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On many-valued Riemann-Stieltjes integration, II. Integration of bounded functions with respect to functions of bounded variation
Published online by Cambridge University Press: 24 October 2008
Extract
The principles and general properties indicated in the first part of the present theory are those which belong most strictly to the province of the “algebra of many-valued quantities,” by their practically exclusive reliance on notions of equality, union, and passage to the limit in this domain.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 27 , Issue 3 , July 1931 , pp. 345 - 380
- Copyright
- Copyright © Cambridge Philosophical Society 1931
References
* The properties given in this section are of the nature of lemmas for later use.
* This does in fact mean that to each t there corresponds an N = N e, such that t does not belong to for any n > N e, i.e. such that
for all n > N e.
* p. 335, above.
† Part I, § 2.
* Proof of W. H. Young's test of “integrability” with respect to a monotone function, modelled on that of Riemann's own test for g (t) = t.
The mode of presentation adopted here is almost identical with that in my paper on “Riemann integration with respect to a continuous increment,” Math. Zeitschr. 29 (1928), §§ 3, 4 (pp. 224–5).
† The oscillation (or fluctuation) of a function f(t) at a point in (a, b) is the difference between the largest and the least possible limits of f(t′) as t′ tends to t by values, in (a, b), less than, greater than, or coincident with, the value t. It is thus the breadth of the limit
lim f(t′),
t · → t
where t′ → t combines all possible modes of approach to t in (a, b), including the identical mode t′ = t.
The oscillation of f in an interval is not less than that at any point of the interval.
‡ Thus g(Q (ε)) is the content of Q (ε)with respect to g in (a, b). Cp. § 3 below.
* A “neighbourhood” of a set of points E is any set which contains all points within a certain distance δ of points of E. A descending sequence of closed sets none of which is entirely contained in this neighbourhood has always some point belonging to all the sets whose minimum distance from points of E is not less than δ.
* I.e. |f n(t) − f(t)| < K for all n and t. The functions are not then necessarily all bounded, unless f(t) is.
* This definition may easily be seen to coincide with the usnal one by means of containing open sets, and a repeated passage to the limit.
† To see this, note that for any subdivision of (a, b), each point of Q, other than a or b, is either interior to a mesh of the subdivision, or belongs to two abutting meshes.
* That there is no kind of equivalence between the two conditions may be seen from the following example, where G-content is ordinary Lebesgue measure Divide the unit interval into 2n equal parts, and let be the kth part. Then m (Q i) → 0 with i, and lim Q i = null, lim̲ Q i = unit interval. The complementary sets provide a similar sequence, for which however m (Q i) → 1.
* Cp. my note on “Non-uniform convergence and term-by-term R-S integration,” Journal London Math. Soc., 6 (1931), not yet published.
† I.e. if |f n − f| remains bounded for all t and n.
* “Non-uniform convergence…,” loc. cit. § 2.
† It is shown in the paper quoted that outside the sum of the sets of points of discontinuity of all the functions f nin (a, b), (which has content zero with respect to G when all these functions are “integrable” with respect to g) there is no distinction between unsteady and irregular convergence of the sequence at a point in (a, b), and either is equivalent to discontinuity of the limiting function f (t) at the point.
‡ Ibid. § 3.
* This may be either inferred directly from the general symmetry of functional limits referred to in Part I (Theorem of W. H. Young) or deduced independently from the hypothesis of bounded variation.
† See however Hahn, Reelle Funktionen, p. 467.
‡ See e.g. Lebesgue, , Leçons sur l'intégration…, 1st Ed. (1904), p. 53.Google Scholar For convenience of the reader, the argument may be indicated here: We consider a sequence of subdivisions, of norms tending to zero, for which the sums (1) tend to the total variation G(b) − G (a) in (a, b). Each of the sums (1)
≤ [G (t n − G (a)] + |g (b) − g (t n)|,
because the first difference is the total variation in (a, t n).
As the norm of the subdivision tends to 0, its last point of division t n tends to b from the left, and we get
G (b) − G (a) ≤ G (b − 0) − G (a) + | g (b) − g (b − 0)|.
This gives G (b) − G (b − 0) ≤ | g (b) − g (b − 0) |,
while obviously the contrary inequality also holds, and hence
G (b) − G (b − 0) = | g (b) − g (b − 0) |.
By isolating the first instead of the last mesh of the subdivision, we should have got similarly
G (a + 0) − G (a) = | g (a + 0) − g (a)|.
This holds for any points a and b, and proves the statement.
The proof in Hahn is slightly less straightforward.
* It follows also that for any point t,
g 1 (t + 0) − g 1 (t) = larger of [g (t + 0) − g (t)] and 0,
g 1 (t) − g 1 (t − 0) = larger of [g (t) − g (t − 0)] and 0,
g 2 (t + 0) − g 2 (t) = larger of [g (t) − g (t + 0)] and 0,
g 2 (t) − g 2 (t − 0) = larger of [g (t − 0) − g (t)] and 0,
so that in particular, at each point t, one or other of the two functions g 1, g 2 is continuous on the right, and one or other is continuous on the left. And at any point where g (t) lies between g (t + 0) and g (t − 0), in the wide sense, one of the two functions g 1, g 2 is completely continuous.
† Since an expression of the form |a| + |β| − |a + β| vanishes except when a and β have opposite signs, and is then twice the smaller of | a | and | β |, the number e (t) is zero if g (t) is between g (t + 0) and g (t − 0) inclusive, and otherwise equal to twice the absolute difference between g (t) and that one of the limits g (t + 0) and g (t − 0) which is nearest to it. Thus e (t) is double what is occasionally called the external saltus of g (t) at t.
‡ Or also zero, when we take the “sum” to be 0.
* Finite, infinite, or null, as above.
* The argument is the same as that used for the selection of the subsequent from the numbers {βi} on p. 363. The principle is indeed a general one for selecting a sequence of objects with regard to a countable number of characteristics. We confine our attention of course to those of the points τ which are among the countable number of points of division of all the subdivisions of the considered sequence.
* The points involved ultimately as points of division by any given direct sequence of subdivision are also those thus involved by any subsequence of the given sequence.
* The fact that the least possible value of the limit of the sums (1) is is implied in a statement given by E. W. Hobson, Real Variable, 1, § 247, when combined with that at the end of Ibid. § 246. It is used e.g. by J. Hyslop, loc. cit. p. 375 below.
* We may also omit the assumption that a (t) = lim a (t′, t″) is one-valued and only retain that of boundedness of the factor. The modifications required in the results for this case will be indicated in footnotes. The one-valuedness of a (t, t″) itself is of course unessential in any case.
† If a (t) were many-valued, we should here substitute for the word “exactly” the words “some value or values of.”
* In the case of a many-valued a (t), we have here to say “the sums (4) have a (usually many-valued) limit included in. …” The first part of the theorem is of course unchanged.
* In this connection, the note by J. Hyslop, Proc. Edinburgh Math. Soc. (2) 1 (1929), 234–240, completes the classical accounts.
* In particular if g has no exceptional discontinuities.
* In point of fact it is well known that no discontinuous function f has a one-valued integral with respect to all positively monotone functions g″ (it becomes many-valued as soon as integrand and integrator have a common point of discontinuity), and hence the argument shows that f would have to be continuous.
* a − 0 and b + 0 are taken to be a and b for the fundamental interval (a, b).
* Fundamenta Math., 13 (1929), 240–260; L'enseigncment Math., 26 (1927), 63–77. Since we are still working with functions of the undivided point t as integrand and integrator, this adoption of the language of directed points is of course a purely formal one here.
† The retention of a neutral interval as fundamental region is of course optional.
‡ F (t) is continuous at t if lim F (t′) = F (t).