 inclusive, and so actually coincides with