* 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 {β_i} 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).