1-differentials on 1-cells: a further study

1-differentials on 1-cells: a further s t u d y Solomon Leader The Kurzweil-Henstock concept of integral as gauge-limit of approximating sums over tagged-cell divisions should make the traditional approaches to measure and integration on ~ " obsolete. It yields a definition of differentim as object of integration that can clarify and expand elementary calculus. The m-differentials on an n-cell introduced and developed in [1] yield rigorous differential formulations for various concepts associated with integration. An expository outline of tile case for m = n = 1 was presented in [2]. It is this case that we examine further here, although many of our results are valid in higher dimensions. Let K be a 1-cell [a,b]. A tagged cell (I,t) is a 1-cell I with a selected endpoint t. Our initial objects of integration are " s u m m a n t s " which provide the summands in our approximating sums. A summant S is a real function S ( I , t ) on the set of all tagged cells (I,t) in K. Each function x on K defines a summant A~ given by A x ( I , t) = A x ( I ) = z ( q ) - x ( p ) for I = [iv,q]. Traditional sumnmnts are of the form S = zAx defined for functions z, z on K by S(I, t) = z ( t ) A z ( I ) . W e shall be interested also in zlAz[ and in some novel sumnmnts such as Q ( I , t ) = I for t a left endpoint, -1 for t a right endpoint of I. A figure F is a nonvoid union of finitely many 1-cells in K. A division of F is a finite set of nonoverlapping tagged cells whose union is F. A gauge a is a function on K with a(t) > 0 for all t. (I,t) is a.fine if the length of I is less than a(t). An a-division is a division whose members are a-fine. Given a summant S each division ~" of F yields the sum ~ ( S , jc) of S(I, t) over all members ( I , t ) o f ~ ' . For each gauge a let ~ ( S , a ) be the infimum, ~ ( S , a ) the suprenmm, of ~ ( S , ~ ) over all a-divisions ~" of F. Define the lower integral f F S = s u p o ~ ( S , a ) and upper integral "fFS = infa ~ ( S , a) taken over all gauges a on F. If these two integrals are equal their common value defines fF S. S is integrabIe on F if the integral fv S exists and is finite. We shall sometimes use f b or just f for fK" As a function space the summants on K form a Riesz space (vector lattice) S. The summmlts with f IS] = 0 form a Riesz ideal T , a linear subspace d S such that S E S, T E T , IS[ < IT[ imply S E T. Thus D = S/:T is a Riesz space with the linear and lattice operations transferred homomorphically from S to l). A differential tr on K is any element of 7). Explicitly a is an equivalence class [S] of sumnmnts under the equivalence S ,-, S' defined by f IS - S' 1 = O. For p = [R] and a = [S] we have p + a = [R + S], ca = [cS] for any construct c, and

83

Ial = [IS I]. For the lattice operations these give p A a -- [R A S], p V a = In V S], a + = IS+I, and a - = [S-]. T h e definitions f a = f S a n d ? a = f S are effective for any s u m m a n t s S representing a. W h e n these two integrals are equal their c o m m o n value defines f a. a is integrable if f a exists and is finite. For 1E the indicator of a subset E of K the definition l ~ a = [1ES] for a = [S] is effective because 1E is bounded. E is a-null if l e a = 0. a (or any of its representing s u m m a n t s S) is tag-nuU if every point is a-null. A condition holds a-everywhere {or at a-all points t) in K if it holds on the complement of some a-null set. If y is a function defined a-everywhere on K we can define ya = [uS] where a = [S] a n d u is any function on K such t h a t u = y a-everywhere. T h a t this is effective is nontrivial [1]. Each function x on K induces an integrable differential dx = [/kx] with J'z dx = A x ( I ) for every 1-cell I in K . A differential a is integrable if a n d only if a = dx for some x. A differential a is ~umrnable if the n o r m u(a) -- f l a ] is finite, a is tag finite if lpa is s u m m a b l e for every point p. Each representative S of a tag-finite a effectively defines Sp = [SR] for every differential p = [R]. If b o t h p and a are tag-finite t h e n the definition ap = [SR] is effective. T h e convergence I ~ t will always m e a n t h a t the ( I , t ) are tagged cells and the length of I goes to 0. Our work here centers a r o u n d s u m m a b i l i t y and some conditions related to it. We shM1 show t h a t the s u m m a b l e differentials on K form a B a n a c h lattice under the n o r m v. We shall prove some convergence theorems for products of a differential with Borel functions under appropriate s u m m a b i l i t y conditions. We shall characterize damper-summability, defined below as in [1], and investigate some conditions it implies, namely the archimedean and weakly archimedean properties. We shall give lattice-theoretical formulations for two M~solute continuity conditions involving indicator summm~ts. We shall s t u d y s u n m m n t derivatives. Finally we shall give some applications to the calculus: the product formula, iterated integration-by-parts, Ta~clor's formula, finite expansions of differentials, a n d a converse to the f u n d a m e n t a l theorem of calculus.

1. S U M M A B L E DIFFERENTIALS. The s u m m a b l e differentials on K form a RSesz subspace of the Riesz space ~ of all differentials on K . Indeed, under the Riesz n o r m v they form a Banach lattice. ~Te need only prove completeness. P r o p o s i t i o n 1. Given a sequence of differentials al on K such t h a t ~ i v(ai ) < oo there exists a s u m m a b l e differential a = ~ i ai where the series is convergent in the n o r m u. P r o o f . We. m a y assume al # 0. Let Si be a s u m m a n t representing ai.

84 Choose a decreasing sequence of gauges al > ai+l such that

(x)

~(l&l,aO

< 2v(~,).

Let Pi indicate the ai-fine tagged cells. That is, P d I , t) = 1 if ( I , t ) is ai-fine, and 0 otherwise. By (1)

(2) for every division/(7 of K. So IP~&I(I,t) < 2v(~q) for every tagged cell (I,t). Thus we can define S ( I , t) = ~ i P i & ( I , t) since the defilfing series is a b s o l u t d y convergent. Let ~ = IS], the differential represented by S. For /C any ajdivision of K ,

= =

E(IS - E~_<~&l,~:) E(IE,>¢P,&I,~) Ei>¢E(IPi&I,/c)

= <

E(IS - E~<~ P,&I,/C) E(E,>¢IP~&I,X:

<

2Ei>jr'(tri)

by (12). So r,(a - Ei___jai) < 2E~>jr,(a~) --+ 0 as j --+ oo. By the triangle inequality, v(a) < ~ i r,(ai). So o, is summable.

2. L I M I T S OF I N T E G R A B L E D I F F E R E N T I A L S . The next result is elementary but noteworthy because it does not demand absolute integrability. It allows r,(cri) = u(ct) = c¢. P r o p o s i t i o n 2, If ~/ is integrable and t~(~r - al) --* 0 as i --~ oo then a is integrable and f trl --~ f o'. P r o o f . Let 7ri = ~r - ai. Since -u(Irl) < fTrl < flri < u(~'i) mad u(~rl) ~ O, both ~ f l and flri converge to 0. Since tri is integrable flri = f a - f ~rl and frci = fo" - f o'i. So f ~'i converges to the finite value f a = f a . O P r o p o s i t i o n 3. If ~ri is absolutely integrable and r,(a - ~i) ~ 0 then a is absolutely integrable and f ai ~ f a. P r o o f . Apply Prop. 2 to ~ri, ~' and to I~,1, I~1

3. I M P L I C A T I O N S OF 70~ < oo A N D 17~1 < oo. P r o p o s i t i o n 4. If'fK O" < OO then f F ~ < oo for every figure F in K. P r o o f . Let S represent a. By hypothesis there exists c in ~ and a gauge a such that (3) ~ ( S , / C ) < c for every a-division K7 of K. Given complementary fgures F, G in K fix an a-division ~ of G. For any a-division ~" of F the union/C = ~ ' U G is an a-division of K . So by (3)

<4)

~<s,~)

= ~(s,~)

- ~s,~)

_< c - ~ ( s , ~ ) .

85

Hence, fro" _< c - Z ( S , ~ ) < c~. 0 P r o p o s i t i o n 5. If 17= 1 < oo then < for every figu:e F in K. P r o o f . For complementary figures F, G Proposition 4 gives f r a < ov and f a ( r < or. Thus neither of these integrals can equal - o r since ?F~r+fa~r = ?KCr which is finite by hypothesis. P r o p o s i t i o n 6. Let If/ral < or. Given S representing a and e > 0 let a be a gauge such that (5)

~--~(S,K:) _< [ ~r + e for every a-division K: of K. J K

Then for each figure F in K

<]

(6)

F

for every a-division ~ of F. P r o o f i (5) gives (3) for c = f K a + e which gives (4) in the form E(S,.,'v) < r-fKa + e - ~ ( S , ~) for all a-divisions .T, ~ of complementary figures F, G. That 1S~

_<

+ Ifo - z<s, )l

with the two upper integrals finite by Proposition 5. The bracketed term can be made arbitrarily small by appropriate choice of ~. Hence (6). O --t P r o p o s i t i o n 7. Given ftf(r finite define x(t) = f,0,. Then: (i) for every cell I in K, A x ( I ) = f i e , (ii) dx _> a, (iii) if y is any function on K with dy > ~, then dy k dz, (iv) :dz = O, (v) for every hounded function u _> 0 on K, f_ud~ <_fuc' <_7udz. P r o o f . z(t) is finite by Proposition 5. (i) follows from the additivity of both sides. To get (ii) take any S representing a. Given e > 0 take a gauge a satisfying (5). Given a.n a-division /C of K let 5r consist of those meln1)ers of/C for which S >__ Am. Then ~ ( ( S - A x ) + , K ) = ~ ( S - Ax,~-) = ~ ( S , 9v ) - E ( A z , ~ ' ) = ~ ( S , ~ ' ) - - ~ F a <_ e b y ( i ) and (6). S o ( a - d ~ ) + = 0 which gives (ii). To get (iii) let dy >_ a. Then (i) gives A x ( I ) < fxdy = Ay(I) so dx <_ dy. We get (iv) from_.fdx - a = j'dx - f a = Ax - f a = 0 by (i). To get the first inequality in (v) let 0 < u < c. We may assume f_udx > - o o and -'fucr < c¢. Then both of these integrals must be finite since ~udz--]ua < f u ( d z - ~r) < c ~ d z - a = 0 by (iv). This gives the first inequality in (v). The second follows from (ii) for u >_ 0.

4. C O N V E R G E N C E T H E O R E M S F O R S U M M A B L E D I F F E R E N T I A L S . For dx > 0 a function z on K is dx-measurable if given c in ~ and

86 E the set of all t with z(t) > c, 1Edx is integrable. (See section 12 on p.173 in [1].) Every Borel function z is dx-measurable. P r o p o s i t i o n 8. Given a summable on g let ~(t) -- T, lal for all ~ in K. Then v(za) = f ]z]dx for every dx-measurable function z on K. P r o o f . Let Zr, = IzlAn. Then z,~dx is integrable. So v(z,,a) = f z,,d~ by (v) in Proposition 7. Hence $ z,,d~: <_ v(za) < f Izld~ since _< dx by Proposition 7. By the monotone convergence theorem (Theorem 10 in [11) $ z , , d z / " f Izldz.

O v(1Ea) is the "full variational outer measure" [3] induced by a on the subsets E of K. Proposition 8 gives v(1Ea) = f 1Edx for Ml dx-measurable E, hence for all Borel sets E. So x is the distribution function for the induced Boret measure. P r o p o s i t i o n 0. (Dominated Convergence) Let a be a differential on K. Let y l , y 2 . . , and v be Borel functions on K such that lYl,I -< v for all k in .A/, v a is summable, and yk ~ 0 a-everywhere. Then v(yka) --* 0 as k --+ oo. P r o o f . Propositions 7, 8 give z such that dx >_ via I and v(zva) = f ]z]dx for every Borel function z on K. Applying this identity with z the indicator of a Borel set we conclude that every a-null Borel set is dx-null. Similarly the set A of points where v = 0 is dz-null since it is va-null. Each ]ykt/v is a Borel function on K \ A bounded by 1. Moreover lykl/v --* 0 except on a a-null Borel set, hence d~-everywhere. Thus v(yj, a) = f tyk/v[da: ---* 0 by Proposition 8 and the bounded convergence theorem (Theorem 10 [1]). O P r o p o s i t i o n 10. Let a be a differential on K. Let v , v l , v 2 , . . , be nonnegative Borel functions on K such that v = ~ i ~ ¢ vi a-everywhere mad va is summable. Then ~ieJ¢ via is v-convergent to va. P r o o f . Apply Proposition 9 with yk = v - ~i
Note that the required summM)ility of v a is ensured if •i v(via) < oo since this sum dominates v(va) by Theorem 7 [1].

5. D A M P E R - S U M M A B L E A N D A R C H I M E D E A N D I F F E R E N T I A L S . An important consequence of summability is the archimedean property. A member a of a lattice group is archimedean if lr = 0 is the only member such that kl~r I ___ lal for every k in Af. Differentials do not have to be summable to be archimedean. Indeed, damper-smmnable differentials are archimedean. A damper is a function u with u(t) > 0 for all t in K. a is damper-summable if u a is summable for some dmnper u. The next two results relate damper-smnmability to finiteness of derivates. P r o p o s i t i o n U . For a a differential on K the following are equivalent:

87 (a) There is a representative S of o. and an increasing function x on K such that (i) every dx-nuU set is o.-null, (ii) at dx-a.ll t l i m l S ( I , t ) / A x ( I ) l < oo as I~t. (b) K is the union of countably many sets E with leo- summable. (c) o. is damper-sunnnable. (d) There is an increasing flmction x on K such that (i) holds, x is continuous wherever o. is tag-null, and (ii) holds for all S E o.. P r o o f . (a) ~ (b). Let E~ consist of all t where IS I < k A x at ( I , t ) ultimately as I --* t. Then 1Eklo.I < kdx. So each l~,o. is smnmable. By (ii) there is a dx-null set A such that A, Ez, E 2 , . . . cover K. 1.4o" is summable since it equals 0 by (i). (b) @ (c). We may assume K is covered disjointly by A 1 , A , , . . . with each 1Aho. summable. Choose ak > 0 such that the series ~ k al, v(1A~o') < oo. Define the flmction u on K by letting u = ak on Ak. uo. is sumnmble by Theorem 7 [1]. (c) =~ (d). Let To. be summable with damper w. Apply Proposition 7 to get dx > wlo. I. Then (i) is trivial since ]o.I < d x / w . x is continuous at p if l~o. = 0 by Proposition 8 with z = 1~. By adding the identity flmction to x we may assume that x is everywhere increasing. By adding a unilateral saltus at each point of discontinuity we may assume x is continuous wherever it is either left or right continuous. That is, dx is balanced [1]. Each of these additions preserves (i) and (ii). Take a representative S' of o. with wlS' I < Ax. Given S representing o. let P indicate Ax _< tS - S'[. Then 0 < P A x < IS - S'[ ,-~ 0. So P d x = O. By Theorem 16 [1] the indicator summant P is tag-null dxeverywhere. So at da:-all t, P ( I , t ) = 0 ultimately as I --~ t. Nov,- P = 0 means I S - S'I < Ax which implies I S t / A x < 1 + tS']/Ax < 1 + 1/w. Hence (ii) holds for each S representing o.. (d) ~ (a) afortiori. O

(For higher dimensional cells K the equivalence of (a), (b), (c) remains valid, but the implication (c) =~ (d) fails. ) Proposition 12. Let y be continuous on K. Then dy is danlper-sulnnmble if and only if there exists a continuous, increasing function z on K such that (i) every dz-null set is dy-null, and (ii) the derivates of y with respect to x are finite dx-everywhere. Proof. Apply Proposition 11 to o. = dy with S = Ay.

Our next result does not explicitly demand damper-summability but requires only the archimedian property. A set P of tagged cells in K is o.-ne#li#ible if Po. = 0 for the indicator P of P. Proposition 13. Let o., r be differentials on K with o. a.rchilnedean. Let

88 z be a function on K . T h e n r = zo- if and only if (a) every o.-negligible set of tagged cells is r-negligible, and (b) given S, T representing o-, r a n d ~ > 0 the set of tagged cells at which

(7)

IT - zSl ~ vlSl

is o.-negligible. P r o o f . Let r = zo.. If P is an indicator s u m m a n t with Po. = 0 t h a n P r = zPo. = 0. So (a) holds. To get (b) let R indicate (7). T h e n eRISI < IT - zSI so eRIo. I < Ir - z a I = 0. Thus R~r = 0 which gives (b). Conversely let (a), (b) hold. Choose S, T representing o-, r. Given e > 0 let R indicate (7). T h e n Ro. = R r = 0 b y ( b ) , (a). Let Q = 1 - R . QindicateslT-zSI < elSI. So Q I r - z(r[ < ~[o.I. Since Qo. = o. a n d Q r = r, Q ( r - zcr) = r - zo.. Hence Ir - za[ _< ¢[a[ for all e > 0. T h u s r = z a since o. is archimedean. Q)

We remark t h a t (b) is a differentiation condition. It says S ¢ 0 and for each e > 0, I T / S - z] < ¢ except on a o.-negligible set of tagged cells in K .

6. W E A K L Y A R C H I M E D E A N D I F F E R E N T I A L S . Some results on integration of derivatives hold under a condition ostensibly less stringent t h a n archirnedean. We call o- weakly archimedean if Per = 0 for every tagnull s u m n l a n t P . Every archimedean differentiM is weakly archinmdeall since tPI < e u l t i m a t e l y u n d e r gauge refinement for P tag-null. So IP~r t < ~]o-I for all > 0. Since o. is archimedean this implies Po. : 0. For summa.nts R, S define R = o(S) to m e a n given e > 0 there ezists a gauge c~ such that ]RI < ~ISI at all a-fine tagged cells in K . P r o p o s i t i o n 14. o. is weakly archimedean if and only if given S representing o. a n d R = o ( S ) t h e n R --~ 0. P r o o f . Let o. be weakly a r c h i m e d e a n a n d R = o ( S ) for some S representing o.. Define P = R / S if S ¢ 0, P = 0 if S = 0. P is tag-null since R = o(S). So Po. = 0. T h a t is, R ,~ P S ,~ O. Conversely given P tag-nuU define R = P S . T h e n R = o ( S ) so R ,-~ 0, t h a t is, Po. = O. Q)

We conjecture t h a t there exist weakly archimedean differentials t h a t are not archimedean, a n d archimedean differentials t h a t are not d a m p e r - s u m m a b l e . For weakly archimedean differentials the only conclusion we can draw in the direction of s u m m a b i l i t y is tag-finiteness, o- is tag-finite if lpo" is s u m m a b l e for every point p in K . [1] P r o p o s i t i o n 15. If o. is weakly archimedean then o. is tag-finite. P r o o f . Suppose v(lpo.) = oo for some p. T h e n given S representing o. there is a sequence Ik ~ p such t h a t IS(Ik,p)l ~ oo. Let R indicate the set of tagged

89 cells which occur in this sequence. Then R = o(S) so R ~ 0 1)y Proposition t4. But f R > 1, a contradiction. Q)

The next result generalizes Theorem 15 [1] in that "weakly archimedean" replaces "damper-surnnmble", and the demand that x be continuous is dropped. Proposition 16. Let z = ( x l , . . . ,zm) be an m-flmction on K such that each dxi is weakly archimedean. Let .f be a 1-function on a neighborhood of x ( K ) in ~"~ such that for some subset A of K: (i) f is differentiable at z(t) for all t in A, (ii) lady = dy for y defined on K by y(t) = f(x(t)). Define the m-function z on g by z(t) = grad f ( z ( t ) ) for t in A, z(t) = 0 for t in g \ A. Then dy is weakly archimedean and dy = z • dz. P r o o f . Ildzl[~ = Idz~l + . . . + ldz,,,I is weakly archilnedean. Hence, so is IId ll in the Euclidean norm since all norms on ht~"~ are equivalent. C,learly aAz, y - z . z , x -_ o ( l l A x l l ) . So dy = l a d y --- z . d ~ b y Proposition 14. This implies dy is weakly arehimedean since each dzi is weakly archimedean.

T. S L I M M A N T D E R I V A T I V E S . We say that the derivative of a summant T with respect to a summant S exists in the narrow sense at p with value c if S ( I , p ) # 0 ultimately and (8)

T ( I , p ) / S ( I , p ) ~ c as I ~ p.

(Thomson [3] calls this the "ordinary sense" .) The derivative exists in the broad sense if S ( I , p ) does not ultimately vanish and (8) holds for S ( I , p ) ~ O. Our definitions include the cases where c = co, c = - c o . For c finite the broad sense derivative equals c if and only if (T - cS)lp = o(S) and lvS # o(S). The latter condition assures uniqueness of c since it gives a (non-void) filterbase for (8). Note that the set E of all p where lvS = o(S) is ~r-null for a = IS] since I ~ S = 0 ultimately under gauge refinement. The (broad or narrow) derivative of T with respect to S is generally not an invariant of the differentials represented by T, S. With this caution in mind we can retain the traditional notation dy/dx of Leibniz for the (broad or narrow) derivative lim A y ( I ) / A x ( I ) as I ---* t. We shall similarly use ldyl/dx, dy/ldx h ]dyl/tdxl as in [1]. For weakly archimedean differentials results on derivatives follow readily from Proposition 14. Our first such result generalizes Theorem 18 [1]. Proposition IT. Let a, r be differentials on K with a weakly arcllimedean. Suppose they have representatives S , T for which the derivative (8) exists in the broad sense and is finite at r-all points p. Then r is weakly archimedean. Let A be a subset of K such that l a r = v and the finite derivative (8) exists in the broad sense at eeazh p in A. Define z(p) to be this derivative for p in A, and 0 for p in K \ A. Then r = za.

90

P r o o f . 1aT - z S = o(S). So v = l a y = za by Proposition 14. Since a is weakly archimedean so is z~r. P r o p o s i t i o n 18. Let a, v be differentials on K with a weakly archimedean. Given representatives S, T let E be the set of all p where the broad sense derivative {8) equals 0. Then E is v-null. P r o o f . I ~ T = o(S). Hence 1Ev = 0 by Proposition 14. P r o p o s i t i o n 19. Let a, v be differentials on K with a weakly archimedean. Given representatives S, T then the set E of all p where the broad sense derivative limx_~p t S ( I , p ) / T ( I , p ) I = ~ is v-null. P r o o f . Given p in E and a positive integer k, ISt > k i T I ultimately at ( I , p ) as I ~ p. So l E T = o(S). Thus 1Ev = 0 by Proposition 14. O

The last three results have interesting consequences for the case where S = A x , T = A y , cr = dz, v = dy. For instance given dx weakly arctlimedean, Proposition 17 (or 18) implies that if d y / d x = 0 in the broad sense dy_ everywhere then y is constant. There are conditions under which sumlnant derivatives are differential-invariant almost everywhere justifying the term "differentiM quotient" for the derivative. A differential a is dampable [1] if u a is absolutely integrable for some damper u. a is balanced [1] if lpa = 0 wherever there is a tagged cell ( I , p ) with fI lt,~" equal to zero. The next result is valid on 1-cells. P r o p o s i t i o n 20. Let a be a dampable, balanced differential on the 1-cell K. Let z be a function on K. Given T, S representing za, a respectively then the narrow sense derivative (8) exists and equals z(p) at a-all points p in K. P r o o f . Given e > 0 let R indicate tT - zSl > ¢ISI. R a = 0 by Proposition 13. For some damper u, u[o"l is integrable and has the same null sets as ~r. So Theorem 16 [1] applied to ula ] implies that R is tag-null cr-everywhere. That is, R ( I , t ) = 0 ultimately as I - ~ t at a - a l l t . N o d R = 0 means S # 0 mad IT~S- z I < e. Since a countable union of a-null sets is a-null the proof is done.

O For integrable, tag-null a Thomson [3] proved that [a] dampable implies a darnpable, answering a question posed in [2]. An alternate proof comes from the following differentiation theorem. P r o p o s i t i o n 21. Let z be a continuous function on tile 1-cell K with Idol dampable. Then the narrow sense derivative dz/Idzl exists dz-everywhere (with values 1 or -1 of course). P r o o f . (All derivatives here are narrow sense.) Given a damper z for ]dzl there is an isotone, continuous function y on K such that dy = z t d z I. By Proposition 20 with a = [dzJ, T = Ay, and S = tAxi

(9)

dy/Idxl

= z > o

91 dz-everywhere. There exists a gauge a on K such that at each t where (9) holds both A y ( I ) > 0 and l£xx(I)] > 0 for all a-fine ( I , t ) . Since x is continuous the intermediate value property implies that as I --~ t - either A x ( I ) > 0 ultimately or A x ( I ) < 0 ultimately. A similar statement holds as I ~ t+. So (9) implies the existence of the left and right derivatives ( d x / d y ) _ , ( d x / d y ) + with values restricted to 4-1/z. These unilateral derivatives ctua differ only at countably many points ([4]p.359). A countable set is d~-null since d~ is tag-null by continuity of x. So d x / d y exists and is finite dx-everywhere. Since (9) also holds d~-everywhere so does d x / I d x ] = z d x / d y . O

~Te can now get Thomson's result. P r o p o s i t i o n 22. Let • be continuous on the 1-cell K with [dx[ dampable. Then dx is dampable and every damper for Idxl is a damper for dx. P r o o f . Extend x to the right of K = [a,b] by defining x(s) = x(b) for s > b. Define w ( t ) = limn_.oosgn[x(t + l / n ) - x(t)] for a < t < b. Since x is continuous w is a Borel function. Also Iwl < 1. Let z be a damper for Idxl. By Proposition 21 d x / I d x I = w dx-everywhere. So dx = w l d x [ by Proposition 17. Hence z d x = wz]dx I = wdy. Since dy > 0 and w is a bounded Borel function w d y is integrabte. T h a t is, z d x is integrabte. O

8. A B S O L U T E C O N T I N U I T Y C O N D I T I O N S I N V O L V I N G IND I C A T O R S U M M A N T S . For a, r > O in any lattice group ~Dthe condition Ak~C(r -- ko') + = 0 iml)lies the condition

(10)

pAr=0forallpin~DsuchthatpA~r=0.

Tile converse holds if ~r is archimedean. For the Riesz space ~D of differentia.Is on K (10) is just the absolute continuity (a) in Proposition 13. We prove this next. P r o p o s i t i o n 23. For differentia.is ~r,v > 0 on K (10) is equivalent to

(11)

Every a-negligible set of tagged cells is r-negligible.

P r o o f . Given (11) and p A a = 0 pick representatives R , S of p, 0, with R A S = 0. This can be done by replacing arbitrary representatives R, S by ( R - S ) +, ( R - S ) - since p = p - p A a = (p--~r) + mld o" -----o" = o'--pAo" = (p--o')-. Let P indicate R > 0. Then P S = 0 so Po, = 0. By (11) P r = 0. So we can choose T _ 0 representing r with P T = 0. Since P R = R , RAT = ( P R ) A T = R A ( P T ) = O. S o p A r = 0. Hence , (11) implies (I0). Conversely, given (10) and an indicator smnmant P with P a = 0 we

92

contendPr=0. by (10). 0

Since(Pr) A~=rA(Pa)=rA0=0,

Pr=(Pr)Ar=0

For ~,,r >_ 0 the v-convergence of r A ( k a ) to r as k --+ c~ is stronger thaa (11). Since r - r A ( k a ) = ( r - k a ) + it is just (12)

v ( r - k a ) + --* 0 as k ---+ or.

P r o p o s i t i o n 24. Let a,'r _> 0 be differentials on K. Then (12) holds if and only if both of the following conditions hold: (13)

(r - m~r) + is summable for some rn in Af,

(14)

given ~ > 0 there exists ~ > 0 such that

v ( P r ) < ¢ for every indicator summant P with v ( P a ) < ~. P r o o f . Let (12) hold. Then (13) is trivial. To get (14) apply 0 < r < (r - ka) + + ko" to get 0 _< P T <: ( r -- k a ) + + k P a . Hence

(15)

v ( P r ) <_ v ( r - k ~ ) + + k v ( P a ) .

Given ~ > 0 apply (12) to get k with v ( r - k(r) + < ~/2. Take $ = e / 2 k . Then (14) follows from (15). Conversely, let (13) and (14) hold. Choose m from (13). Let S, T > 0 represent a, r. Let Pa indicate T > k S . Then (16)

0 <_ ( T - k S ) + = P k ( T - k S ) = P a T - kPhS.

So k P a S < PaT, hence 0 < k P k a < Pkr. Thus (k - m)Pko" <_ Par - mPa(r = P~('r - m a ) <_ (r - m a ) +.

So 0 <_ (k - m)v(Pao') <_ v ( r - m a ) + < ¢x~

for k _> m. Thus u(Paa) --* 0 as k --* oo. So u ( P a r ) --* 0 by (14). (r - k a ) + < Pkr by (16). So v ( r - k a ) + <_ v ( P a r ) giving (12). 0

Now

9. T H E D I F F E R E N T I A L O F A P R O D U C T , I T E R A T E D I N T E G RATION-BY-PARTS, A N D T H E T A Y L O R F O R M U L A . For each tagged cell ( I , t ) define the summant Q ( I , t ) to be 1 if t is the left endpoint of I, -1 if t is the right endpoint. If t + h is the endpoint of I opposed to t then Q ( I , t) = sgn h. So

(17)

A x ( I ) = Q ( I , t ) [ z ( t + h) - ~(t)] for z a function on I.

93

Proposition 25. (18)

For u, v functions on the l-cell K A(uv) = u A v + v A u + Q A u A v .

If u, v are bounded then (19)

d(uv) = udv + vdu + Qdudv.

If u is continuous and dv weakly archimedean then (20)

d(uv) = udv + vdu.

P r o o f . In the identity u(~)v(~)

,~(t)[~(~) - v ( t ) ] + . ( t ) [ , ~ ( ~ )

- u(t)~(t)

=

- u ( t ) ] + [,,(8) - u ( t ) ] [ v ( ~ )

- ~(t)]

set 8 = t + h, multiply through by Q = sgn h, and apply (17) to get (18). Since dudv is well defined for u and v bounded, (18) gives (19). For u continuous du is tag-null. So A u A v .~ 0 for dv weakly archimedean giving (20) f,'om (18).

O Validity of (20) and integrability of udv imply integrability of vdu through integration-by-parts. For example, if u is continuous and v is of bounded variation then vdu is integrable. In particular, if u is continuous then vdu is integrable for every polynomial function v. Equivalently, every continuous u has finite moments f ~ t k d u ( t ) for k = 0,1, .... For u , v sufficiently smoothly differentiable (20) extends to the iterated integration-by-parts fornmla (22). Proposition 26. Let u , v be functions on the 1-cell K both of class C'~ for some n >_ 1. Define the continuous function n

(21)

w :

]E(-1)J~(%(~-j). j=O

Then ( 22 )

dw = udv(") + ( - 1 )~vdu (~).

P r o o f . For 0 ~ j < n Proposition 17 gives du (j) = u(J+l)dt. Thus, since u (j+l) is continuous, du (j) is absolutely integrable, hence archilnedean. Also, v ('~-j) is continuous. Similarly dv = v(1)dt is archimedean and u ('~) is continuous. So by Proposition 25 the classical product rule (20) applies to each term on the right side of (21). Thus dw

= = :

E'~=o(-1)~d(u(~)v ("-j)) ~j"=o(- 1 )Ju(J)dv (~-J) + ~j"=o(- 1)Jv(~-J)du (j) udv(n) + [Ej~=l(-1)Ju(J)v('~-J+l)+ ~y=~(-1)Ju(~+l)v("-~)]dt + ( - 1 )"vdu (~)

94

The quantity in brackets vanishes since for 1 < j < n term j in the first sum negates term j - 1 in the second sum. -Hence (22). C)

Proposition 26 gives a general Taylor formula. P r o p o s i t i o n 27. Let u be a function of class C" on a 1-cell J containing the points p, q. Then (23)

. j u(q) = 2_, ~ t q j=o .7.

- P)J + r .

where (24)

r,~ = fp' (q - t "

Thus

(25)

Ir.I <

Iq-pl"

fs Id"<")l'

P r o o f . Define v ( t ) = (t - q ) " / n ! . By repeated differentiation (26)

v(n-J)(t) = (t -- q ) J / j ! for j = 0 , . . . ,n.

In particular v(")(t) = 1, so dr(") = O. Hence (22) in Prop 26 reduces to d w = ( - 1 ) " v d u ( " ) . Integrating this from p to q we get w(q) - w ( p ) = r , with the implied existence of the integral f ~ ( - 1 ) " v ( t ) d u ( n ) ( t ) in (24) defining r,,. So w ( q ) = w ( p ) + r,,. This is just (23). Indeed w ( q ) = u{q) by (21) since v(~-J)(q) = 0 for 0 < j _< n by (261). w ( p ) is just the sum in (23) by (21),(26). (25), of use for u(") of bounded variation, comes from (24). O

10. D I F F E R E N T I A L F O R M U L A S F O R F U N C T I O N A L L Y REL A T E D V A R I A B L E S . For z bounded (19) ~ v e s d ( z 2) = 2 z d z + Q ( d x ) 2 This is a special case, u(z) = z 2, of our next result. For brevity we introduce the notation / k z ( I , t ) = Q ( I , t ) ~ z ( I ) and d z = Qdx. Since Q2 = 1 we have Az = Q ~ z and dz = Q d z . Also dz = [~z]. P r o p o s i t i o n 28. Let z be a bounded flmction on K and y = u ( z ) where u is a polynomial of degree n > 0. Then

(27)

dy = ~ j=l

u(~)(z)

(dz) j

P r o o f . The Taylor identity + h) -

=

j=~

J!

95

with z -- x(t) and h = / ~ z ( I , f) gives

.(~)(x)

=

(h~(x, t))J

,/=1

This gives (27) since $ is bounded, making (dz) j well defined.

©

With suitable restrictions on z we can get (27) for functions u more general than polynomials. Our next result does this. P r o p o s i t i o n 29. Let x be a continuous function on K such that (dx)" is weakly archimedean for some n > 1. Let y = u(x) where u is a function on z ( K ) of class C" with u(") of bounded variation. Then (27) holds. P r o o f . Given I with endpoints t, t + h let p = z ( t ) , q = z(t + h). Then h z ( I , t) = q - p and b y ( I , t) = u(q) - u(p). So Proposition 27 gives (28)

Ihv(I,z)- ~

,/=I

.(J)(p) 5! (h~,(I,*))¢t < IA~(/)I"f3 Id,~(")t

where J is the interval joining q,p. As I ~ t, h --, 0 and q ~ p by continuity of x. That is, J -* p. Therefore f j tdu(") l -* 0 since the variation of a continuous function is continuous. So (28) gives

2~v - ~ u(J)(x)(h~)~ = o(IAx(I)l").

~=t J!

This gives (27) by Proposition 14 since (dz)" is weakly archimedean.

11. G E T T I N G d y / d x = z F R O M generalizes the classical result that

dv = z d s .

O

Our final proposition

+/." z(t)dt = z(x) for z continuous. It is strong enough to yield L'H6pital's rule. The condition (i) is trivial for z strictly monotone on each side of c. P r o p o s i t i o n 30. Let z, y, z be functions on K and c a point in K such that: (i) x is of bounded variation and its variation function v has finite derivates in the narrow sense with respect to x at c, (ii) z is continuous at c, and (iii) dy = zdx. Then d y / d x exists in the narrow sense at c and equals z(c). P r o o f . By (i) there exists M < co such that (29)

Av(I) <

MIA~(I)I

96

ultimately as I ~ c in K. Let p = z(c). Given e > 0 choose * > 0 such that for all I with (I, c) *-fine (29) holds and by (ii)

(30)

Iz - p [ < e / M on I.

For such I, ](Ay-pAx)(/)]

= [fldy-pdzl < (~/M).fidv < ~lAz(I)[

= [fz(z-p)dm I = (~/M)Av(I)

by(30) and (29). That is, ] A y ( I ) / A z ( I ) - p[ < e.

O

REFERENCES

1. S. Leader, A concept of differential based on variational equivalence under generalized Rienmnn integration, Real Analysis Exchange 12 (1986-87), 144-175. 2. - - , What is a differential? A new answer from the generalized Riem a n n integral, Amer. Math. Monthly 93 (1986), 348-356. 3. B.S. Thomson, Some remarks on differential equivalence, Real Analysis Exchange 12 (1986-87), 294-312. 4. E.C. Titchmarsh, Tile Theory of Functions, London (1950). Rutgers University New Brunswick New Jersey 08903 U.S.A.