広義のDenjoy積分に同値な一般化されたPerron積分
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(2) Vol. 14, No. 1 Journal of Hokkaido Gakugei University (Section II A) Aug. 1963. The generalized Perron integral equivalent to the general Denjoy integral Y6to KUBOTA The Department of Mathematics, Hakodate Branch, Hokkaido Gakugei University. ^ftffll^A : J^»® Denjoy ^^-^Ifflfa^-^^?^-/; Perron ^^ § 1. Introduction. S. Saks CD showed without using the transfinite induction that the ordinary Perron integral is equivalent to the restricted Denjoy integral. J. Ridder C23 has given the Perron scale of integration equivalent to the Denjoy integral in the wide sense (D-integral). The aim of this paper is to define the generalized Perron integral which is equivalent to the general Denjoy integral in more natural style than that of Ridder C23 and to develope tKe theory without using the transfinite induction. In § 2 we shall define the P-integral by the method of Ridder C33 and prove its fundamental properties. The equivalence between the P-integral and the D-integral will be established in §3. The author expresses his sense of gratitude to Dr. G. Sunouchi and Dr. T. Tsuchikura fo their suggestions and criticisms.. § 2. Generalized Perron integral. Definition 1. The finite function f (x) is said to be AC below CAC above^l on a set E if to each positive number s, there corresponds a number 8 such that. S{f(b0-f(ak)}>-s,. CS{f(bk)-f(aO}<e; for all finite non-overlapping sequence of intervals {(ap, bk)} with end points on E and such. that 2(bk-ak)<<?.. If f(x) is both AC below and AC above on E, then we say that f(x) is AC on E. Definition 2. If the set E is the sum of a countable number of sets Ejr on each of which. f(x) is AC below CAC above^, then f(x) is termed ACG below CACG above'] on E. If f(x) is both ACG below and ACG above on E, we say that f(x) is ACG on E. Lemma 1. (Ridder C221). A function ivhich is AC 6e/ow CAC above} on an interval Ca, b') is differentidble almost everywhere. Lemma 2. Let f(x) 6e a function ivhich is AC beloiu on a closed set E and let {(air, bjr)} (k=l, 2,......) 6e <^e system of complementary intervals of E with respect to the smallest. ( 1).
(3) The generalized Perron integral equivalent to the general Denjoy integral. closed interval Ca, b^l luhich contains E. 7"/ we ^>2(<. J(x)=f(x) (xeE) =f(aO+^^Cf(bk)-f(a,)) (xe(a,, b,)) ik—. then the function f(x) is also AC below on Ca, b^. Proof. Since f (x) is AC below on E, for a given e>0 we can find 8>0 such that. S{f(d,)-f(c,)}>—|for all finite non-overlapping sequence of intervals {(ck, dk)} with end points on E and S(dk-Ck)<3. Let N be a natural number such that S(bk-a0<<5. k-N+l. Since f(x) is linear on Caiii bk3 and is therefore AC below on Cafc, bk3, for a given s/2k+l(k =1, 2,. ..... N) we can find 8\c such that. S{f(d0-f(cv)}>-^r v. for all non-overlapping sequence of intervals {(d,, dv)} contained in Cak, bn^ and S.;(d,,-c,,)<5k. If we put 5o=min (3, 5i, 8^, . . . . . , 8s) then 5g is a required 8, that is, for a given e>0 we can prove that for all non-overlapping intervals {(cv, dn)} contained in Ca, b~) and S(dv-cv)<3o,. we have. S{f(dv)-f(cv)}>-e. Definition 3. A function U(x) CL(XD is termed upper Clower^l function of a measurable. f(x) in Ca, b^, provided that. (i) U(a)=0 CL (a) =01 (ii) U (x) (;L (x) 3 is continuous on Ca, b^],. (iii) U (x) CL (x) ^ is ACG below CACG above:) on Ca, bl (iv) AD U(x) ^ f(x) a.e.'CAD L(x) ^ f (x) a.e.^. Definition 4. If f(x) has upper and lower functions in Ca,b^ and. inf U(b)=sup L(b) then f (x) is termed P-integrable on Ca, b]. The common value of the two bounds is called the definite P-integral and is denoted by. (P).)':f(0dt. a. Lemma 3 (Ridcler C33). If f(x) t's AC below on Ca, b^l fl%rf D f(x) ^ 0 a.e. then f (x) is non-decr easing on Ca, b^).. Theorem 1. For any upper function U and loiuer function L, the difference U (x)-L (x) is non-decr -easing on Ca, b~].. Proof. Let CD (x)= U (x)— L (x) and let E be the set of points of Ca, b~] throughout no neighbourhood of which (0 (x) is non-decreasing. Then E is a closed set.. ( 2).
(4) Yoto Kubota Suppose that ffk<cik<0k<bk where (an, bk) is any complementary interval of E. Then it can be shown by the method of repeated bisection that 10 (x) is non-decreasing on C^k, ^~). Since co (x) is continuous on Ca, b^, it follows that <u(x) is also non-decreasing on Cak, bk3. The set E can therefore contain no isolated points, and must be perfect or null. Suppose that E is not null. By Definition 2, Ca, b^] is expressible as the sum of a countable number of sets Eic on each of which U (x) is AC below and L (x) is AC above. It follows from Baire's thorem that there exists a portion a of E on which some Eic is everywhere dense.. Since U (x) CL(x)^l is AC below CAC above;] on Ek and is also so on a. Therefore (D (x) is AC below on a. Let [^c, d~) be the smallest closed interval containing a. We define the function <0y (x) on Cc, cT) as follows, a)o.(x)=ffl(x) (xetf) = linear (x is in any complementary interval of a}. Then (Oy (x) is AC below on Cc, cQ by Lemma 2. Similarly we can define Uo.(x) CL<r(x)^) on Cc, d) which is AC below CAC above'] on Cc, d~], and we have ^(x)=U.(x)-L,(x). Since a (x) is non-decreasing on Can, bk^ and i»<r (x) = co (x), it holds that for any inner point of complementary interval of Dtt)<r(x)^0. Next we shall show that D a><r (x) sS 0 almost everywhere on a. Since Uo. (x) is AC below on Cc, d.'), it follows from Lemma 1 that D (»(r(x) is finite almost everywhere on a. Hence D U<r(x) = D; Uo-(x) = Df U(x) a.e. The set a is measurable and therefore almost all points of <s are points of density of a. Consequently we have. D U,(x)^f(x) a.e. tf: Similary we have D L,,(x)^f(x) a.e. a.. It follows that D <y,,(x)=D Uo.(x)-D L^(x)^0 a.e. a. Thus for almost all points of D a)o.(x)^0. Since a>o-(x) is AC below on Cc> d'), it follows from Lemma 3 that a);r(x) is non-decreasing on Cc, d^, and co (x) is so on a, which is a contradiction for <s contains points of E. Thus we have proved that E is null. We can devolepe the Perron-scale of integration as usual using Theorem 1, and prove the following theorems.. Theorem 2. If f(x) ?'s P-integrable on Ca, b^ </2ew f(x) is also so on Ca, x^ (a<x<b). Proof. For a given e>0, there exists an upper function U and a lower function L such that. U(b)-L(b)<e. By Theorem 1 we have. U(x)-L(x)<s (a<x<b),. (3).
(5) The generalized Perron integral equivalent to the general Denjoy integral. which proves P-integrability of f (x) on Ca> x^). Definition 5. We define the indefinite P-integral as. F(x)=(P)J;f(t)dt for a Sa x 5a b.. Theorem 3. For any upper function U (x) (^loiver function L (x) ~), the difference U (x)—F(x) CF(x)—L(x)] is non-decr easing on Ca, b^). Proof. We prove the case for U(x)—F(x), the other case being similar. Let a^Sxi <X2^b. The function U(x)—U(xi) is an upper function of f (x) on Cxi, Xg^. It follows that. (P)J'2f(t) dt^U(xa)-U(xi) that is,. F(x^)-F(xi)^U(x2)-U(xi) which proves the theorem. »x. Theorem 4. The indefinite integral F(x)=2(P) | f(t) dt ;s continuous on Ca, b~S. a. Proof. It follows from Theorem 1 and Theorem 2 that for a given number n(n=l, 2, .. . .) there exists an upper function Un (x) such that. 0<Un(x)-F(x)<l/n (a^x^b). Hence Un (x) uniformly converges to F (x) on Ca, b), and F (x) is continuous on ^a, b~).. Theorem 5. The function F (x) is approximately differentiable at almost all points of Ca, b~) and AD F(x)=f(x) a.e. Proof. For a given e > 0, we can find an upper function U (x) such that. U(x)-F(x)<82. We put. R(x)=U(x)-F(x). This function is non-decreasing by Theorem 3. It has a finite derivative R' (x) almost everywhere and. (L) J"R (t) dt ^ R (b)- R (a) = U (b)- F (b) < s\ We set. A (e) = {x: AD F (x) < f (x)- e}, A={x:AD F(x)<f(x)}, and denote by M the set of points where R'(x) exists and is finite. Then we have | A | =0 and | M|=b—a. If x e A (e)—A then we have. AD F (x) < f (x)- e ^ AD U (x)- e, from which it follows that. AD U(x)-AD F(x)>e. If x 6 M, then. R'(x)=AD U(x)-AD F(x). Hence putting. (4).
(6) Yoto Kubota. B(e)={x:R/(x)>e} x e (A (e)-A) • M implies x e B(s). But. e I B(e)|^(L)(R'(x) dx^(L)) R'(x) dx < e", B(e). */. a. from which it follows that. |B(e)| <e and hence. |A(e)| <e. Since. {x: AD F(x)<f(x)}=S^:AD F(x)<f(x)-^} we have. |^:ADF(x)<f(x)}|^S^=e,. Consequently we obtain that. ADF(x)^f(x) a.e. Introducing a lower function, it can be proved analogously that. ADF(x)^f(x) a.e. We have thus proved that. AD F(x)=f(x) a.e. § 3. Equivalence between P-integral and D-integral We shall prove by the method of Ridder C3^) that P-integral is equivalent to D-integral. Theorem 6. The P-integral is equivalent to the ^-integral. Proof. Suppose that f(x) is D-integrable on Ca> b^]. We put. F(x)=(D)j"~f(t) dt. a. Then it follows from a descriptive definition of (D)-integral that F (x) is continuous, ACG on. Ca, b^) and AD F(x)=f(x) a.e. Hence the function F(x) is an upper function and at the same time a lower function of f (x) in Ca, b^). Thus f (x) is P-integrable and. (P)|~f(t)dt=F(b)=(D) |~f(t) dt. a. •/. a. Next we shall show that the D-integral includes the P-integral. Suppose that f(x) is P-integrable on Ca, b^ and that. F(x)=(P)J"f(t) dt. Then F (x) is continuous and. ADF(x)=f(x) a.e. by Theorem 4 and Theorem 5. We must show that F(x) is ACG on Ca, b~]. Since f(x) is P-integrable, there exists a sequence of upper functions {Up (x) } and a sequence of lower. functions {Lk(x)} such that (1) lim Uk(x)=lim Lk(x)=F(x).. (5).
(7) The generalized Perron integral equivalent to the general Denjoy integral. Then Ca, b^I is expressible as the sum of a countable number of sets EI( such that any Uk is AC below on any Eii and at the same time any U is AC above on any Ek. It is sufficient to prove that F(x) is AC on Ek. For this purpose we shall show that F(x) is both AC below and AC above on Ek. Suppose that F(x) is not AC below on E]{. Then there exists a e>0 and a finite, nonoverlapping intervals {(ay, bv)} with end points on Eii such that for any small 8. S(bv-av) <8 implies. (2) S{F(bv)-F(a.)}SS-e. Since we can find a natural number p such that. Up(x)-F(x)<l/2-e, and Up(x)—F(x) is non-decreasing on Ca, b^l by Theorem, 3, we have. (3) S<Up (bv)-Up (av) }-S{F (bv)-F (av)} = SC<Up (bv)-F (by) }- {Up (av)-F (a.) }D ^Up(b)-F (b) <-|-e. It follows from (2) and (3) that. S{Up (bv)-Up (av) } < S{F (bv)-F (av) }+-|-e. --y. This contradicts the fact that U (x) is AC below on Ep, and therefore F (x) is AC below on Ei... Similary we can prove that F (x) is AC above on E]{. Thus F (x) is AC on Ek and also ACG on Ca> b^. This completes the proof.. References. CO S. Sales, Theory of the integral (Warsaw) 1937. C2D J. Ridder, Ueber den Perronschen IntegralbegriflE und seine Beziehung zu den R-, L -und D - Integralen, Math. Zeit., (1931), 234-269. C3] J. Ridder, Ueber appoximativ stetiege Denjoy-Integrale, Fund. Math., 21 (1933), 1-10,. ( 6- ).
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