ADDITIVE SET FUNCTIONS IN EUCLIDEAN SPACE. II

(1)

ADDITIVE SET FUNCTIONS IN EUCLIDEAN SPACE. II

BY

C. A. R O G E R S and S. J. T A Y L O R

London

l . Introduction

I n our previous paper [11], we discussed various decompositions of additive set functions in E u c l i d e a n space. Our m a i n object was t o show h o w a s y s t e m of Haus- dorff measures could be used to analyse a given set function, as far as is possible, into components, which were uniform in a certain sense. I n t h e present work, we use t h e results of a series of papers [5, 8, 9, 12, 13] to correct a n d e x t e n d some of t h e results o b t a i n e d in [11].

W e continue to restrict our a t t e n t i o n to t h e s y s t e m :~ of those continuous com- pletely additive set functions F, h a v i n g a finite value

F(E)

for e v e r y set E in the field B of all Borel subsets of a fixed closed rectangle I 0 in k-space. I t is clear t h a t t h e analysis extends i m m e d i a t e l y to a-finite set functions, defined for Borel sets in E u c l i d e a n k-space.

I n t h e first three sections of [ l l ] , we w o r k e d with a single H a u s d o r f f measure f u n c t i o n

h(t),

a n d we o b t a i n e d a u n i q u e decomposition of a set function F of :~ into three components, one strongly continuous with respect to h-measure, one, n o t only absolutely continuous with respect t o h-measure, b u t also c o n c e n t r a t e d on a set of a-finite h-measure, a n d one c o n c e n t r a t e d on a set of zero h-measure. T h e extensions a n d refinements of this work, which we m a d e in [13] will be vital for the sequel.

I n t h e last t h r e e sections of [11] we i n t r o d u c e d a s y s t e m l: of H a u s d o r f f measure functions

](t),

which was t o t a l l y ordered b y t h e relation - ( , defined b y :

/•g,

if

g(t)//(t)--->O,

as t - + + 0 .

W e first studied the special case, when 1~ is the s y s t e m of functions t ~ ( 0 < a < k ) ,

(2)

208 C. A . R O G E R S A N D S. J . T A Y L O R

and then turned to the case when E is maximal. I n these studies, we analysed a set- function F of :~ into a sequence of components, corresponding to certain sections of E, and a residual component with a "continuous C-dimension spectrum". While our main theorems (Theorems 3 and 6) are correct, the second of t h e m (Theorem 6) cannot be regarded as satisfactory. Unfortunately, there is an error in the proof, t h a t F(1 s) is continuous on the right, given in L e m m a 10. The faulty lemma led to our introducing an unsatisfactory definition for a set-function with a "continuous E-dimension spec- t r u m " ; a set-function satisfying our conditions would not, as far as we know, necessarily have the essential p r o p e r t y of having no component which should naturally be associated with a section of E.

I n this paper, we correct and extend the analysis we made previously. The results we obtain are rather more complicated t h a n those t h e y replace. We find t h a t t h e y depend essentially on the various properties of the system E, and on the nature of the sections of E considered. We defer a n y detailed description, until after we have introduced the necessary concepts. The results m a y be described, in general terms, b y saying that, provided t h a t the system E has appropriate density properties, we can analyse a set function F of ~ into:

(a) a sequence of isolated components corresponding to functions h of E;

(b) a sequence of isolated components corresponding to sections of E having no countable basis;

(c) a residual set function having no component belonging to a n y function or section of E.

We give an example to show t h a t the components corresponding to sections of E having no countable basis m a y be non-zero; b u t we have to leave unanswered some interesting questions concerning these components.

2. Scales o f functions

L e t ~ be the system of all positive functions h(t), defined for t > 0 , continuous monotonic increasing, and with limt-~+0 h ( t ) = O. When h and g belong to ~ , we write

h ~ g ,

if 0 < lim inf g(t)/h(t) <~ lim sup g(t)/h(t) < + oo ;

t--~.+O t--~+O

and h < g,

if lim g(t)/h(t) = O.

t---~ + 0

(3)

A D D I T I V E S E T F U N C T I O N S I N E U C L I D E A N S P A C E . I I 209 W e s a y t h a t h a n d g in ~4 are c o m p a r a b l e , if

h-<g, or h,.~g, or g-<h;

a n d t h a t t h e y are m o n o t o n i c a l l y c o m p a r a b l e if, in addition, the ratio h(t)/g(t) is m o n o t o n i c for all sufficiently small positive t.

W e list some properties, i n t r o d u c e d in [5] a n d [8], which m a y hold for a subset s of ~4.

(P1) I f g, h E ~, t h e n g a n d h are c o m p a r a b l e .

(P*) I f g, h E s t h e n g a n d h are m o n o t o n i c a l l y c o m p a r a b l e . (P2) T h e function ~ is in L.

(P3) I f g , h ~ s a n d ~, fl are real n u m b e r s , a n d g ~ h ~ E ~ , t h e n g ~ h ~ E s (Pa) I f g, h E C a n d g 4 h , t h e n g is n o t e q u i v a l e n t to h.

(Ps) I f h E ~ a n d h is c o m p a r a b l e w i t h each e l e m e n t of i:, t h e n h is e q u i v a l e n t t o a t least one e l e m e n t of l:.

(Pc) I f h E ~ , t h e r e is a t least one g in I: w i t h g-<h.

(PT) I f hE:// a n d h-<t ~, t h e r e is a t least one g in l: w i t h h-<g-<t k.

(P1.) I f 11, l~ . . . ul, u~ . . . . E ~ a n d 11-<12-< ...-<l~-< . . . -<urn-< . . . - < u 2 - < u 1, there is a g in i: w i t h ll -<12 -< .. . -< l~ -< .. . -< g..< . .. -< u,~ -< . . . -< u2 -< u 1.

(1)11) I f h E ~ a n d ll, 12...EF~ a n d ll -<12 -< ... -< l~ -< ... -< h , there is a g in C w i t h

l~ -<12 -< . . . -< l~ -< . . . -< g -< h .

These conditions do n o t correspond e x a c t l y with those given in [5] a n d [8], since t h e r e we were concerned w i t h functions which m i g h t t e n d to 0 or t o + ~ or t o a finite limit as the variable t e n d e d to + co; b u t t h e differences are trivial.

A set I~ of functions of ~ will be called a m o n o t o n e scale of functions, if it satisfies (P1), (P*), (P2) a n d (P3)- A scale w i t h p r o p e r t y (P4) is irreducible; one w i t h (1)5) is m a x i m a l ; one w i t h (Pc) a n d (PT) is dense; a n d one w i t h (Pc), (1)7), (1)10) a n d (Pll) is strongly dense.

T h e m a i n result which we o b t a i n e d w i t h t h e help of P. E r d 5 s in [5] a n d [8] is:

T ~ E O R E M A. T h e c o n t i n u u m hypothesis i m p l i e s the existence of a monotone scale which is irreducible, m a x i m a l a n d strongly dense,

R e m a r k . I t is also shown in [8] t h a t e v e r y m a x i m a l scale I: has t h e p r o p e r t y (P10), a n d the s a m e simple proof shows t h a t such a scale i: has the p r o p e r t y o b t a i n e d f r o m (P10) b y replacing one or b o t h of the sequences /1, 12 .. . . . or ul, u 2 .. . . b y a single e l e m e n t of 1:.

(4)

210 C. A. R O G E R S A N D S. J . T A Y L O R

At this stage we require one further definition. Two subsets L, R of an irreducible scale s are said to form a section of l~, if:

L has no maximal element, L N R = o , L U R = ~ ,

and hiE.L, h~ E R implies hl'~ h ~.

We return to study and classify the sections of strongly dense irreducible monotone scales in w 5.

3. B a n d s a s s o c i a t e d w i t h a s e c t i o n in a s c a l e o f m e a s u r e f u n c t i o n s

Let s be an irreducible maximal scale of functions of ~4. Let s be a section of

~: into two sets L~, R~. I n [11] we associated two pairs of complementary bands(1) in ~ with the section s. Let R~ denote the set obtained by removing from R~ its least element, if it has one. Then as in [11] we introduce:

1. C~ is the class of set functions F of :~, which are s-continuous; that is, those set functions such t h a t F ( E ) = 0 , for a n y E E ]~, for which there is at least one h E s with h - re(E) = 0;

2. St is the class of set functions F of :~, which are s-singular; t h a t is, those set functions, for which there is some E 0 in ]~, such t h a t

.F(E) = F ( E N E0), (1)

and E 0 = O ~ l E i , where, for each E~, there is some h~ in L~, for which h ~ - m(E~) = 0;

3. C* is the class of set functions F of :~, which are strongly s-continuous; t h a t is, those set functions such that, if E EB, and h - m ( E ) = 0 , for every hER~, then F(E) = 0 ;

4. S* is the class of set functions F of :~, which are almost s-singular; t h a t is, those set functions, for which there is some E o in ]~, such that (1) holds, and h - m ( E o ) = 0 , for every h in R~.

I n [11], we showed that C~, ,~ and C*, $* are pairs of complementary bands in ~. I n order to obtain a more complete analysis, we now define two further classes:

5. C** is the class of set functions F of :~, which are hyper s-continuous; t h a t is, those set functions which can be expressed as F = ~ 1 F,, with

~i~176 IF tl(I0)

convergent, and each Fi being hrcontinuous, for some h, in R~;

(1) This concept is explained later in this section.

(5)

A D D I T I V E S E T F U N C T I O N S I N E U C L I D E A N S P A C E . I I 211 6. S** is t h e class of set functions F of :~, which are h y p o s-singular; t h a t is, for each h in R~, there is a set E 0 in B, such t h a t h - m ( E o ) = O , a n d (1) is satisfied.

The s y s t e m ff of sections s of l: is ordered in a n a t u r a l way; we write s < t, if the corresponding left-hand sets Ls a n d Lt satisfy L s ~ L t , t h e inclusion being proper.

I t is clear, from t h e definitions, t h a t , for a n y sections u, s, t, of 9", with t < s < u, we h a v e

C~ ~ Cs** ~ Cs* ~ C -

~tr" n**. },

S~ ~ S~** ~ S* ~ Ss ~ S~*.

(2)

Our first object is to prove t h a t C** a n d S** f o r m a pair of c o m p l e m e n t a r y b a n d s in :~. This could be done directly for t h e space :~, b u t we prefer to prove a more general t h e o r e m a b o u t collections of b a n d s in a UMB-lattice, a n d t h e n o b t a i n o u r results as a p a r t i c u l a r case. To this end, we recall t h e concept of a B a n a c h lattice, with a u n i f o r m l y m o n o t o n e norm, a n d show t h a t :~ is such a lattice.

As explained in [ l l ] , :~ is a v e c t o r lattice, t h a t is a partially ordered linear space with a lattice structure. F u r t h e r , :~ is conditionally complete: t h a t is, every subset ~ c :~, which is b o u n d e d above, has a least u p p e r b o u n d F 0 in :~. I f R is a n y conditionally complete vector lattice, we call a subset ~ of R a band, if:

(i) ~ is a vector subspace of R;

(ii) if GE'U, t h e n every F of ~ with

JFJ~<JGJ

is also in ~ ;

(iii) ~ is a sublattice of R, which is conditionally complete; t h a t is, every subset of ~ , which is b o u n d e d above in R, has its least u p p e r b o u n d in lying in ~ .

Here, in (ii), the element I F I is defined(1) b y

I '1 u { - F } .

I n a slightly more general context, Birkhoff ([1] p. 232) calls such a set a closed/-ideal.

I f a vector lattice is also a B a n a c h space, with t h e same v e c t o r structure, a n d with a n o r m related to the lattice structure b y t h e condition t h a t ,

if

iF j<IGI,

^then

IIFJI<JJGJJ,

(3)

it is called a B a n a c h lattice (see [1]). I t is easy t o v e r i f y t h a t :~ is a B a n a c h lattice, u n d e r t h e n o r m

IIFII = IFI

(z0). (4)

(1) Note added to proof The same symbols have been used for the lattice operations of join and meet for as the set operations of union and intersection; but this should cause no confusion.

(6)

Following Birkhoff [1], we call a norm in a Banach lattice uni[ormly monotone when, given s > 0 , one can find ~ > 0 , so t h a t if F>~0, G~>0, and H F I I = I , then [IF+Glib< [[FI]+(~ implies []GII-~<~. I t is clear t h a t :~ satisfies this condition, for, in :~, if F~>0, G~>0, then

IIF+ GII = G(Z,,)= IIFII + II GII.

Any Banach lattice, which is a conditionally complete lattice, and which has a uniformly monotone norm, will be called a UMB-lattice.

I n [11] we used some general theorems a b o u t conditionally complete lattices, which we now repeat for reference purposes. We recall t h a t two elements F, G in a vector lattice are said to be complementary, if

I '1 n IGI

= 0 .

THEOREM B. I / ~ is a conditionally complete vector lattice, and 11 is a subset o/ ~, the set ~ o/ all elements o/ ~, which are complementary to every element o/ 11, is a band in R. I / 11 is also a band, then 11 consists o/ all the elements o/ ~ which are complementary to every element o/ ~.

This theorem follows almost immediately from the definitions; see N. Bourbaki [3], or S. Bochner and R. S. Phillips [2]. I f 11 and ~ are bands in ~, each consisting of all the elements in }~ complementary to all the elements of the other, we say t h a t t h e y are complementary bands in ~. The following fundamental decomposition theorem follows:

THEOREM C. Given two complementary bands 111, 11~ in a conditionally complete vector lattice ~, each element F o/ ~ can be expressed uniquely as

F=FI+F ,

where F 1 E l l I and lv ~ E 112.

Further IF[ = IF1[ + IF21.

The existence and uniqueness of the decomposition is essentially due to Riesz [7].

I f F1, F 2 are c o m p l e m e n t a r y elements of a n y vector lattice, then

IF,+&I=J&I+IF I.

(This result is well known: a proof can be easily constructed using the identities on page 19 of Bourbaki [3].)

(7)

A D D I T I V E S E T F U N C T I O N S I N E U C L I D E A N S P A C E . I I 213

so t h a t for all i.

THEOREM D. A n y band in a conditionally complete Banach lattice is closed under the metric.

Proof. Suppose t h a t ~/ is a b a n d and t h a t ~ is the complementary band. L e t F be a n y limit point of "U. Then there is a sequence $1, F 2 .. . . in ~ converging in n o r m to F. L e t G belong to ~. Then as G, _~ belong to complementary bands,

IGI n I ~ l : l l o l n i v i - I o l n ,v, ll

<IiFI-IF~II<-<IF-V~I,

II IOI n IEI I1~ II~'- ~,11,

Hence IGI n I F I = O and F is complementary to G. Thus V belongs to the

band complementary to ~, i.e:, to ~ . Thus ~ is closed under the metric.

We also need

THEOREM E. I / a set ~ in a UMB-lattice R satis/ies the conditions:

(i) ~ is a vector subspace o/ ~, which is closed under the metric;

(ii) i/ G E ~ , then every F o/ ~ with ]F[~<IG] also belongs to "tl;

then it is a band in ~.

Remark. I n Birkhoff's terminology, this says t h a t an /-ideal in a UMB-lattice, which is closed under the metric, is a closed /-ideal.

Proo/. I t suffices to prove t h a t ~/ is conditionally complete. Now, b y (i), the set ~/ is a sub-Banach space of }~. Further, if F and G belong to ~ , then, b y (ii), we have IFI and IGI in ~/. Thus I F I + I G [ E ~ / b y (i). B u t

IFnol~<lFl+lol, IFU OI~<IFI+IGI,

so t h a t b y (ii) we have

F N G E ~ and F O G E Y . Thus ~ is a sub-lattice of ~.

Now suppose t h a t ~ is a subset of ~ which is bounded above b y an element R 0 of ~. L e t V 0 e ~ and consider the system ~ of finite joins

V oo V l o . . . U V~ (k>~O).

is a directed set contained in ~/ with

Vo <~ W <~ R o,

(8)

214 C. A . I ~ O G E R S A N D S. J . T A Y L O I ~

for all WET~O. Hence, by [1], Theorem 11, page 249, the directed set ~0 converges metrically to some element, U 1 say, of }~. As ~ c ~/, and ~ is closed under the metric, UIE ~/. Further U 1 is the least upper bound of ~0 in ~. Consequently ~/ is conditionally complete, as required.

These two results enable us to prove

TH]~OR]~M F. Suppose ~ is a UMB-lattice and ~ is a band in ~ /or each ~ in an index set Y. Then

is a band in ~. Further, the closure under the metric o/ the vector space generated by the union

U~=

is a band in ~, and is the least band in ~ containing this union.

Remark. We shall only need the special case when the bands ~ with ~ E :7 are nested (i.e., are totally ordered by the relation of set inclusion). In this case, the result follows almost immediately from Theorems B, D and E; but the general result seems to be sufficiently interesting to justify its special proof.

Proo/. I t follows immediately from the definitions t h a t

n ~

is a band in ~.

Let Q be the closure under the metric of the vector space generated b y the union

U ~ .

g e y

I t follows, from Theorem D, t h a t any band in ~, containing this union, must also contain Q. Hence it suffices to prove that Q is a band in ~. B u t Q satisfies the condition (i) of Theorem E. So it suffices to prove t h a t Q satisfies the condition (ii) of Theorem E.

We first prove that, if ~/ and ~0 are two vector spaces in ~ satisfying condition (ii) of Theorem E, then the vector space ~Lq generated by the union of ~ / a n d ~O also satisfies this condition. Suppose t h a t W E W and F in ~ satisfies IFI ~< IW]. Then

W = U I + V 1 with U 1E~/ and V 1E~. Further

F+ ~ I F I < [ U I + V I I ~ I U I I [ - I V I ] , and

U2=]UllEU, V2=[VIIEV.

(9)

A D D I T I V E S E T F U N C T I O N S I N E U C L I D E A N S P A C E . I I 2 1 5

Hence F + = F + N ( U s + V s ) = U s + ( F + - U s) NV s.

W r i t e u 3 = { ( F + - Us) n V~} n 0

V3 = { ( F + - U2) n Vs} u o.

Then F + = U s + U s + V 3.

Also 0 < - U~ = ( ( U s - F +) U ( - V~)) U 0 < U~, and 0 ~ V 3 = { ( F + - U 2 ) N Vs}U0~<V2, so t h a t U 3 E ~ , V 3 E ~ and F + E ] 0 . Similarly F - E ~ 0 and so does F = F + + F -.

Applying this result inductively we see t h a t the vector space generated b y the union of any finite collection of the sets ~ with a E :1 satisfies the condition (ii) of Theorem E.

Now suppose t h a t G E Q, and t h a t F in ~ satisfies IF[ ~<]G I. As G~ Q, we have

G = lim Gi,

i-->~

j(i)

where for each i, Gi = ~ G~k,

k = l

G~ e N~(~.~) (k = 1, 2 . . . j(i)),

~(i, k) e Y (k = 1, 2, ..., i(i)).

Since 0~<F + ~<]G], 0~< - F - ~< [GI, we have

I F - F + NIG~I+(-F

) N I G ~ i I = I F + N I G t - F

+ N I G , I - ( - F - ) N I G t - ( - F - ) N I G , I I

< ~ 2 1 1 G I - I G , I I < ~ 2 I a - G , I .

So I I F - F + n IG, I + ( - F ) n IG, I l l ~ < 2 ] l a - ~ , l l . Thus F is the limit in norm of the sequence

H~=F +NIG~I-(-F-)NIG~I

( i = 1 , 2 . . . . ).

But, for each i, the vector space generated b y the sets

~ . ~ (]~

= 1, 2 . . . i(i)), satisfies condition (ii) of Theorem E.

(10)

So

F+nlGil

and

(-F-)nlG, l

lie in this vector space and thus belong to Q. Consequently Hi, i = l, 2 . . . . an(

F = lim~_.~ Ht belong to the closed vector space Q. This completes the proof.

We now return to the s t u d y of the space :7 of additive set functions. We prow T H ~ . O ~ G. For any section s o/ ~, the sets C** and S** are complementar~

bands in ~.

Proo/. Suppose s is the section Ls, Rs of ~/. For each h in R's, the class C(h)oJ h-continuous set functions, and the class $(h) of h-singular set functions form com.

p l e m e n t a r y bands in ~7. So, b y Theorem F, the set S**= fl S(h)

a~n d is a band in ~.

L e t (~ be the closure under the norm of the vector space generated b y the union

O

C(h).

aeas"

Now every element in this union is complementary to each element of S**. I t follows t h a t each element of Q is also complementary to each element of S**. B u t each element of ~, which is complementary to each element of Q, lies in $ ( h ) f o r all hER's, and so lies in S**. Thus S** is the band c o m p l e m e n t a r y to the set Q. But, b y Theorem F, the set Q is a band in ~. Hence S** and Q are complementary

Cs 9

bands in ~, and it remains to identify (2 and **

Since every F of ~ of the form

oo

F =

YG,

i = 1

with ~$=1 ^{I I i}F

II

convergent, and with F, e C(h~), for some hi in R;, is clearly in Q, we have C** c Q.

Now, if F E Q , we have

F = lim G ,

t-~Oo

for some sequence union

G 1, G~, ... of set. functions in the vector space generated b y the

U

C(h). (5)

h~R~

Replacing G1, G 2 .. . . b y a suitable sub-sequence H1, H 2 .. . . . and writing H 0 = 0, we have

(11)

A D D I T I V E S E T F U N C T I O N S I N E U C L I D E A N S P A C E . I I 217 F = ~ ( H i -

Ht-1),

i = 1

w i t h ~ H H , - H~-lll

convergent. Now, for each i,

F , =

H,

-

H,_I

lies in the vector space g e n e r a t e d b y t h e u n i o n (5), a n d so has a representation as a finite sum

i ( t ) l k f f i l

w i t h F~k e C(h~k) (k = 1, 2 . . . . , ](i)),

a n d h~kER's ( k = 1, 2 . . . . , ](i)).

T h u s Q Cs , ** a n d t h e t h e o r e m is proved.

4. The first decomposition theorems

As in section 3, let /: be an irreducible m a x i m a l scale of functions of ~//. I n this section, we show h o w a function F of :7 can be expressed as a s u m of components, associated with sections s of the s y s t e m 9' consisting of all sections of E, a n d a residual c o m p o n e n t h a v i n g w h a t w e : s h a l l call a continuous dimension spectrum.

W e first prove

THEOREM 1. Given any set /unction F E ~ and any section s o/ F~, there is a unique decomposition

F = F~ ~) + F~ 8) + W ) + F~ ~), ₃

where F~ s) is hyper s-continuous, F~ ~ is strongly s-continuous and hypo s-singular, F~ ~) is s-continuous and almost s-singular, and F~ ~) is s-singular. Further

IFI

= [Fi~)[ + IFh~)l + F (~)~ + ]Fs

Remark. T h e decomposition of this t h e o r e m is a p p a r e n t l y finer t h a n t h a t of T h e o r e m 5 of [11]; t h e set function F~)+F~ ~) above corresponds to t h e first compo- n e n t of the decomposition given in [11]; b u t we k n o w of no example with F~s)#0.

T h u s we h a v e been unable to decide w h e t h e r or n o t the class C* t3 $** is always void.

N o t e that, if ~ 6 L~, t h e n necessarily

F = F(4 ~ F~ ~) = F (~) - F (~) - 0 ₂ _{- -} ₃ _{- -} ₉ See also the r e m a r k at the end of section 8.

1 5 - - 6 3 2 9 1 8 A c t a mathematica 1 0 9 . I m p r i m 6 l e 1 4 j u i n 1 9 6 3 .

(12)

Proo/. B y the definitions, the results of [1], and Theorem G, the pairs {Cs, S~}, {C*, S*} and {C**, S**}, are pairs of complementary bands in :~, and satisfy

C~ cC~ =C~,

S** ~ S* ~ S~.

The result now follows, on applying Theorem C three times.

We now need to know how the decomposition of Theorem 1 depends on the section s of 1:. We can define left and right continuity in the norm topology of :~

relative to the order topology of ~. If F(s) is a set function of :~ defined for s in ~r, we say t h a t F(s) is continuous on the right at s, if, for each c > 0 , there is a u in ~r, such t h a t

[[F(s)-F(t)l[<~, for all t in ~ with s < t < u .

Continuity on the left is defined similarly, and the discontinuities on the left and right of a function with a simple jump discontinuity are defined in the natural way.

T H ~ O R E ~ 2. I n the decomposition o/ Theorem 1,

-[[F7)I[ and [IF[~)l]

are monotonic increasing /unctions o/ s, the /unction F~ s) is continuous on the right, and F(4 ~) is continuous on the le/t. The sections s, w h e r e - F ( 1 ~) and F(4 s) are discontinuous coincide, and at such a section -F(1 s) has a simple jump discontinuity on the felt and F(4 ~) has a simple jump discontinuity on the right, these discontinuities being equal and having the value

Remark 1. This result, as Lemma 10 of [11] was alleged to, follows from use of the methods of w of [11], but, in view of the lacuna we have found in the proof of Lemma 10, it seems best to give the proof in some detail.

Remark 2. Since IIFll=IFI(Io) and IFI is non-negative for all $'e:~, the fact t h a t -I[F(I~)[[ and [IF~)]] are monotonic increasing implies t h a t -IF(I~' I (E) and [F(4~) I (E) are monotonic increasing for all E E B.

Proo/. Suppose t h a t s and t are sections of E with s < t. Let

IFI = [ Is'I + [ + I + IF(:f,

F ( t )

IF[=[FT)[+]F~t)I+IF~~ , ,

(13)

A D D I T I V E S E T F U N C T I O N S I N E U C L I D E A N S P A C E , I I 219 be the corresponding decompositions of IF[, provided b y Theorem 1. Now, b y the inclusion relations (2), the function ]F(~)I is hyper s-continuous. Further, b y Theorem l, we have a decomposition

where G~ ~), i = 1, 2, 3, 4, are all non-negative, and where G~ s) is h y p e r s-continuous, G~ ~) is strongly s-continuous a n d hypo s-singular, G~ ~) is s-continuous and almost s-singular, and G~ ~) is s-singular. Since the decomposition of

IFI

with respect to the section s is unique, we have the identification

I F~') I = G~ ~) (i = 2, 3, 4).

Since G~ s) is non-negative, we have

II FiS)ll = I Fis)I ( I 0 ) = IFi"l (x0) + Gi s, (I0)

Thus - I I ~ I s ' I I is monotonic increasing. Similariy IIF(:II is monotonic decreasing.

Now we s t u d y the behaviour of F(1 t), for t in ~ to the right of a section s E ffl The function F(~ s) itself is h y p e r s-continuous and so has a representation

r 1 6 2

F{ ~) =t~lG~,

with ~~ convergent, and with each Gt being hi-continuous, for some h, in the class R~ associated with the section s. For each e > O, we can choose a positive integer k, so t h a t

o o

L e t h = min {h 1, h2 . . . . , hk}.

Then h e R~.

I f Rs has no minimal element, t h e n R~ = Rs has no minimal element. On the other hand, if Rs has a minimal element, then Rg was formed b y removing it, and it follows from the m a x i m a l i t y of s t h a t R~ has no minimal element.

Thus we can choose g in R~, with 9Kh, and we shall still have elements r in R~, with r-<g. L e t u be the section of 1: formed b y taking the left class to be all the functions r of /: with r-<g. Then s<u, and g belongs to the right class of u.

(14)

2 2 0 c . A . R O G E R S A N D S . J . T A Y L O R

Consider a n y section t in T, with 8 < t < u .

As g belongs to t h e right class of t, a n d

g < h ~ ( i = 1 , 2 . . . k),

k

the set function ~ Gt

5 = 1

is h y p e r t-continuous. The set function

H = ~ Gt

t = k + l

has a decomposition in t h e canonical f o r m

H = H(~ ) + H(~ t) + H(~ ) + H(4 t).

T h e set function ~ - F (s) + F (s) + F(4 s) i x - - $ 3

is h y p o s-singular a n d so /-singular for a n y 1 in t h e modified r i g h t class R~ of s.

T a k i n g 1 to he in R~ a n d also in t h e modified left class of t we see t h a t K is t-singular. T h u s c o m p a r i n g t h e decomposition

F = G, + H f ) ~ ~- 1~2~(t) + H(t,s + { H f ) + K}, (6)

with t h e decomposition with respect to t h e section t p r o v i d e d b y T h e o r e m 1, we h a v e

k

F i ') = 7 G, + H i ~).

i = l

H e n c e HFI t) - FT)II = IIH~ t) + H (t)+3 H(t), '<~ I[HII < e.

This proves the c o n t i n u i t y of F(1 t) on t h e right at t = s.

B u t it also follows f r o m the decomposition (6) t h a t F(t) _ ~ ( t ) + K . 4 - - 1 ~ 4

H e n c e I[F(4 t) ^- F (s)2 - F ( s ' - Fs = llH(?l[ < llHIl <

This proves t h a t F(4 t) has a simple d i s c o n t i n u i t y F (s) ~-F (s) 2 " 3 on the right at t = s , or is continuous on the right at t = s in the case when $'(28)+ F~ ~)= 0.

We r e m a r k that, if it is possible to choose h in R~ so t h a t h-Kh~ ( i = 1 , 2 , 3 . . . . ),

(15)

t h e n t h e a b o v e p r o o f shows t h a t

F(1 ~ ) : ~<,s), ~ ) = 0, ~(~> = 0, F<~> = F i ~) + F<3 ~> + F + ,

for all t w i t h s < t ~ u . Thus, in t h i s ease, F(1 t) a n d F(4 t) a r e c o n s t a n t a n d F(~ t) a n d F(s t>

v a n i s h in a n o p e n i n t e r v a l t o t h e r i g h t of t = s.

W e n o w s t u d y t h e b e h a v i o u r of F(4 t), for t i n 7 . T h e f u n c t i o n F(4 ~) is itself s-singular. So F~ ~) is c o n c e n t r a t e d o n a set E o in B a n d

E o = (~ E~

where, for e a c h i t h e r e is some h~ in Ls for w h i c h h i - m ( E ~ ) = 0. W i t h o u t loss of g e n e r a l i t y we m a y a s s u m e t h a t t h e sets E~ a r e d i s j o i n t B o r e l sets. N o w , w r i t i n g

o~ (E) = F(~ s> ( ~ n E,),

we h a v e F(4 s)= ~ Gi,

i=1

a n d ~I1 = = I FiS~L (E~) = IIFi~'[[ < + ~ .

z = t = '

T h e a r g u m e n t now p a r a l l e l s t h a t u s e d in t h e discussion of F(~ t). F o r e a c h e > 0, we c a n choose a p o s i t i v e i n t e g e r k so t h a t

T h e n h = m a x {h I, h 2 . . . . , hk}

belongs t o L~. F u r t h e r L~ h a s no m a x i m a l e l e m e n t . So we can c o n s t r u c t a section u in ~ w i t h u < s h a v i n g a f u n c t i o n g w i t h h-~ g in i t s l e f t class.

Consider a n y t in ~ w i t h

u < t < s . A s g belongs t o t h e left class of t, a n d

h~-<g (i = 1, 2 . . . lc),

k

t h e set f u n c t i o n ~ G~

i - 1

is t-singular. T h e set f u n c t i o n

H = ~ Gi

i = k + l

(16)

2 2 2 c . A . ROGERS A N D S. J . TAYLOR

h a s a c a n o n i c a l d e c o m p o s i t i o n , s a y

H ---- La lrJ'(t) ~- ~2rr(t) + H(t)a + H(~

w i t h r e s p e c t t o t h e s e c t i o n t. T h e set f u n c t i o n K = F i ~) + F ? ) + F~ ~)

is s - c o n t i n u o u s . So K is / - c o n t i n u o u s for e a c h l E L s. Since t < s, K i s / - c o n t i n u o u s for a t l e a s t one l in R~. H e n c e K is h y p e r t - c o n t i n u o u s . C o m p a r i n g t h e d e c o m p o s i t i o n

F = { H f ) + K } + H~ ~ ~ - - a J- u(t) + H(4 t) + a ,

w i t h t h e d e c o m p o s i t i o n of T h e o r e m l , we o b t a i n t h e i d e n t i f i c a t i o n F i ~ = H i ~ + K ,

F~t)= H(~ t),

k

F(, ') = H(, ') + ~ a , . t = l

I t n o w follows, as a b o v e , t h a t F(4 ~ is c o n t i n u o u s on t h e left a t t = s, a n d t h a t F(1 t) h a s a s i m p l e d i s c o n t i n u i t y - F ( ~ s) - F ( 8 s) on t h e left a t t = s, o r is c o n t i n u o u s on t h e left a t t = s in t h e case w h e n F(~)+ F(a ~)= O.

F u r t h e r , if i t is p o s s i b l e t o choose h in L~ so t h a t hc<h ( i = 1 , 2 , 3 . . . . ) t h e n F(lt) = F(lS) + ~(s) + ~(s) _{.L' 2} _{z ' $} _,

for all t w i t h u<~t<s.

C o m b i n i n g t h e a b o v e T h e o r e m 2.

F~ ~) = 0, ~ ' = 0, F', ~ = F(, ~),

r e s u l t s we see t h a t we h a v e c o m p l e t e d t h e p r o o f of Two n e w c o n c e p t s a r e n o w n e c e s s a r y .

D E F I N I T I O N . I / S E ~ and F E ~ , then F is said to have the exact C.dimeusion s, i~ F is s.continuous and hypo s-singular.

D E F I n I T I O n . A set /unction F o/ :~ is said to have a di//use C-dimension spec- trum, i/ there is no set /unction G o/ :~ with 0 <IGI <, IF[ having an exact C-dimension.

(17)

These definitions are intended to supersede the definitions given in section 6 of [1], which we now regard as being inappropriate. B u t the examples of section 5 of [1] are still relevant, and provide examples of set functions having exact C-dimension s, and of a set function having a diffuse C-dimension spectrum, on taking l~ to be any irreducible maximal scale of functions.

Tl~e following theorem characterizes the set functions, with a diffuse JC-dimension spectrum, in terms of their behaviour under decomposition.

THEOREM 3. A set /unction F o/ :~ has a di//use L-dimension spectrum, i/, and only i/, its decomposition

F = Fi ') + F ~ ~) + F ~ ~) + F~ ~),

/or t in [~, satis/ies one o/ the /ollowing equivalent conditions:

(1) F~ t) is continuous /or all t in •;

(2) F (t)~ --_F (t)=a 0 /or all t in if;

(3) F~ 0 is continuous /or all t in ft.

Proo/. I t follows immediately, from Theorem 2, t h a t the three conditions are equivalent.

If the condition (2) fails, there is a t in if, with, either

~ ( t ) ~.

F~t)~=0, or ~'3 ~--.

Then 0 < IF~ t)] + I.F(st)[ <~ IF(it) l + IF~t)l + (t)

~ ( t )

while IF~t)l + -~8 is t-continuous and hypo t-singular. Hence F does not have a diffuse E-dimension spectrum.

On the other hand, if F does not have a diffuse s spectrum, there is a t in ff and a G in ~ with

0<lal<l~l,

G being t-continuous and hypo t-singular. Now we can write

IFI~IGI+IHI,

where H = I F ] - ] G ] . Decomposing IF], IGI and IHI with respect to the section t, a n d comparing the two resulting decompositions of / F I, we have

[F~t)[ + ~.~(t)~ = iGht)[ + ~,8~-(~) + I H(~)[ + ]H (~)

~>la~)l+ a(o~ =IGI>0.

Thus the condition (2) fails. This completes the proof.

We are now in a position to state the main theorem of this section.

(18)

T ~ O R ~ M 4. Given any irreducible maximal scale s o/ /unctions o/ ~4 and any set /unction F in ~, there is a finite or enumerable sequence sl, s 2 .. . . o/ distinct sec- tions o/ E, and a decomposition

F = F (d) + F(~ ~') + F (~') + F (~') + F ('~) + ₃ ₂ ₃ _{- . . ~} (7)

where F <~) has a di//use C-dimension spectrum, F~ ~ is strongly s~-continuons and hypa s~-singular, and F~ ~) is s~-continuons and almost s~-singular, /or i = l, 2 . . . The set of the sections s~, and the decomposition (apart /rom the order o/ its terms) are uniquely determined by F.

Proo/. The result follows, m u t a t i s m u t a n d i s , from the proof of T h e o r e m 3 of [11].

The changes required are clear; in particular, it is necessary to work with the function II_F~t)ll+[]F(at)l[ of t h e section t i n ~r in place of t h e function IF(2 ~)] ( I 0 ) o f t h e real variable ~.

This t h e o r e m should be regarded as a more a p p r o p r i a t e version of T h e o r e m 6 of section 6 of [11]. I n the earlier version, the c o m p o n e n t s F~ ~) were lumped t o g e t h e r with the c o m p o n e n t F (d), with the result t h a t t h e c o m p o n e n t , said to have a diffuse E-dimension spectrum, did n o t deserve this description.

The r e m a i n d e r of this paper will be concerned with the f u r t h e r s t u d y of t h e decomposition of (7). W e shall investigate the t y p e s of sections s, for which the c o m p o n e n t s F(~ s) a n d F(s s) can be non-zero; we shall also show that, in certain circumstances, c o m p o n e n t s of the decomposition are isolated in a certain sense.

5. Sections in a scale o f functions

L e t E be an irreducible m a x i m a l scale of functions of ~ . L e t ff be the s y s t e m of sections of E. A set Q of functions of E will be said to form a right basis f o r a section s in ~r, if L~ is precisely the set of functions h of E, which satisfy h-<q for all q in O. N o t e t h a t every section s has the corresponding set R~ as a r i g h t basis. Also, if a finite set of functions is a right basis for a section s, t h e n t h e smallest of the functions of t h e set is the minimal element of R~, a n d t h e set containing just this one function is a right basis for s. Further, if a countable set, s a y ll, 12 .. . . , of functions of E forms a right basis for a section s with no finite r i g h t basis, the sequence Ul, u 2 .. . . , given b y

u ~ = m i n ( l l , l~ . . . l~} ( i = l, 2 . . . . ),

(19)

A D D I T I V E SET F U N C T I O N S I N E U C L I D E A N SPACE. I I 225 h a s no m i n i m a l e l e m e n t , a n d h a s a subsequence, vl, v~, ... say, w i t h

f o r m i n g

(F) (E) (N)

W e

fication, W e

Vl ~-V2>- . . . .

a r i g h t basis for s. T h u s t h e r e a r e t h r e e m u t u a l l y e x c l u s i v e possibilities:

t h e s e c t i o n s h a s a r i g h t basis consisting of a single f u n c t i o n ;

t h e s e c t i o n s does n o t h a v e a f i n i t e r i g h t basis, b u t does h a v e a n e n u m e r a b l e r i g h t basis whose e l e m e n t s f o r m a s t r i c t l y m o n o t o n i c sequence;

t h e section s h a s n o e n u m e r a b l e r i g h t basis, b u t does h a v e a n o n - d e n u m e r a b l e r i g h t basis.

c a n i n t r o d u c e a s i m i l a r d e f i n i t i o n for a left basis, a n d m a k e a s i m i l a r classi- b u t , as L~ h a s no m a x i m a l element, o n l y t h e t w o cases (E) a n d ~(N) occur.

s a y t h a t a section s is of t y p e (AB), if i t s l e f t h a n d b a s e s fall u n d e r t y p e (A) w i t h A = E o r N , a n d its r i g h t h a n d b a s e s fall u n d e r t y p e (B) w i t h B = F , E or N . So a p r i o r i a s e c t i o n s belongs t o one of six t y p e s . B u t we shall soon e l i m i n a t e t w o of t h e s e t y p e s .

F i r s t s u p p o s e t h a t s were a section of t y p e ( E F ) . L e t h be t h e m i n i m a l e l e m e n t of Rs a n d let v 1, v~ . . . . b e a left basis for s w i t h

vl"< v~'< ... ~ h.

B y t h e r e m a r k a f t e r T h e o r e m A, t h e r e is a n e l e m e n t k of • w i t h Vl -< V ~ ( . . . ~ k ~ ( h.

B u t n o w k can b e l o n g n e i t h e r t o L~ n o r to Rs. T h i s c o n t r a d i c t i o n p r o v e s t h a t t h e r e is no s e c t i o n of t y p e (EF).

A p r e c i s e l y s i m i l a r a r g u m e n t shows t h a t t h e r e is no s e c t i o n of t y p e ( E E ) . W e are left w i t h t h e sections of t y p e s (NF), (EN), (NE) a n d (NN); i t is e a s y t o c o n s t r u c t e x a m p l e s of sections of e a c h of t h e s e t y p e s . W e n o w go o n t o discuss e a c h of t h e s e t y p e s s e p a r a t e l y . H o w e v e r t h e t y p e s ( N F ) a n d (NN) h a v e a p r o p e r t y in c o m m o n w h i c h h a s i m p o r t a n t consequences, a n d we c o n s i d e r t h i s first. I n b o t h cases t h e s e c t i o n h a s no e n u m e r a b l e l e f t b a s i s a n d f u r t h e r R~ h a s no c o u n t a b l e co- f i n a l sequence i n l~, i.e. t h e r e is no c o u n t a b l e sequence rl, r 2 . . . . i n l~ w i t h

. . . ~( r m ' ~ . . . ~ r2 ~ rl,

such t h a t t h e m e m b e r s of R~ are t h o s e r in IZ such t h a t r m ' ~ r for s o m e i n t e g e r m.

T h i s l a s t p r o p e r t y follows, in t h e case of a s e c t i o n of t y p e (NF), b y t h e a r g u m e n t u s e d a b o v e t o e x c l u d e sections of t y p e ( E F ) .

(20)

226 C . A . R O G E R S A N D S . J . T A Y L O R

We f o r m u l a t e the results for sections of these two t y p e s in terms of 'isolated c o m p o n e n t s ' .

~'(~) + F ~) is said to be isolated, when D E F I N I T I O N . For a n y s e r f , the component ~.~ 8

there are sections u, t o/ ff with u < s < t such that, i/ G is any s e t / u n c t i o n o / ~ , which is u-continuous and t-singular and satis/ies IGI~<JF[, then G is o/ exact E-dimension s and satis/ies I GI < [Fh ~) + F(~)I.

L E ~ M A 1. Suppose s is a section o/ F~ o/ type (NF) or (N-N). T h e n the component F~ ~) + F~ ~) is isolated.

Proo/. I f s is such a section, t h e n we k n o w there is no sequence rl, r~, ..., in R~, such t h a t a n y h in l: satisfying h<r~, i = l, 2, ... c a n n o t be in R~; a n d there is no countable sequence lx, l~ . . . . in Ls, such t h a t a n y h in ~: satisfying l~<h, i = 1,2, ...

c a n n o t be in L~. U n d e r these circumstances, the r e m a r k s m a d e in t h e proof of Theo- r e m 2 apply, a n d there will be sections l, r in ff with

l - ~ s ~ r such t h a t F~ t) = F(1 s) + ~'(s) + F (~) _• _{3 ~}

F(} ) = 0, F(~ t) = 0, p(,t) = F(p for l ~< t < s, a n d F i t) = F 7 ), F(2 t, = 0, F(8 t) = 0,

F(4t) = F(~) + F~s) + F(4 ~) for s < t <~ r.

Consider a n y G in :~, which is /-continuous a n d r-singular, a n d which satisfies I G I < I F I . Since l a l is /-continuous, a n d

Iol < IFI = IFi" + Fh" + F~"I + IF(4"/,

where IF(4 t)j is /-singular, it follows t h a t

Iol < IF, (`' + ~(') + F (',~ - ^{~ , ~} 3 j - I F T ) I + I F ( @ + F T I 9

is)

F u r t h e r , since IGI is r-singular, a n d

-- ..x.. 1 7 - . [ ' 2

.F~'I+IF(,'[,

where I F(~ ') +F(2')+ F~)[ is r-continuous, it follows t h a t

Io1~< IF(PI = IF~S' + F(~8'I + IF(P[ 9 <9)

(21)

A D D I T I V E S E T ] ~ U N C T I O N S I N E U C L I D E A N S P A C E . I I 227 Since

IFis) I

a n d [F(4s) I are complementary it follows from (8) and (9) t h a t

~t' 2 T ~ , 3 .

This implies t h a t G is of exact E-dimension s. Consequently the component F ~ ) + F~ ~) is isolated.

6. Sections of type (NF)

L e t I~ be an irreducible m a x i m a l scale of functions of :H and let s be a section of ~ of t y p e (NF). L e t h be the minimal element of Rs. Then s is uniquely determined b y h, and we write s = s(h). We recall t h e definitions of the sets C(h), C* (h), S(h) and S*(h) given in [1].

1. C(h) is the class of set functions F of :~, which are h-continuous; t h a t is, those set functions such t h a t F(E)=0, for a n y E in B of zero h-measure.

2. $(h) is the class of set functions F of :~, which are h-singular; t h a t is, those set functions, for which there is some E 0 in B with zero h-measure, such t h a t

F ( E ) = F ( E f3 E0) for all E in B. (10)

3. C*(h) is the class of set functions F of :~, which are strongly h-continuous;

t h a t is, those set functions such t h a t F(E)= 0, for a n y E in B of a-finite h-measure.

4. S*(h) is the class of set functions F of 5, which are almost h-singular; t h a t is, those set functions, for which there is some E 0 in B with a-finite h-measure, such t h a t (10) holds for all E in B.

I n [11] we proved t h a t C(h) and S(h) and C*(h) and $*(h) are pairs of com- p l e m e n t a r y bands in :~. We r e m a r k here t h a t the results could have been obtained a bit more directly, b y use of results of H a h n and Rosenthal [6], rather t h a n b y use of the theory of bands.

I n the special case, when ~ is dense, we can establish connections between the four bands associated with h and the six bands associated with s(h), b y using the results of [13].

LEMMA 2. 1[ ~ is dense, h EF~ and s(h) is the section o / s

of

type (NF)associated with h, we have

C(h) = C~h), $(h) = $~h),

C * ( h ) = Cs(h) = * Cs(h), * *

$ * ( h ) = * * *

(22)

Proo/. Comparing the definitions of C~(h) and C(h), we see immediately t h a t Cs(h) ~ C(h).

So consider a n y F in Cs(h). L e t E be a n y set of ]g with zero h-measure. Then, by a result of Besicovitch {see L e m m a 6 of [13]}, there is a function i of ~ with j ~ h and j - re(E) = O.

Since iZ is dense, it follows from (Pc) t h a t there is a g in l~ with g'<h] -1. Then, b y (Pa), the function l = ~ J -1 is in s and satisfies

Thus we m u s t have 1 in L~(h), and

j < l < h .

l - m(E) = O.

As F is in Cs(h), this implies t h a t F ( E ) = 0. Hence F is in C(h). Consequently Cs(h) = C(h).

I t follows t h a t the complementary bands Ss(h) and S(h) m u s t coincide.

Cs(h), C,(h) and C*(h), we see immediately t h a t Comparing the definitions of * **

9 f ' * * C / " *

Now consider a n y F of C*(h). B y Theorem 5 of [13], there is a function g of with h-<g and F E C * ( g ) . Since 1~ is dense, it follows t h a t there is an l in 1~ with

h<l.<g.

Then 16 R~(h), and F 6 C* (1) c ~s(h) ~s(~).

~ * * c 1 " * c C * ( h ) .

Thus C*(h) c~s(h) ~s(h)

Hence C* (h) = Cs(h)* = Cs(h)** ⁹

Consequently S* (h) = Ss(h)* = Ss(h)**, as well. This completes the proof.

The following theorem provides considerable information a b o u t the components in Theorem 4 corresponding to sections of t y p e (NF) in a dense scale.

T H E O R E M 5. Let s be a section o/ type (NF) in a dense irreducible maximal

scale s and let h be the minimal element in Rs. Then, in the decomposition o/ a /unc- tion F o/ ~, provided by Theorem 4,

F(: ) = O,

(23)

A D D I T I V E S E T F U N C T I O N S I N E U C L I D E A N S P A C E . I I

E(~) is h-continuous and almost h-singular, and has a representation ₃

229

[ "

F(3 s) (E) = J En S /(x) d h - re(x),

where /(x) is a Borel-measurable point /unction which does not vanish on S, and S is a Borel set o/ a-/inite h-measure. Further the component F(3 ~) is isolated.

Proo/. Using L e m m a 2, we have

F(~ ~) e * Cs = Cs ** and F(2 ~) E Ss**.

Hence F(~ s) = O.

Using L e m m a 2 repeatedly, we have

F i ~) + F~ ~) e r = C* (h),

~(~) e C~ n S* = C(h) n S* (h), 3

F(~ ~) ~ S~ = S ( h ) .

So ~'3~(s) is h-continuous and almost h-singular. Further, comparing the decomposition F = F i ~) + F~ ~) + F(3 ~) + F(~ ~)

with t h a t of Theorem 7 of [13], we identify F(~ ) with the second component of t h a t theorem, so t h a t it has a representation of the required form. Alternatively the representation m a y be obtained (as in Theorem 7 of [13]) b y a direct application of t h e Radon-:Nikodym theorem.

The fact t h a t F(~ ) is isolated follows from L e m m a 1.

7. Section of type (NE)

L e t s be an irreducible m a x i m a l scale of functions and let s be a section of C of t y p e (NE). I n order to obtain significant results for sections of this t y p e we shall h a v e to assume t h a t l: is strongly dense. B u t first we prove a lemma.

LEMMA 3. Let hi, h 2 .. . . be a sequence o/ /unctions o/ ~ with hl >-h~ >- ....

and let E be a F,.set with h ~ - m ( E ) = O, i = l, 2 . . . T h e n there is a /unction g o/

with g-<h~, i = 1 , 2 . . . . , and g - m ( E ) = O .

(24)

230 C . A . R O G E R S A N D S . J . T A Y L O R

Pro@

W e first consider the case of a b o u n d e d closed set E . Since E is c o m p a c t , each covering of E b y o p e n sets contains a finite covering of E. Since

h i - m(E)= O,

for i = 1, 2 . . . it follows t h a t for each p a i r of positive integers i, ?" we can choose finite s y s t e m of open c o n v e x sets

C~t.j), fv(i.j) / ~ ( i . / ) ' J 2 ~ . . . ~ v / j ,

where # = # ( i , j), covering E, a n d such t h a t

ht (d( C<ri'J)) ) < 1/],

r = l

a n d m a x

d(C~ ira) < 1/i.

l~<r~<~

W r i t e d <~'j)= rain

d(C~id)).

l ~ r ~ < #

W e n o w choose i n d u c t i v e l y a sequence of functions k 1, k 2 .. . . . a sequence of integers Jl, ]a . . . a n d two sequences of real n u m b e r s ~1, ~ . . . a n d */1, */3 . . . so t h a t :

(1)

kl(x)=hl(X )

for all x>~O, a n d A = j ( 1 ) = I ;

(2) w h e n r/> 1 a n d

k,(x)

a n d j ~ = j ( r ) h a v e been chosen, ~r is chosen so that.

h,+x(x)>h,(x)

for all x w i t h O < x < ~ , , a n d also ~T<dr

(3) w h e n r~> 1 a n d

k,(x), jr

a n d ~, h a v e been chosen, ~/, is chosen, with 0 < ~ , < ~ , , so t h a t h , + l ( ~ ) = h r ( ~ , ) ;

(4) w h e n r ~ > l a n d

k,(x), iT, ~,

a n d */~ h a v e b e e n chosen,

k,+l(X)

is defined b y

kr+l(x)=kT(x)

for ~T<x,

k~+l(x)=kr(~,)=hr+l(~r)

for ~ < x < ~ , ,

kr+l(x)=h~+l(x)

for 0 < x ~ < ~ ;

(5) w h e n r>~ 1 a n d

kr(x),

?',=j(r), ~ , ~, a n d

k~+l(x)

h a v e been chosen, j~+l = ~ ' ( r + 1) is chosen to be a n integer so large t h a t

, ( 1 )

- - < m i n ~/r,

?'r+l r - - ~ Consider t h e m e a s u r e function

g(x)

defined b y

g ( x ) = lim kr ( x ) .

I t is clear t h a t

g(x)>~hr+l(x)

for 0 < x < ~ T + l .

(25)

A D D I T I V E S E T F U N C T I O N S I N E U C L I D E A N S P A C E . I I 231 A s h r + l ~ h T , i t follows t h a t g ~ h r for r = l , 2, . . . . F u r t h e r , for each i n t e g e r r>~2, t h e set of o p e n c o n v e x sets

C~ r'j<~>>, C <~'j(T'2 , ...,

C~r'J~>>,"

w i t h /x =Ix(r, j(r)), f o r m s a covering of E w i t h

d/~(r'j(r))~ ~* ~ 1 < ? J r - l )

m a x ~xvn ^I

l~<n~</x

m i n d(C~ "j<~))) = d r > ~ .

l~<n~#

T h u s ~. g(d(C(~.J(r))))= ~ k,.(d(C~,~(r))))= ~ hr (d(Cn <,.,i(,.)) )) < l/j(r) < l / r .

n = l n =1 n =1

H e n c e E is o f zero g-measure.

N o w s u p p o s e t h a t E is a F , - s e t . T h e n we h a v e

E = ~. UE,

J=l

for s o m e sequence Et, E 2 . . . . of b o u n d e d closed sets.

A s h~ - m(E~) < h~ , m(E) = O,

we can, for e a c h j, choose a f u n c t i o n gj of ~H w i t h gj'<h~ for i , j = l , 2 . . . . a n d g j - m ( E j ) = O for j = 1 , 2 . . .

Then, b e c a u s e ~ is s t r o n g l y dense, we c a n choose a f u n c t i o n g of :H w i t h gj'<g'<hi for i, j = 1, 2, . . . .

T h e n g - m(E) <~ ~= g - m(E~) <~ ~= g j - m(Es) = O.

j = l j = l

T h i s c o m p l e t e s t h e proof.

Remark. T h e p r o b l e m of e x t e n d i n g t h i s l e m m a t o cover a n y B o r e l set E seems t o us t o b e o p e n a n d i n t e r e s t i n g .

W e c a n n o w e a s i l y p r o v e

L~.MMA 4. I1 s is strongly dense, and s is a section o/ s o/ type (NE), then C~ = C* = C ~ , ** L = S~ *=S**.

(26)

2 3 2 c . A . R O G E R S A N D S. J . T A Y L O R

P r o o / . I t suffices t o p r o v e t h a t

**S, S*=**

⁼ ^{$ ~}** ^,

i n d e e d , since S s = S* ~, ,

i t suffices t o p r o v e t h a t $ * * c $ s .

Ss 9 L e t v v be a r i g h t basis for s w i t h C o n s i d e r t h e n a n y f u n c t i o n F of :~ in ** v 1 . . . .

v l > - v ~ > - . . . .

As F E$**, we can choose sets E l , E 2 . . . . o f B, such t h a t F ( E ) = F ( E N E i ) for all E of B,

v~ - m ( E i ) = O,

for i = l , 2 . . . T h e n E o = A ~ = I E i is a set of B w i t h F ( E ) = F ( E N E o ) for all E of B, a n d vt - m ( E o) = 0 (i = 1, 2 . . . . ).

Since F is a n a d d i t i v e set f u n c t i o n of 5 , t h e r e is a n F a - s e t G, c o n t a i n e d in E o, w i t h

IFI(O)=IFI(Eo).

Since vi - m ( G ) = 0 (i = 1, 2, . . . ) ,

i t follows f r o m L e m m a 3, t h a t t h e r e is a f u n c t i o n g of :H, w i t h g - < v i ( i = 1, 2 . . . . ),

a n d 9 - m ( G ) = O.

Since i~ is s t r o n g l y dense, t h e r e is a n 1 of E w i t h g-<l-<v~ ( i = 1, 2 . . . . ).

Since v 1, v~ . . . . is a r i g h t basis for s, i t follows t h a t 1EL~. Now, we h a v e 1EL~, l - m ( G ) < g - m ( G ) = 0,

a n d , for e a c h E of B,

I F ( E N G) - F ( E ) I ~< ] F ( E N Eo) - F ( E ) ] + I F ( E N Eo) - F ( E N G) I

< [F(E

n Eo) - F ( E ) I + [I F I (E0) - I FI (o) l = 0.

(27)

Thus F is concentrated on the set G of B, with zero /-measure, with l in JLs, and so F E $s. Consequently

S** c S . a n d the required result follows.

As an immediate corollary of this l emma, we have

THEOREM 6. Let s be a section o/ type (NE) in a strongly dense irreducible maximal

~cale F~. Then, in the decomposition o/ a /unction F o/ ~, provided by Theorem 4,

F(2 ~) = F(~ s) = O.

8. Sections of type (EN)

Let I: be an irreducible maximal scale of functions and let s be a section of s ,of type (EN). I n order to obtain significant results for sections of this type, as for

~ections of type (NE) we shall have to assume t h a t the scale I~ is strongly dense.

B u t first we need two lemmas.

Our first lemma is a result which was proved, but not explicitly stated in [13]

it introduces the concept of a set function G, which is uniformly Lip h(t), i.e. a set function G ot :~, such that, for some constants K and (~ > 0 , we have

]GI (I) <~ gh(d(I)), for all sub-intervals I of I 0 with d(I)< (~.

LEHMA 5. I / h E ~ and F in :~ is h-continuous, then there is a sequence E~,i= 1,2 ...

v/ disjoint sets o/ B, such that the set /unctions

G~(E)=F(ENE~) ( i = 1 , 2 . . . . ), are each uni/ormly Lip h(t), and

t = l oo

lIFIl= 5111G II.

Proo/. The result follow immediately, from part (d) of the proof of Theorem 1 of [13], on taking

E i = K ~ - K ~ - I ( i = 1 , 2 . . . . ).

LEHMA 6. Let hi, h~, ... be a sequence o/ /unctions o/ "~ with high2-< ...

16 - 632918 A c t a mathematica. 109. I m p r i r a ~ 17 j u i n 1963.

(28)

234 C . A . R O G E R S A N D S. J . T A Y L O R

and let F be a set /unction o/ 7, which is h~.continuous /or i = 1, 2 . . . Then there i~

a /unction g o/ ~ , with h~-<g /or i = 1, 2 . . . such that F is still g.continuous.

Proo/. L e t e be given with 0 < e < 1. B y L e m m a 5, for each ], we can choose sequence E} j), i = 1, 2, ... of disjoint sets of B, such t h a t the set functions

G~ j) ( E ) = F ( E N E} j)) (i = 1, 2, ...), are each u n i f o r m l y L i p hs(t), a n d

Ila ')ll'

F = ~. G~ j), I I F I I - - =

for j = 1, 2 . . . F o r each ], we choose a n integer k(]), such t h a t

i ~ k U ) + t

k(]) k(])

a n d t h e n write Qj = I.J E} j), H j = ~ G~ j).

i ~1 i = 1

T h e n H j belongs to 7, a n d is u n i f o r m l y L i p hi(t). F u r t h e r

a n d

for ] = 1, 2 . . .

W r i t e K = [~ Qj,

j = l

T h e n H belongs to 7 , a n d satisfies

oo

H ( E ) = F ( E ~ K),

for all E in B. As K c Qj, it follows t h a t H is u n i f o r m l y L i p hi(t), for ~'= 1, 2, . . . . A l s o 2~

I I F - H I I < I ] F - H I I I + ~ IIH,+I - H , II< l ~ e 9

J = l

Since H is u n i f o r m l y L i p hi(t), for j = 1, 2 . . . a n d h~"< h~'< ....

we can use t h e usual technique to f o r m a function g of ~ , with h~ < h~-< . . . < g,

such t h a t H is still u n i f o r m l y L i p g(t).

(29)

A D D I T I V E S E T F U : N C T I O N S I:N E U C L I D E A N S P A C E . I I 235 R e p e a t i n g this construction, for different values of s, we can choose a sequence H 1, H 2 .. . . of functions of :~, with

IIF-HjlI<2-1 (i=1, 2, ...),

a n d a sequence of functions gl, g2 . . . . of ~/, with h i ' g 1 for i , ] = 1 , 2 . . .

such t h a t H i is u n i f o r m l y Lip gt(t), for ~= 1, 2 . . . N o w choose g in ~/, with h~-<g'<gj for i, j = 1, 2 . . .

E a c h function H i is u n i f o r m l y Lip g(t). F u r t h e r

F -I+

i=1

with ~ IIHj+I- Hjl I

i=1

convergent. Hence, b y T h e o r e m 1 of [13], /~ is g-continuous, as required.

L E M M A 7. 1/ s is strongly dense, and s is a section o/ s o/ type (EN), then

=C~=

CS , S ~ = S ~ = S s 9

Proo/. As Cs ~ Cs* ~Cs**,

it suffices ~o prove t h a t C** ~ C ~ . Suppose t h e n t h a t F is a n y set function of C~.

L e t v I, v 2 .. . . be a left basis for s with

vl ~ v~-< . . . .

T h e n F is v~-continuous, for i = 1, 2 . . . So, b y L e m m a 6, there is a function g of

~/, with

v~<g ( i = 1 , 2 . . . . ),

such t h a t F is g-continuous. Since s is strongly dense, there is a function l of s with v ~ < l < g ( i = 1 , 2 . . . . ).

As s is of t y p e (EN), t h e function 1 belongs to R~. F u r t h e r /r is /-continuous. T h u s F is h y p e r s-continuous. This shows t h a t C ~ C * * , a n d the proof is complete.