On some classes of finitely additive set functions

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE X I (1968)

ANNALES SOCIETATIS MATIIEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X I (1968)

W. O

(Poznań)

On some classes of finitely additive set functions

1. Throughout th is paper we shall denote a collection (a space) of elements by E and an algebra of sets in E by S. The symbol {t} means one-element subset of E consisting of the element t. Fo r a given sequence of sets from E en \ denotes that en c en+1 for n — 1 , 2 , . . . ; en \ e denotes that en t and e = (J en. The symbols en \ or en\ e w ill be used to denote a sequence w ith the property en

en+1 for n — 1 , 2 , . . . or w ith the property en \ , e — f') en , respectively. The letters ?](:), A(:) w ill always stand for extended-valued set functions. A set function rj(:) w ill be called a subadditive measure on S if the following conditions are fu lfille d :

1) rj(0) = 0 where 0 denotes the empty set, 2) v (ex) < rj(e2) if ex a e2,

3) г] (ex w e2) < г}(е1)-\-г}(е2) where ex e2 = 0 . I f instead the condition 3) the stronger condition

3') rj (ex w e2 w ... w en w ...) ^ rj (ex) + rj (e2) + ... + rj (en) + . . . , pro­

vided that ex w e2 w ... w en w ... c <f,

is fu lfille d , rj(:) is said to be cr-subadditive on S. The term “ an additive (u-additive) measure rj(\) on «?” w ill be used in the customary sense, i.e.

it refers to the case when the sign < in 3) (in 3') is replaced by = .

Fo r a given algebra & and a finite subadditive measure r\(\) on &

an algebra i n j (called, the Jordan algebra generated by i and rj(:)) can be defined by the requirement that e e E belongs to if and only if for any e > 0 there e xist two sets ex, e2e ё such that ег c e <= e2, rj(e2 — e1)

< e. Setting X(e) = inf?](a), where the infim um is taken w ith respect to a ll a belonging to $ and containing e, we define a set function A(:) on S j . I t is a subadditive measure on S'} (called the subadditive Jordan measure generated by ??(:)) and represents an extension of the measure rj(:) from S to S } [6].

In what follows X always denotes a real or complex Banach space, and x(\) a set function on an algebra S w ith the values in X . I f x ( e x w e2)

— #(ei) + ^(e2) whenever ex, e2e S, ex e2 = 0 , %(:) is said to be additive

318 W . O r l i c z

on ё ; it is called в -additive on ё if

x ( e 1 < j e2 w ••• •••) — ^ (^ i) + ж(е2) + я?(еи) j

provided that en e ё for П = 1, 2, U еп е ё .

The variation (called “ semi-variation” in the terminology of the book [4]) v(e, x) of x{:) on e (with respect to ё) is defined by

v{e, x) = sup||A1tf(e1) + -.. + 4aKew)||,

where ее ё, and the supremum is taken w ith respect to a ll d isjoint ei in ё, contained in e, and a ll scalars A*? \h\ < 1 (real or complex according as 1 is a real or a complex space). The set function x(:) is of bounded variation on a set e if v (e, x) < oo. Let ns assume that besides the set function x(:) a finite subadditive measure r\{:) is defined on ё. I f for any en in ё , r){en) 0 as n -> oo im plies x(en) -> 0 as n -> oo, then x{:) is called absolutely continuous on ё w ith respect to y(:) (shortly: a.c. with respect to у (:))j it is weakly absolutely continuous on ё with respect to y(:) (w.a.c. w ith respect to ??(:)) if en \ , ene ё , y{en) -> 0 as n oo im ply x( en) ->■ 0 as n -> oo. As it is easily seen, if x(:) is additive and of bounded variation on E, then it s variation v ( : , x ) is a finite subadditive measure on ё and x{:) is a.c. w ith respect to v(:, x). I t can be shown that if x(:) is additive on ё and a.c. w ith respect to a bounded, additive measure rj(:) on ё , then x(:) is bounded on «?, or equivalently, of bounded variation on E (cf. for example [5], where a special case of th is theorem is considered).

Moreover, it can be shown that for a real valued x{\) the weak absolute continuity of x(:) on ё w ith respect to y(:) implies its absolute continuity w ith respect to y{:), provided that x(:) is of bounded variation on E.

The same s t ill holds if X is a fin ite dimensional space, but not in the general case when X is an arbitrary Banach space [6]. Fu rth e r, whenever we speak on the absolute continuity, or on the weak absolute continuity of a subadditive measure with respect to y{:), the meaning of these notions is to be understood analogously to that given above for a set function x{:).

1 .1 . In th is paper we shall deal w ith the convergence of sequences of set functions of a special type which are defined on the algebra S j generated by an algebra ё and a finite, subadditive measure r\{\) on it.

As an application of the general Theorem I , a generalization of a theorem by G. Pólya [7] w ill be given. Given a system of n elements

x x, x 2, .. . , x n where X i e X for i — 1 , 2 , . . . and a system of n distinct elements (points)

where Це Е fo r i — 1, 2, ...

let us define the set function x(:) as

( "f") x ( e ) — ffiiXe ( t i ) X 2 %e ( I2) 4 “ • * • ~t“ Xe ( I n ) ?

Classes of fin itely additive set functions 319

for ее S’}-, Xe{’) means the characteristic function of the set e. Obviously, x(\) is cr-additive on S } . Le t us assume that the one-element subsets of E {tt} , {t2} , . . . , {tn} belong to S } . Under th is assumption v ( E , x )

= s u p а>1 + А2®8+ . . . + Япжп| | (variation w ith respect to S}) , where the supremum is taken w ith respect to a ll scalars A* for which |А*| < 1.

I f X is the space of real or complex numbers, then v ( E , x) = \xx\-\- \x2\-\- + . . . + |a?n|; we obtain the variation v ( e , x ) taking sup^A*#*), where

a

I Ail < 1 , and the summation is extended over a ll indices i for which Uee.

Le t the system of elements

(o) Xi , x 2 , . . . , Хцп) where x f e X , and the system of distinct points

(oo) h J

) • • •

lk(n)

be given for n — 1 , 2 , . . . and let us denote by x n(:) the set function associated with these systems by the definition ( + ). Let vn (e) = v(e, x n) for в e and let

y{e) = limsup®M(e) if ее S'}.

П—>0O

Evidently, y{:) is a subadditive measure on S } .

1.2. Using the notions and notations given above we can formulate

the following.

T

I. Let each one-element subset {t} belong to the algebra A. Suppose:

(a) The sequence x n(e) converges for any ее S,

(b) y(:) is weakly absolutely continuous (absolutely continuous) on

& with respect to

Under these assum ptions the sequence x n (e) converges fo r any e in S j and its lim it x(e) is an additive set fu n ctio n on S } which is weakly absolutely continuous (absolutely continuous) with respect to A (:), i.e. with respect to the subadditive measure generated on by y(:).

B . I f the relation

(*) x n ( e ) -> x(e) as n -> oo

holds fo r any ее x(e) = 0 f or each one-point set, then y{:) is w.a.c. on

£ } with respect to A(:).

Ad A. Let e belong to « fj. From the definition of follows the

existence of sets eh, el e S’ such that е \ \ , e l - l , eh c e <= el for n — 1 , 2 , . . . ,

У ( el — eh) -> 0 as n -> oo. In virtue of (b) for any e > 0 there exists an

index к such that у (el — el) < e. Since el — e a e l — el we obtain for

320 W . O r l i o z

П ^ n 0(Jc)

\\xn {e) — x n {el)\\ < vn {e\— e\) < y { e l — e\) + e < 2e, and

II®»(e) — coa{e)|| < \\xp (e) — xp (el)\\ + \\xp{e2 k) — x Q(el)\\-{-\\xq(e)-X q(el)\\ < 5e for sufficiently large p , q, so the existence of the lim it x(e) is proved. The additivity of x{:) on follows from the additivity of xn(:), its absolute weak continuity (absolute continuity) w ith respect to A(:) from the in ­ equality y{e) ^ ||я?(е)||.

Ad B . The proof w ill be based upon two lemmata,

a) Let у ni eA f or n , i = 1 , 2 , ... and let © = (e j denote a sequence the terms o f which are 0 or 1. I f fo r any sequence © = {e4} the series

oo

Уп{е) = ^ ~ f j 2 , ...

г=1

converges and y n {s) ->y(©)eX as n oo,

sup ||yn4| | 0 as _i -> OO П

(see [1]).

(3) Le i y{e) > e > 0 where ее S’} , let a natural number n 0 be given and let a0 = { t i , t 2, . . . , tr} be a fin ite set of elements belonging to e. Under these assum ptions there exists a fin ite set am = {tf, t™, .. . , t™}, where t™ cz e fo r i = 1 , 2 , . . . , $ , such that

Uffi ^ «О == Ь , Vm(am) ^ ||«Гт(^т)11 55"

and m ^ n 0.

Indeed, choose an index m, m ^ n 0, in such a manner that Г

W^m ({^ г})|| "'C V m { e )

i—l

T h is is possible owing to the inequality y(e) > e and the hypotheses we made on x m(:) and x(:). Because of the inequality

sup||a?m(a)|| > ±vm{e), acc,aeSj

a finite collection am of elements tmJ belonging to e can be found such that ||a?m(gm)|| ^ e/4. Let us set am = am— a0. Bo r the set am the in ­ equality

^ ^ ll^!m(<^m)|| (||*®m(0i})IH“ • • • ~f" 11^т({^г})11)

holds and besides am ^ a0 — 0.

Classes of finitely additive set functions ^{3 2 1}

Passing to the proof of B , let us firs t assume that ее <*} and A(e) = 0.

Suppose y(e) > e > 0. B y Lemma (3) there exists a sequence of sets ame Ś 1 } and an increasing sequence of indices nm w ith the following properties:

a) the sets am are d isjo int,

b) am is finite and is contained in e,

c ) I K m ( « m ) l| > £ / 8 .

Fo r any © = {si}, et = 0 ,1 , the set a(s) = у e4a4 is in cff, for each subset of e belongs to S j . B y the cr-additivity of x n{:) we have

OO

(i) ^n(^(®)) = j

г = 1

and since the lim it (*) e xists for ее we have by Lemma а), sup||a?n(a4)||

n

0 as i -> oo — th is contradicts the inequality \\xn (am)\\ > e/8. Hence y(e) = 0 if A(e) = 0.

Suppose en e $ for n = 1 , 2 , ..., en\,, rj(en) -> 0 as n -> сю. Let us set a ~ en ; evidently A(a) = 0. F o r Ic — 1 , 2 , . . . the following inequality (ii) lim in f vk {ep — eq) > v k{ep — a)

p^q—y oo

holds. Indeed, let e1, e 2, . . . , e r czep — a 1 ei e S 1f, and ё* are disjoint, let Au A2, . . ., Ar be scalars, |A*| < 1, and

{ef) -|- A^xk (e2) “L • • • “Ь Arx k (e,.)|| ^ Vki^p &) £•

B y the (7-additivity of x k {\) and because of ё4 лл {ep — ea) t ё4 r\ {ep — a)

— ё4, AiXhfei (ep — eq)) -> AiXk {ef) as q ->

; hence for q ->

(®i ^ £g)) ^

*®Л

^ (®

> £g)) "~Ь • • • “b Arx k (er r\ {ep ^qi)\ I

— >

^

^ (

) + A^X^ef) -j~• • . ~b ^r*^A:(£r)ll ^ ®к{вр O') £ J and (ii) follows. F o r any s > 0 there e xists /(e) such that

(iii) sup®ft(ep— eg) < s for p ^ q ^ r .

к

I f th is is not so, there e xists e0 > 0, Aq < Aq < .. ., р г < p 2 < ...

such that

®*4( S ~ e^+i) > £o for » = 1, 2, ...

I t is always possible to choose щ е contained in ePi— ep.+1 in such a manner that

. \\оок1(щ)\\ > e0/4.

Fo r any © = {si}, ^ = 0 ,1 , the set a(s) = (J belongs to « fj.

As regards a(©)e«f} we w ill make the following remarks.

R oczn ik i PTM — P r a c e M a tem a ty czn e XI.2 21

3 2 2 W . O r l i c z

Evid e ntly щ is reduced to a finite set of points. Fo r a point in E — say t — the one-point set {t} is either in «?}— S or in S. In the firs t case only the empty set 0 from S belongs to {t} and (by the definition of S}) Ц Щ ) = 0. I f { t } e S then, possibly, it could be A({/}) = > 0.

B u t even in the extreme case, when a ll points of a ll щ belong to S } — S, the inclusion a ( s ) e S y is not a consequence of the fact A (a*) = 0, i = 1 , 2 , . . . , only, because A(:) is not, in general, a сг-subadditive set function on S } and so some further properties of щ are required to prove the inclusion. We have

M * ) == er a r w er + i% + i v-' • • • *— (врг 6pr + l ) w (врг + 1 - 6pr + 2 ) ^ . . . c: eP r ,

Qjy.

(®j

^2

w

S

and we can choose sets e ' , e n € S such that e' <= ar(e) a e", щ{е" — e')

< e/2. Hence e' <= a (e) <= e" w ev and we can choose р Го such that v ( ePr0) < £/2- T lie ri

7]{e" w eP r - e ' ) < ц { е " - e' ) - \ - r\ {eV r - e " ) < e

and consequently a(s)e S }. The equality (i) holds and the lim it xn [a{e)} ->

-»■ ж («(©)) e xists for any ©. As previously Lemma (3) yields a contradiction w ith | | х к.(щ)\\ > sj 4.

We get the inequality

y(en) < y{en — a) + y{a).

B u t, by (ii), ( iii) , vk(en— a) ^ s for ifb ^

*(cj and 1c — 1 , 2 , .. ., hence y(en — a) < s and besides we have у (a) = 0, for A (a) — 0. Consequently y(en) < e for n > r(e) and the w.a.c of y(:) on S is proved.

R e m a rk . From the weak absolute continuity of y(:) on S w ith respect to r\{\) it follows immediately that y{\) is w.a.c. on S n j w ith respect to A(:).

2. In th is section we shall denote by /(•), 2/(*)> ••• real valued func­

tions defined on E. Given an algebra S we denote by R ( S ) ' the linear space of functions which can be uniform ly approximated on E by simple functions, i.e. by functions of the form

( + ) M*) = ^l^e1(') + ^a^e2( ‘) H~**- +AnZen ( ' ) ,

where S and A* are real scalars. Equipped w ith the norm ||/(-)||

= sup 1/00 [, R ( S ) is a Banach space.

UE

2.1. Take for E the interval .<a, by and for S the algebra J of sets

winch consist of fin ite unions of the intervals of the form (c, d), a < c

< d < b , <c, b y , a < c < b. Denote by p(e) the Lebesgue measure of

Classes of finitely additive set functions 3 2 3

the L-measurable set e in <a, by. The algebra generated by J and y{\) is evidently the algebra of sets measurable in the classical Jordan’s sense, and it is known that a function belongs to R { £ j ) if and only if it is integrable in the classical Eiemann sense. T h is class of Eiemann integrable functions can be generalized in the following way. Suppose that an additive vector valued set function x(:) is defined and a.c. on J with respect to /a(:). I t is known that x{:) can be uniquely extended to an additive set function on the algebra of L-measurable sets in <u, by w ith the preservation of the absolute continuity w ith respect to y(:), for J is dense in in the sense of the Badon-№kodym metric д(ег, e2)

= ju(e1 — e2)-j-Ju(e2— e1). Let ij(e) be the variation of x(:) on e e S L , £}

the Jordan algebra generated by J and r\{\), A(:) the subadditive Jordan measure generated on i f by rj(:). Let us remark that a ll closed intervals (c, dy, a < c < d < b and, more general, a ll sets which are measurable in the classical Jordan’s sense are in S j , but it may happen that &

—

— ф

, S n j — Ф 0. Denote by df the set of points of discontinuity of the function/(•). Then, the following statements are true: a) %e(-) e_R( <f}) i f and only i f ее <

‘, (3) a fu n ctio n /(•), bounded on < a , by, belongs to R { S j )

i f and only i f X{df) = 0 (cf. for instance [6]).

2.2. B y means of the set function x ( :) which has been introduced in 2.1 we define the vector valued function x(t) = a?(<a, t)) for a < t < b.

Le t A = <<x, by, let n denote the partition a — t0 < tx < < tn = b and for a scalar function/(•) let

where r i e<^_1, t i >. A function /(•) bounded on A is said integrable in the Riem ann-G ow urin sense over the interval A if for every normal sequence of partitions л п and arbitrary choice of r™ in idh interval of the n t h partition the corresponding sums a( nn , f ) converge to the common element y, which w ill be denoted by

and w ill be called here in short integral of /(•) over A ([2], [3]).

2.3. Any function belonging to R (£”}), where <fj denotes the Jordan algebra generated by «/ and the variation y(:) of x(:), is integrable over A in the Eiemann-Gowurin sense. In particular:

F o r e e i j the integral П

Ъ J f { t ) d x a

b ( + )

a

324 W . O r l i c z

exists. y (:) is a.c. on <?j with respect to A(:) and y(e) —

(e) if ее i Tj гл $ n j . ь

The integral f f ( t ) d x is an additive operator on R ( $ j ) and bounded;

a

more precisely, the following inequality holds

b

( + + ) \\j f (t )Xe(t )dx\ ^ < \\f\\rj(e), where e e S .

a

Because of the inequality

(i) H ^ ) I K I I / ( - ) l b M ) ,

and the definition of the class R ( $ j ) it is sufficient to show that the Biemann-Gowurin integral e xists for any %e('), where e e $ } . Let nn:

a = t o < t i < . . . < t^n) — b be a normal sequence of partitions of A, т” e<<”_ u t t y for i = 1 , 2 , . . . , Jc(n) and let a( nn, f ) be the corresponding Biemann sums 2.2 (i), yn(e) = а ( л п , %e) for S’} . Let e e J , then e can be represented as the union e1 w ez w ... w ek , where e*, i = 1, 2, .. ., &, is an interval of the form <c, d), a < c < d < b or <c, by, a < c < b, and the closed intervals ё* are m utually disjoint. I t is a simple matter to prove that for a set e e J the variation rj(e) is equal to the supremum of P 1£c(ó1) + A2&(ó2) + ... + AsaKA)ll for a ll |Ai|<l and for a ll decomposi­

tions of e into subintervals дг, ó2, ..., ds . Hence we obtain the inequality

к к

(ii) v a r( e , y n) £ r j { d ' in) + J£4(<C ),

i = l i = l

where d'in = or ó " = < ^ b l, $ (i)>, respectively, is the interval of the partition nn which covers the left end-point or the rig ht- hand one, respectively, of the interval e*. Let us remark that the varia­

tion v (e, y n) is taken w ith respect to i j and the a ll one-point sets belong to S j . B y the absolute continuity of x(:) on S L (which implies that Yj(:) is a.c. on $ L w ith respect to /л(:)) it follows from (ii) the inequality

y(e) = lim supvar(e, yn) < y(e), e e J

n - >00

and the condition 1.2 A(b) for the sequence of the set function yn(:) is fulfille d . B y a.c. of rj(:) w ith respect to /л(:) it is also evident that the lim it y n (e) as n ->

e xists and is equal to x(e). B y Theorem I y n(e) converges for any e in i j as n ->

, and, as it is easily seen, th is lim it y(e) equals x(e) if e is in ^ S'j and it is independent of the choice of n a. The in ­ equality ( + + ) is an easy consequence of the inequality

\ K( n, f xe) \ \ < \\f(’)\\r)(e) where ее/.

3. In th is section the algebras </, S’} , the set functions x ( : ) , r/(:)

etc. have the same meaning as in section 2 , x n (:) denotes, as in section

1.1, a set function associated w ith a system of elements 1.1 (o) and a system

Classes of finitely additive set functions 3 2 5

1.1 (оo) of distinct reals in ( a, &>. Setting

( * ) Q n ( f ) — x i f { t i ) + #2/ ( ^2) + • • • + x n f ( t k ( n ) )

for f ( - ) e B ( S j ) we define the quadrature formula corresponding to the

“knots” 1.1 (oo). Qn(f) is an additive operator on B ( S j ) and it is continuous, for

(i) Ш / )1 К Ш 1 1 / 1 1 ,

where \\Qn\\ = sup ||$n(/)||, and the supremum is taken either for a ll func- 1

tions in B ( £”j) for which ||/|| < 1, or — it is the same — for a ll continuous functions subject to the condition ||/(| ^ 1. W rite Qn(xe) = ®»(e) for ее S vj, and denote by vn{e) the variation on e of the set function ocn {:). I t is readily seen that ||ф „|| — vn (z1). As previously, let us define for ее S}

y(e) = lim su p

>n(e).

n— >oo

3.1. I f y{e) is w.a.c. on S j w ith respect to A(:), then the sequence

WQnW

= ®n(^) is bounded.

I t follows from the subadditivity of y(e) that if y( A) = oo, then there exists a sequence of intervals A = <32 з ...such that p ( di+l) — y(di)l2 for i = 1 , 2 , . . . and y(<5*) = oo, which evidently contradicts the weak absolute continuity of y{\) w ith respect to r)(:), for y{en) -> 0 as n oo implies Я(еп) 0 as n oo.

3.2. T

I I . Necessary and sufficient conditions fo r Qn (f) to converge to j b af { t ) d x f or all f ( - ) e B ( S j ) are the following:

(a) QniXe) ->w(e) as n -> oo for any interval in A,

(b) y(:) is weakly absolutely continuous on J with respect to y(:) Assume (a), (b) to be satisfied. Applying Theorem I to QniXe) = ®n(e) we ve rify that the lim it

lim QniXe) = *(e) n— >co

exists for е е «?}. B u t by (a) z{e) = x(e) = y(e) = faXe(t)d<v for any e e J , and owing to the w.a.c. of y{e), z(:) on S } and by 2.3( + )? the equation z(e) = У(е) holds for every e in I t follows

ь

Qn(f) -> / f ( t ) d x as fl — > OO

a

for any simple function of the form 2 ( + ), w ith and by 3 (i),

3.1 and by definition of the class B ( S } ) th is relation is also true for

an arbitrary function belonging to В ( S } ) .

3 2 6 W . O r l i c z

If QnU) -> faf(t)d% as n oo for /(•)■ € R{ S}), then by the definition of the class R{S}) Шъ condition (a) is fu lfille d for any interval in A (the closed one as well, for any one-point set belongs to S}). Since %eel%(S})

if ее S’}, by Theorem I В we get the condition (b).

R e m a rk . I f y{:) is w.a.c. w ith respect to then it is also with respect to for /л(еп) 0 as n -» oo implies v(en) -> 0 as n ->■ oo.

3.3. In Theorem I I we can replace the condition (a) by the following condition

b

(a') Qnif) ff{t)dx as n -> ^oo for any function /(•) which is con- tinuous on A. a

In fact, assume conditions (a), (b) to be fulfille d , and /(•) to be con­

tinuous in A. As the function /(•) can be uniform ly approximated on A

by simple functions 2 ( + ) where ex, e2, .. ., en belong to J and in virtue of the inequality 3 (i), 3.1 it follows that the sequence of the quadrature formulae Qn(f) converges to j bf(t)dx, and (a') follows.

Now, assume conditions (a'), (b) to be fulfille d . Take a half-closed interval e = ( c , d ) , a < c < d < b, and a < c < c, d < d < b. B y (b) we can always choose c, d in such a manner that y(e') < e, y(e") < e

and besides such that rj(e') < e, y{e") < s where e' = <c, c), e" — <d, d ^>.

Consequently, vn (e') < e, vn (e") < e ^for n ^ n0(e). Define a continuous function /(•) setting f(i) — 0 for a < c , f(t) = 1 for c < t

< d, and extending it linea rly in intervals e', e" . We have

ь ь b

|| ff(t)dx — л?(в) || < || (t)%e>(t)dx ||-f-1| Jf (t)ye"(t)dx^ < rj(e') + y ( e f') < 2e,

a a a

\\Qn(f)~Qn(Xe)\\ < vn ( e' ) +v n {e") < 2s for n ^{^} n0.

B u t by (a') the inequality

b

\\Qn(f)— ff(t)dx\\ < e

a holds for n ^> h and consequently

WQn(Xe) — %(e)\\ < 5e for n > max(&, n0).

In the same way we prove 3.2(a) for intervals of the type (a, d), ^<c, by

and for one-point sets.

3.4. Suppose now X to be the space of real numbers. Let

have the same meaning as before in 2 and 3, let S } denote the algebra of sets in <A, by which are measurable in the classical Jordan sense, let /(•) denote a real valued function integrable over <a , by in the classical

b

Riemann sense and let ff(t)dt be it s Riemann integral. B y Qn(f)i

a

Classes of finitely additive set functions 3 2 7

n — 1 , 2 , ... the quadrature formula for /(•) w ill be denoted, as in 3 — 3.3.

Assume x(:) = p{:). Under such circumstances we have for the variation vn(e) over of Qn(Xe)

where denotes that the summation is to be extended over a ll tth for which tniee. I f y{:) is w.a.c. on J with respect to rj(:), then it is also w.a.c.

on w ith respect to p{\) and conversely, for in the case under considera­

tion r\{e) = /л(е). Applying Theorem I I and 3.3 we can formulate the following theorem, in which part A has been proved firs t by G. Polya [7].

T

I I I . A. Necessary and sufficient conditions for a sequence of quadrature form ulae Qn (f) to converge to j b af ( f ) d x f or any Eiemann-integra- ble fu n ctio n are the follow ing:

(a) Qn(f) f a f ( t ) dt f or any continuous fu n ctio n /(•), (b) I f i t => i 2 ••• where in e«/, ju(in) 0 as n -> oo, then

[1] A. A le x ie w ic z , On sequences of operations I, Studia Math. 11 (1950), pp. 1-30.

[2] M. G ow u rin , Tiber die Stieltjesche Integration abstrakten Funktionen, Fund. Math. 27 (1936), pp. 254-268.

[3] N. D u n fo r d , Uniformity in linear spaces, Trans. Amer. Math. Soc. 44 (1938), pp. 305-356.

[4] — and J. S c h w a r tz , Linear operators, part I, New York 1958.

[5] S. L ea d er, The theory of iP-spaces for finitely additive set functions, Ann.

Math., 58 (1953), pp. 528-543.

[6] W. O rlicz, Absolute continuity of vector valued finitely additive set functions, Bull. Acad. Polon. Sci., Ser. sci. math., astr. et phys. 16 (1968).

[7] Gr. P o ly a , fiber die Konvergenz von Quadraturverfahren, Math. Zeit. 37 (1933), pp. 264-286.

У {in)

0 as n -> o o .

B . I n A we can replace condition (a) by the following one:

(a') Qn(Xe) f*{e) f or апУ interval e contained in <a , by.

R eferences