ROCZNIKI POLSKIEGO TOWA RZY STW A MA TE MA T YC ZN EG O Serin I: PRACE MA TEM AT YCZ NE XXVII (1988)
M. K. Bo s e and S. Ba i d y a (Darjeeling, West Bengal, India)
Proximal Perron-Stieltjes integral
Abstract. In the paper we introduce the definition o f (a>)sparseness o f a set at a point with respect to a non-decreasing function w and then, using this concept, we introduce the definition o f (ry)proximal continuity and (ofiproximal derivative. Then an integral o f Perron type is defined and some o f its properties including an integration by parts formula are studied.
1. Introduction. Sarkhel and De [9 ] have introduced the concept of sparseness o f a set at a point. With the help o f this, they have introduced proximal limit, proximal continuity and proximal dérivâtes. In this paper we define the (co)sparseness o f a set at a point with respect to a non-decreasing function со. W e establish some results on the (co)sparseness of a set at a point.
Next we define (co)proximal limit, (co)proximal continuity and (co)proximal dérivâtes. Then we introduce the (co)proximal Perron-Stieltjes integral, the (PPS)-integral. W e study different properties of the (PPS)-integral. In particu
lar, we establish an integration by parts formula for this integral.
2. Preliminaries. Let со be a finite non-decreasing function defined on R, the set of real numbers. Let S denote the set o f points, of continuity o f со and D = R — S. Then D is countable. Let S0 denote the union o f pairwise disjoint open intervals (a,-, bt) on each of which со is constant. Let
= {л 19 b u a2, b 2, . . . } , S2 = S nS 1 and S3 = S — (S0u S 2). Further, let S2 and S2 denote those points o f S2 which are respectively the left-end points and the right-end points o f (a,-, b t) . Let [a, Ь]<=Я be a finite closed interval.
Let 'V' [2 ] denote the class of real-valued functions / such that / is defined on S and, for any point x0eD, f ( x ) tends to finite limits as x tends to x 0 + or x0— over the points o f S. W e denote these limits by / (x 0 + ) and f (x 0 — ) respectively. For the function we write
, + , , _ ] / ( * ) if x e S , f / ( x ) if x e S , f W l / ( x + ) if x e D , f | / ( x - ) if x e D .
A M S subject classification (1980): Primary 26 A 39.
K e y w o r d s an d p h r a s e s : (co)density, (oj)sparse, (a>)proximal limit, ((o)proximal continuity, (to)proximal derivative, (P P S )-integral, (A P S )-integral, (PS)-integral, (D *S)-integral.
We denote by ;//0 the class of functions / o f class тГ which are continuous at each point of S relative to S. Further, 1 ш ( c# 1)) denotes the class of functions / o f class V ( a//0) for which f ( x ) = i ( / ( x + ) + f ( x — )) for x e D .
Following the definitions of Jeffery [6], the outer (co)measure and the (co)measure of a set E are denoted by co*(E) and \E\W, respectively. The complement o f a set E is denoted by E'.
W e require the following definitions and theorems:
Definition 2.1 (cf. [3], [6]). Let / e У . For x, £ eS (x Ф £), the function ф(х, Ç) is defined by
l / ( £ ) - / ( * )
<A(x, £) = < co(£)-co(x)’
( o,
co(£)-co(x) Ф0, co(£) —co(x) = 0.
The upper limit o f ф(х, £) as Ç ->x is called the upper (co)derivate of / at x and is denoted by D f a (x). The other (co)derivates Df^ix), D + f w(x), D+ /w(x), D~ /ю(х), D_ /ю(х) and the (co)derivative /^(x) are defined in the usual way.
Definition 2.2 ([8], p. 130). Let AczR and x e R . Let v = [x , x + h] and v° = (x, x + h), h > 0. We write
со* (Ar^v) d(x, h) = I^L
if No, Ф 0,
if |u|w = 0 and Arw° Ф 0 , otherwise.
Then lim supd(x, h) and lim inf<i(x, h) are called, respectively, the upper
h->0+ h ->0 +
right ((o)density and the lower right (co)density of A at x and are denoted by d*{A, x) and dw +( T , x ), respectively. Similarly, d~(A, x), the upper left (co)density and dw- (A, x), the lower left (co)density o f A at x are defined. If all these four (co)densities are equal, then their common value is the (co)density of A at x and is denoted by d(a(A, x).
N o t e 2.1. If x eAc\D, then dw(A, x) = 1 and if x e D - А , then dw{A, x) = 0.
Theorem 2.1 ([8], Theorem 2.1). The (co)density of a set J e R exists and is equal to 0 or 1 (со) a.e. on R.
Theorem 2.2 (cf. [5], Lemma 2.3). Given a set E c= [a, b~\ n 5 3 with \Е\Ы
= 0 and any positive number e, there exists a function p e °UQ such that p is non-decreasing on [a, b] n S and
p ( a - ) = 0, p(b + ) < s , Dp^ix) — + o o for x e E .
3. (m)sparse sets.
Definition 3.1. A set E c R is said to be (afisparse at a point x on the right if, for an arbitrary set F <= R,
d„+ (F, x) = 0 = » 4 + ( £ u F , x ) = 0.
Analogously we define the (co)sparseness of the set E on the left o f the point x.
The class o f sets which are (co)sparse at x on the right is denoted by
£<,>(* + ) and the class o f sets which are (w)sparse at x on the left is denoted by Sw(x — )• A set E is said to be (co)sparse at x if E e S w(x), where Sa (x) = 5ю(х + ) п 5 ы( х - ) .
N o t e 3.1. Let x e D . Then £ e S w(x + ) (and also E e S w{x — )) if and only if x $ E .
Definition 3.2. Let E cz R. A (co)measurable set A => E is said to be a (i(immeasurable cover of E if со* (E n M ) = \A n M\w for every (w)measurable set M.
Lemma 3.1. Let E be a {œ)measurable set and x e S 3 u S T h e n ( F , x) + da+ {E, x) = 1.
P r o o f. For all y > x we have
IE' n [x , у]|ю + 1E n [x , y ] L = |[x, y ] L which implies that
d+(E', x) = 1 -d n + iE , x).
Theorem 3.1. A necessary and sufficient condition for a set E a R to be (a>)sparse at a point x on the right is that for an arbitrary set F <= Я,
da+ (F, x) = 0 and d+ (F, x) < 1
d(a+ { E u F , x) = 0 and < C ( E u F , x ) < 1
P r o o f. Let E be (tu)sparse at the point x on the right. Then (1) d&+(H, x) = 0=>d(O+( H u E , x) = 0
for an arbitrary set H. Let d* (F, x) < 1. W e consider the following cases successively.
C a se (i). x e D . Since d* {F, x) < 1 and E is (w)sparse at x on the right, it follows from Notes 2.1 and 3.1 that x ÿ £ u f whether œ is discontinuous at x on the left, or on the right, or on the both sides. So d*(E u F, x)
C a se (ii). x e S 0 or S2 . Then it follows that x is an isolated point on the right o f both the sets E and F and so o f E и F, which again implies that d£ (E и F, x) = 0 < 1.
C a se (iii). x e S ^ u S ^ • Let A and В be (co)measurable covers of E and F, respectively. Then d£ (B, x) < 1 and so by Lemma 3.1, db}+ (B\ x) > 0.
Now
d(a+ ( E u (A' n jB'), x) = dw+ [A и (A' n B '), x) = dco+ {А и B', x) > 0.
Therefore, by (1),
dw+{A' слB’, x) > 0 and so, by Lemma 3.1,
d£ (A u В, x) < 1 => d£ (E и F, x) < 1.
If further dt0+ {F, x) = 0, then by (1), dto+ {E u F, x) = 0. Therefore the condi
tion is necessary.
T o prove the sufficiency, suppose that for an arbitrary set H c R, if d^+iH, x) = 0 and d£ (Я , x) < 1,
da + ( L u Я , x) = 0 and d £ ( E u H , x) < 1
Suppose for a set F, dw + (F, x) = 0. If d£ (F, x) < 1, then it follows that dw+{ E u F , x) = 0.
Next, let d£(F, x) = 1, which implies that x<£D u S 0 u S J . Therefore x gS3u S2+ . Let A and В be (immeasurable covers of E and F, respectively.
Then dm+ (B, x) = 0 and d£ (В, x) = 1 and so, by Lemma 3.1, doi+ (B\ x) = 0 and d£ (В', x) = 1 Again
(2) d£ (E u ( A n B ), x) = d£ (А и B', x) = 1.
Since da + ( A n B ', x) = 0, from (2), we have
d£ (A’ n B ', x) = 1 =>d(0+ (A u f i , x ) = 0 =>dw+ ( E u F , x) = 0.
This completes the proof o f the theorem.
Theorem 3.2. A necessary and sufficient condition for a set E cz R to be (a))sparse at a point x on the right is that for an arbitrary set F c= R,
d£ (F, x) < 1 => d£ {E u F, x) < 1.
P r o o f. Let E be (&>)sparse at x on the right. Let d£ (F, x) < 1. If dca+ (F, x) = 0, then by the previous theorem, ^ ( £ u F , x ) < l .
N o w let dbi+(F, x) > 0. Therefore x £ Z )u S 0u S J . So x e S 3 u S £ . Let A
and B be (co)measurable covers o f E and F, respectively. Then d* (В , x) < 1 and so d0)+ (B', x) > 0. Therefore we have
(3) C ( £ u ( T n B ' ) , x ) > 0 .
Now
d,.,+ (F, x) > 0 => d0}+ (B, x) > 0 => d* (В', x) < 1 => d* (A' n В', x) 1.
Therefore, applying the previous theorem, it follows from (3) that d(a+ {A’ n B', x) > 0=>d* (A u В, x) < 1 =>d* (E и F, x) < 1.
Hence the condition is necessary.
Conversely, let d* (F, x) < 1 =>d* (E u F, x) < 1 . Proceeding as in the necessary part o f Theorem 3.1, we show that
da+ (F, x) = 0 and d l (F, x) < 1 imply
d0i+( E u F , x ) = 0 and d* (E u F , x) < 1.
Therefore, by the sufficient part o f Theorem 3.1, it follows that E is (co)sparse at x on the right. This completes the proof o f the theorem.
Theorem 3.3. (i) I f E eSw(x + ), then dw+ (F, x) = 0 and d+ (£, x) < 1.
(ii) I f A, B e S (0(x + ), then A и В е 5 ю(х + ) and for any E cr A, Е е S J x + ).
The proof of the theorem is straightforward.
Each of the above results can be extended without difficulty to its left- hand analogue.
4. (co)proximal continuity and (co)proximal derivative.
Definition 4.1 (cf. [7], p. 218; [9], p. 33). Let / e f . We take / (x )
= f ( x — ) for x e D . W e define the right upper (co>)proximal limit of / at any point x e S as the greatest lower bound o f all the numbers r e R such that the set
{y: y e R and f ( y ) > r } e S 0){x + ) and this limit is denoted by (р)+ (/, x, ш).
The right lower (co)proximal limit (p)+ (/, x, со), the left upper (cu)proximal limit (p)~ (/, x, со), the left lower (co)proximal limit (p)_ (/, x, со), the upper (co)proximal limit (p)(f, x, со) and the lower (cu)proximal limit (p)(f, x, со) o f / at x e S are defined in the usual way [9 ]. In each case we shall take / (x ) = f ( x - ) for x e D .
The corresponding (co)approximate limits (cf. [8 ]) are denoted by {ap)+ (/, x, со), etc.
Obviously we have
(ap)+ (/, x, со) < (p)+ (/, x, со) < (p)+ (/, x, со) ^ (ap)+ (/, x, со), x e S . A similar relation holds for the left-hand limits. If
(p) {f, x, со) = (p) {f, x, со) = / (x), x e S, then / is said to be (co)proximally continuous at x.
Definition 4.2. Let / e i and x, £ eS (x Ф £). Let ф(х, Ç) be defined as in Definition 2.1.
The (co)proximal limit ((m)approximate limit) o f ift (x, £) as £ -> x is called the (co)proximal derivative ((co)approximate derivative) of / at x and is denoted by P D ^ w(x) (A D y w(x)). The extreme dérivâtes P D /ш(х), P D /ш(х), P D + f(o(x), P D + /ы(х), P D " / W(x), and P D _ / W(x) are defined in the usual way.
It is easily seen that
A D + f M ) ^ PD+ L ( x ) < P D + / J x ) ^ A D + f w(x), x c S . A similar relation holds for the left-hand dérivâtes.
Lemma 4.1. Let [a, ff\ c- R. Let A and В be (wo sets such that (i) A u B = [a, /Г],
(ii) A n B = 0 , (iii) аеЛ*
(iv) d~(A, x) < 1 for all x e B n S , (v) d* (В ,x) < 1 for all x e A n S ,
(vi)for each point x e A n D there exists a 3 > 0 such that (x, x + <5) <= A and
(vii)for each point x e B r\D there exists a 3 > 0 such that {x — 3, x) <= B.
Then В is empty.
P r o o f. Let со be constant on the interval (c, d) c [a, ft]. From (v) and (vi) it follows that if there is a point ÇeA in [c, d), then [£, d] c A and from (iv) and (vii) it follows that if there is a point tjeB in (c, d], then [c, rç] с B.
From the above fact, the lemma follows in case (a, jS) <=. S0. W e now suppose that (a, j8) f S 0. If possible, suppose В Ф 0 . We write
A 0 = (x: x e A and d* (В, x) = 0}
and
B0= {x : x e B and d~(A, x) = 0}.
By Note 2.1, we have d^iA, x) = 0 for x e B n D . Therefore {A, x) < 1 for
all x e B. Again since d~( A, x) = 0 for all x eB0, by Theorem 2.1, we have IВ — В0\ы = 0. Therefore the (cu)density o f A is 0 (co)a.e. on A'. Thus it follows that A is (cu)measurable. Similarly \A — A0\m = 0 and В is (unmeasurable.
Let xe A r\(S3 и $2 )• Then d * ( B , x ) < 1 and so, by Lemma 3.1, dœ+(A, x) > 0 which implies that dm + {A0, x) > 0. Therefore, x is a limit point of A 0 on the right. N ow let х е Л n (S 0 и SJ). If x e (c, d) a S0, then since \_x, d] c A, it follows that [x , d) c~ A 0. Therefore x is a limit point of A 0 on the right. Again if x e A n D , it follows from (vi) that x is a limit point o f A 0 on the right. Thus each point o f A is a limit point of A0 on the right.
Similarly we can show that each point o f В is a limit point o f B0 on the left.
Now if x 0 eA0— (Sq и Sj ), y0 e B0 — (S0 u S2 ) and x 0 ^ y0, then for any e > 0, there exist points х г eA 0n ( x 0, y 0) — (S0 u SJ) and yl E Bo >’o) — (So u S2 ) such that
(4) and
\A n [f, Уо]L < e | [f, y0] L for all г е ( х ! , у 0)
(5) \ B n [x 0, t]|w < £ | [x 0, r ]L for all t e ( x 0, yi).
In order to verify these, first we suppose that y 0 is a limit point o f A on the left. Also d ~ (A , y o) = 0. Therefore existence of x t follows from the discussion in the first section. Again if y0 is not a limit point of A on the left, then since x 0 is a limit point o f A on the right, we have
x 0 < s = sup {A n ( x 0, y 0)} < Уо•
Since A n(s, y 0] = 0 , it follows that s$A. Therefore s is a limit point of A on the left. From this it follows that s^Sq uSj k jD . Now we take a point x^eAq—(S0U S2) with
(6) IOi, s]L <e|[s, y0]L-
If t e { x 1, s), we have
IA n [t, y0]L = I A n It, s]L + 1A n (s, y0] L
= \ A n [t , s ]L ^ |[f, s]\m
<fi|[s»>'o]L (by (6))
<e|[t. Уо]\ш- Again, if tG[s, y 0), we have
\A n It, Уо]\а> < И n Is, y0] L = 0 < e |[t, y0]\w.
Thus (4) is verified for this x ^ Existence of y! can be shown similarly.
Now, by (iii), o l eA. Therefore, since В Ф 0 , we have a < /?' for some P 'eB. From the discussion in the first section and from the fact that every
point o f Л is a limit point o f A 0 on the right and every point o f В is a limit point o f B0 on the left, it follows that there are points а0е Л 0 — (S0 U S2) and P0£B0- ( S 0 n S 2 ) such that а0 < fio- Starting with /?0 and applying (4) and (5) alternatively, we select successively olu ol2, fi2, ... such that for n
= 1 , 2...
&n ^ Aq, f i n £ Bq, Ot/i— 1 <:' &n < ' f i n ^ f i n — 1 ’
(7) И n [ f , j ^ - J L < - | [t , / ?„-i]L for all tE(cin, f i n- i ) n
and
(8) \B n [а„, г]|ш < -| [a „, f ] L for all te (a „, ft,).
n Now
(9) |[a„+i, ft,+ i ] L = |4 n [a „+ i, ft,+ i ] L + |£n [<*».+1, ft,+ i ] L
^ \A ^ E^n + 1 ’ fin— J L Л 15 O \_0Cn, fin+1Ц|ct>
1 1
<-|[ая+1» n Pn-ilL+-\l<*n, fin+ilc n
(by (7) and (8)),
< 4 I n f i l l
n Let
pt = lim a„ and p2 = lim fi„.
n ~ * CO «-► C O
From (9) it follows that
either pt = p2 or Pl = an < bn = p2 («и, f t e S ),
'(n„, ft,) being some component interval o f S0. If pt = p2 — p (say), then <x„
<' P < fin for и = 1, 2, ... It is clear that p e S 3. By (8) and (9) we get cL+ {A, p) = 0 and ft,_ (B, p) = 0.
Consequently, by Lemma 3.1,
d+ (В, p) = 1 and d~ (A, p) = 1.
Therefore p $A kjB = [a, fi~\ which is a contradiction. In the other case, we notice that
dw_ (B , Pi) = 0 and dw+{A, p2) = 0
implies
d^{A, py) = 1 and <C (B, p2) implies
рхфВ and Р2ФА
implies
Pi eA and p2eB.
But an = e A => [a„, b„] c= Л and b„ = p2eB=> [a„, b„] <= B. Thus, also in this case we arrive at a contradiction. This completes the proof of the lemma.
Lemma 4.2. Let f E ' t " b e s u c h t h a t
(i) *, со) < f ( x ) ^ (p)+ (/, x, со) /or x e [o , b ]n S , (ii) f ~ ( E ) has v o i d i n t e r i o r, w h e r e
E = !x: x e (a , b )n S 3 am/ max ! PD + /ю(х), P D _ / W(x)l < Oj u
u !x: x g (a, 6 ) n S 2+ , P D + / / x ) ^ 0 f l « r f / ( y ) = / ( x )
/or a l l у e a n i n t e r v a l h a v in g x as r i g h t - e n d p o i n t ] u u \x: x e ( a , b ) n S j , PD _ f a ( x ) ^ 0 am/ / (y ) = / ( x )
/or a l l у e a n i n t e r v a l h a v i n g x as le f t - e n d p o i n t ] u u Jx: x e(a, b) n S0 and f ( y ) = f ( x )
f o r a l l у e an i n t e r v a l h a v in g x as an i n t e r i o r p o i n t ] u u lx: x e(a, b ) n D a n d f ( x + ) = f (x —)],
(iii) f { x + ) > f ( x - ) for x e [a , b] n D and
(iv) / is non-decreasing on each of the intervals of [a, b] n (S 0 u S 2). Then f is non-decreasing on [a, b] n S .
P r o o f. We take / (x ) = f ( x - ) for x e [a , b ] n b . First we show that / is non-decreasing on (a, b). Let a, / (a < /5) be any two points o f (я, b). If possible, suppose that / (/ ) < /(a). Since / (£ ) = f ~ ( E ) has void interior, there is a k ф f(E ) such that /(/?) < к < / (a). W e write
Л = {x : x e [a, / ] and / (x) > к ] и
u {x : x e [ a , / ] n (S 3 u S2 ), / (x ) = к and P D + /w( x ) > 0 } u u { x : x e [a , 0 ] n ( S 0 u SJ), / (x ) = к a n d / (y ) > / (x )
for all y e an interval having x as left-end point} u u {x: x e [a , / ] n û , / (x ) = к and / (x + ) > / ( x —)}
and
в = [ « , Я - л .
Let х е Л n (S 3 u S 2+) and f ( x ) > k . Then we have (P)+ (/, со) > к.
This implies that
{у: y e [ ос, Д] and f ( y ) ^ /c}eSw(x + )
and so B e S (a(x + ). If х е Л n (S 3 u S2) and f ( x ) = k, then PD+ /w( x ) > 0 . Again, for all y eB ,
f { y ) - f ( x ) = f { y ) - k * b 0.
Therefore B e S (0{x + ). If x e A n (S 0 u S J ), then since / is non-decreasing on the intervals o f S0 u S 2, all the points in the right neighbourhood o f x are points o f A in both the cases f ( x ) > к and f ( x ) = k. Therefore B e S ^ x - h ).
Hence, by Theorem 3.3, we see that (В, x) < 1 for all x e A n S . If x eA n D, then f (x + ) > к in both the cases / (x) > к and / (x) = k. There
fore there exists a 6 > 0 such that (x, x + <5) c A.
Since k 4 f { E ), PD _ f w{x) > 0 for every x e B n S 3 at which f ( x ) = k.
Because if P D _ / W(x) < 0 for some x gB n S 3 at which f ( x ) = k, then x e E and so к = f ( x ) e f (E). If x e S n S j with / (x) = k, then there is a point £ in any interval having x as left-end point such that f {if) = / (x). Since / is non
decreasing on the intervals o f S0 и S2, f ( y ) = / (x) for all y e (x , f). So P D _ fajix) > 0, since otherwise, к = f (x )e f (E). Similarly, if x e B n ( S 0 u S 2 ) and / (x) = k, then / (y) < / (x) for all у belonging to an interval having x as right-end point. Now, proceeding as in the preceding section, we have d ~ ( A , x ) < 1 for y e B n S . Again if x e B n D and /c = / (x ), then f ( x + )
t^ / (x —) since otherwise к = / (x )e / (£ ). But for all x e [a , b] n D , f ( x + )
^ f ( x - ) . Therefore it follows that В does not contain any point o f D at which f (x) = к. Thus f ( x — ) = f {x) < к for all x e B r D. Therefore there exists a Ô > 0 such that (x — <5, x) a В .
Since / (a ) > к > /(/?), we have o l eA and fieB. This contradicts Lemma 4.1 and consequently / ( a ) ^ /(/?)• Therefore / is non-decreasing on (a, b).
From (i) and (iii), now it follows that / is non-decreasing on [a, b], Hence / is non-decreasing on [a, b] n S .
Theorem 4.1. Let / e f be such that
(i) / is (a))proximally continuous on [я, b] n S,
(ii) PD /ю(х) > — oo on [я, b] n S 3 except for a countable set, (iii) PD /ю(х) ^ 0 (co)a.e. on [a, b] n S 3,
(iv) /(x + ) ^ / (x — ) for all x e [f l, b] nZ) яга/
(v) / is non-decreasing on each of the intervals of [a, b] n ( S 0 u S2).
Then f is non-decreasing on [я, b] n S .
P r o o f. There is a set A c [я, b] n S 3 such that \A\W — 0 and P D f M(x)
^ 0 on [я, b ] n S 3 —+ . Let e > 0 be chosen arbitrarily. By Theorem 2.2 there exists a non-decreasing function petft0 with the following properties:
g(a — ) = 0, p{b + ) < £, Рцы(х) = + o o for x e A .
Let g(x) = / (x ) + ju(x) + e(cu(x) + x). Then g satisfies the conditions o f Lem
ma 4.2 and so g is non-decreasing on [я, b] n S . Since e > 0 is arbitrary, the theorem follows.
5. The (PPS)-integral.
Definition 5.1. Let / be an extended real-valued (co)measurable function on [a, b]. A function M e У is said to be a major function of / on [a, b] if
(i) M is (cu)proximally continuous on [a, b] n S, (ii) M - (a) = 0,
(iii) P D (x) > — oo on [a, b] n S3 except for a countable set, (iv) P D M w(x) ^ /(x) (eu)a.e. on [я, b ] n S 3,
(v) M (x + ) —M (x —) ^ / (x)(cu(x + ) — со(x — )) for all x e [a , b ] n b and (vi) M is non-decreasing on each of the intervals of [a, b ] n ( S 0 u S 2).
A minor function m e i ^ o f / on [a, b] is defined analogously.
By Theorem 4.1, we can prove that for any major function M and any minor function m o f f the function M — m is non-decreasing on [я, b ]n S . If / has major functions and minor functions on [я, b] and inf M + (b)
M
= supm+ (b), then / is termed proximal Perron-Stieltjes integrable or, in
m
short, (PPS)-integrable with respect to the function со. The common value of the two bounds is called the definite (PPS)-integral o f / on [я, b] and is denoted by
(PPS) j/f/o).
a
If we take w(x) =* x, then we call this integral the proximal Perron
b
integral of / on [я, b] and denote it by (P P ) j f dx.
a
If in the definition of the major function and the minor function the (cu)proximal continuity is replaced by the (^approxim ate continuity and the (cD)proximal dérivâtes by the (co)approximate dérivâtes, then the correspond
ing integral is called the approximate Perron-Stieltjes integral or, in short, the (APS)-integral (cf. [8]).
N o t e 5.1. It is clear from the definition that the (PPS)-integral is a generalization o f the (APS)-integral and so of the (PS)-integral [5]. Again
since the (PS)-integral is a generalization o f the (LS)-integral [6], the (PPS)- integral is a generalization o f the (LS)-integral.
Example 6.1 o f [9 ] is not (APS)-integrable with respect to the function co(x) = x, but it is (PPS)-integrable with respect to the function co(x) = x.
Therefore it follows that the (PPS)-integral is substantially more general than the (APS)-integral.
Standard arguments show that the (PPS)-integral has the following properties:
Theorem 5.1. Let f be a finite function on [a, b] and let F e i ' r . I f P D F <0(x) = / (x ) for all x e [f l, b~] n S — S0, F is constant on the intervals of S0kjS2 and if F ( x + ) — F (x — ) = f (x )(m (x + ) — m(x — )) for all x e [a , b] n D , then f is (P PS)-integrable on [a, b] and
ь
(PPS )$fdcD = F + ( b ) - F - ( a ) .
a
Theorem 5.2. I f a function f is (PPS)-integrable on [a, b] and if /(x)
— g(x) (m)a.e. on [a, b], then g is (PPS)-integrable on [a, b] and
ь b
(PPS ) jgdm = (PPS) \ f dm.
a a \
Theorem 5.3. Let a < a < b. Then a function f is (PPS)-integrable on [a, b] if and only if it is (PPS)-integrable on both [a, a] and [a, b]. Also when f is (PPS)-integrable on [a, b], then
b a b
(PPS) J / dco = (PPS) j f dm + (PPS ) J / dm- f (a) (со (a + ) - m (a - )).
Theorem 5.4. I f f is (P P S)-integrable on [a, b], then f is finite (m)a.e. on [a, b].
Theorem 5.5. I f f and g are two functions which are (PPS)-integrable on [a, b] and if a, f e R , then a f + fig is (P P S)-integrable on [a, b] and
ь ь b
(PPS ) f (a/ + j8g) dm = a (PPS ) J/ dm + 0 (PPS) \gdm.
a a a
Definition 5.2. Let / be (PPS)-integrable on [a, b]. W e define the function F by
(PPS) f f dm
a
F (x ) => F (b)
f(a )(m (a + ) - m ( a - ) )
0
if a < x ^ b, if x > b, if x = a, if x < a.
Then F is called the indefinite (PPS)-integral o f / on [a, b].
Theorem 5.6. I f f is(PPSfintegrable on [a, ft], then its indefinite (PPS)- integral F e i Specifically, F (x + ) = F(x) and F ( x - ) = F(x) — f (x)(a>(x + ) —
- m ( x - ) ) if x e [a , ft] n D .
P r o o f. From the definition of F, we have F(a — ) = 0 = F(a) —
—f(a)(a>(a + ) — aj(a — )) and F(b + ) = F ( b ) . Let г > 0 be chosen arbitrarily.
Then there exist a major function M and a minor function m of / on [a, ft]
such that
(10) M (ft + ) — j£ < F(b) < m(b + ) + j£.
Suppose that a eD . W e have
m(a + ) ^ f (a)(co(a + ) — a)(a — )) ^ M (a + ).
Again
m(a + ) — e < m (x) ^ F (x) ^ M (x) < M (a + ) + e
for all x e S sufficiently close to a from the right. Thus for those x
| F (x )- / (a )(w (a + )- c o (a - ))| < 2s and so
F(a + ) = f(a)(co(a + ) - c o { a - ) ) = F(a).
Now suppose that b e D . W e have
(11) m (ft + ) — m (ft — ) ^ / (ft) (cu (ft + ) — со (ft — )) ^ M (ft + ) — M (ft — ).
Also
(12) m(b — ) — |e < m(x) ^ F (x ) ^ M ( x ) < M (b — ) + j£
for all x e S sufficiently close to ft from the left. From (10), (11), and (12) we get
F(b) — e < F (x ) + f (ft)(co(ft + ) — cu(ft — )) < F(ft) + e for all x e S sufficiently close to ft from the left. Therefore
F ( f t - ) = F(b) — f (ft) (co(ft + ) —co(ft—)).
Again let ote(a, ft) n D . Let F 1 and F 2 be the indefinite integrals o f / on [a, a] and [a, ft], respectively. Then F (x ) = F 1(x) for x ^ a and F (x ) — F (a) + F 2 (x) — / (a) (a> (a + ) — a>(a —)) for x > a . Now
F ( a - ) = F j ( a - ) = F (a )- / (a )(c o (a + ) - c o ( a - ) ) and
F (a + ) = F (a ) + F 2(a + )- / (a )(c o (a + )- t t > ( a - ) ) = F(a ).
This completes the proof.
We can prove the following theorems in the usual way (cf. [5 ] and [8]).
Theorem 5.7. Let F be the indefinite (PPS)-integral of f on [а, b]. Then for a major function M and a minor function m of f on [a, b], the functions
M — F and F — m are non-decreasing on [a, b ]n S .
Theorem 5.8. I f F is the indefinite (P PS)-integral of f on [a, b], then P D F ^ x ) = f ( x ) (ofia.e. on [a, b] n S .
Theorem 5.9. The indefinite (PPS)-integral of f on [a, b] is(wjproximally continuous on [a, b ^ n S .
Theorem 5.10. I f f is non-negative and (P PS)-integrable on [a, b], then f is (LS)-summable on [a, b] and
ь b
( L S ) f / A » = (PPS)J/da>.
a a
With the help o f the above theorem, we can prove the following Theorem 5.11. Let {/„} be a sequence of functions converging pointwise to f on [a, b]. I f for all xe\_a, b] and for n = 1, 2, g(x) ^ f„(x) < h(x), where g, f„ and h are all (P P S)-integrable on [a, b], then f is (PPS)-integrable on [a, b] and
ь b
lim (P P S )\fndco = (P P S )\ f dm.
6. Integration by parts formula. T o prove an integration by parts formula for the (PPS)-integral we need an integration by parts formula for the (D*S)-integral [1 ] which is equivalent to the (PS)-integral [5].
Theorem 6.1. Let f be (D*S)-integrable on the closed interval [a, b], let F be its indefinite (D *S)-integral and let g eû?/0 be AC-co on [a, b]. I f F and g belong to then fg is (D *S)-integrable on [a, b] and
(D * S )ffg d m = - ( L S ) f F.^doj,
a a
where
IFgfa- = F (b + )g (b + ) - F ( a - ) g ( a ~ ) .
The proof o f the above theorem is similar to that for the (DS)-integral ([2 ], p. 44).
N o w we give an example to show that the indefinite (PPS)-integral is not necessarily (PPS)-integrable.
00
W e take co(x) = x and take the set E = (J (an, b„) of Example 3.1 ([9 ],
n = 1
p. 30). The set E has the following properties: E с: [0, 1] and E e S (0).
W e define the function F on [0, 1] in the following way:
я
F (x ) =-
bn- a n я
bn ~ a n 0
sin2 я
sin2 я
x ~ a „
b„ — a„
b „~ x b„ — a„
, . ^ a n + bn
lor an ^ x ^ — - — , n — 1 ,2 ,...,
„ an + bn
for — -— ^ x ^ b n, n = 1 ,2 ,..., for x e [0 , 1] — E.
The function F is proximally continuous at x = 0 and is continuous at other points. It is obvious that F has finite derivative everywhere except at the point x = 0, where it is proximally differentiable. Therefore F is the indefinite (PP)-integral o f / (x ) = P D F (x ). Since the function F is non
negative and
l
(L ) J F dx = oo, о
it follows that F is not (PP)-integrable.
In the integration by parts formula for the (PPS)-integral, we shall suppose that the indefinite (PPS)-integral is (D*S)-integrable.
Theorem 6.2. Let f be (PPS)-integrable and let ge be AC-w on [a, b ] . Let G be the indefinite (LS)-integral of g and G e # 1’ . I f the indefinite (PPS)-integral F of f belongs to iT41) and is (D*S)-integrable, then f G is (PPS)-integrable on [a, b~\ and
ь b
(PPS )\fGdco = l F G ] bat - ( D * S ) l F g d c o .
a a
P r o o f. By Theorem 3.3 of [4 ] g can be expressed as the difference of two non-negative functions g x and g2 belonging to which are also AC- co. It then follows that G is the difference of two non-negative functions Gx and G2. Therefore if the theorem is true for the pairs g lt G x and g2, G2, it will also be true for the pair g, G. Therefore we suppose that g and G are both non-negative. First we prove that f G + Fg is (PPS)-integrable and that
(PPS)f(/G + f 9)«fa) = [FG]‘ i.
a
Let M e f 11* be a major function o f /. W e consider the function H
= MG. Then H e i^ and is (a>)proximally continuous on [a, b] n S and P D H,. > G PD M., + M P D G„
on [a, b ] n S 3. Again since g e ^ \ it follows that Сы = д on S3. It thus follows that
P D H , . ^ G P D M , . + Mfl
