CONCENTRATED AND RARIFIED SETS OF POINTS.

CONCENTRATED AND RARIFIED SETS OF POINTS.

BY

A. S. BESICOVITCH

T r i n i t y College, CAMRRIDG~:.

C H A P T E R I.

Concentrated Sets.

I have arrived at t h e definition of c o n c e n t r a t e d sets f r o m the following two similar problems:

P r o b l e m I. What are the linear sets whose measure with respect to any con- tinuous monotone function q~ (1) is zero?

P r o b l e m I I . What are the linear sets on which the variation of any con- tinuous monotone function is zero?

C o n c e n t r a t e d sets are defined as follows:

A non-enumerable set of points E is said to be concentrated in the neigh- bourhood of an enumerable set H i f any open set containing the set H contains also the set E with the exception of at most an enumerable set of points.

w 1. M e a s u r e w i t h r e s p e c t to a function. L e t ~ (l) be a positive i n c r e a s i n g f u n c t i o n s a t i s f y i n g defined f o r 1 ~ o a n d such t h a t q~ ( + o) ~ o and let E be a l i n e a r set of points. D e n o t e by I ~ ~l~ any sequence of intervals c o n t a i n i n g the whole of t h e set E (li d e n o t i n g b o t h an i n t e r v a l a n d its length). D e n o t e by m ~ E the l o w e r b o u n d of the sum

37--33617. A c t a mathematica. 62. Imprim6 le 12 f@vrier 1934.

290 A . S . Besicovitch.

f o r all possible sets I , f o r which t h e l e n g t h of t h e l a r g e s t i n t e r v a l does n o t exceed ~. T h e limit of m~E, as ). t e n d s to zero, we shall call t h e e x t e r i o r

~0-measure of E , or t h e e x t e r i o r m e a s u r e of E w i t h r e s p e c t to ~.

O b v i o u s l y t h e qg-measure of a n e n u m e r a b l e set a n d t h e v a r i a t i o n of a con- t i n u o u s m o n o t o n e f u n c t i o n w i t h r e s p e c t to an e n u m e r a b l e set a r e b o t h zero. N o w we k n o w t h a t a B - m e a s u r a b l e set E e i t h e r is e n u m e r a b l e or else c o n t a i n s a per- f e c t s u b s e t E'. I n t h e first ease t h e v a r i a t i o n of a n y m o n o t o n e c o n t i n u o u s func- t i o n on E is zero. I n t h e second case E ' m a y be t r a n s f o r m e d into a n i n t e r v a l by m e a n s of a m o n o t o n e c o n t i n u o u s f u n c t i o n f ( x ) . T h e l e n g t h of such a n inter- val is obviously t h e v a r i a t i o n of f ( x ) w i t h r e s p e c t to E ' , a n d h e n c e t h e v a r i a t i o n w i t h r e s p e c t to E is positive. H e n c e

The only B-measurable se~ whose rariation with respect to any eonli~mous function is zero are enumerable sets. 1

W e shall shew t h a t

I n the general case there are ~wn-enumerable sets whose measure is zero n, ith respect to any fu~ction 9~,

b u t t h e p r o o f will be b a s e d on t h e a s s u m p t i o n t h a t t h e p o w e r of t h e conti- n u u m is ~r

w 2. W e shall first p r o v e some p r e l i m i n a r y t h e o r e m s .

T h e o r e m 1. A necessary aJ~d sufficient condition for a li~ear set E to be of measure zero with respect to any f~l~ctio~ qD (l) is that give~ any decreasing sequence of positive numbers

~i, i~--- 1 , 2 , . . . ,

however rapidly te~ding to zero, there always exists a seque~we I of intervals (x~, x'/), x ' / - - x ~ <= li, containing the whole of the set E .

to

The condition is necessary. F o r let E be a set of m e a s u r e zero w i t h r e s p e c t a n y f u n c t i o n a n d li ( i ~ I , 2 , . . . ) a s s u m e t h a t t h e r e exists a n y d e c r e a s i n g

1 L. C. YOUNG. ~Note on t h e t h e o r y of m e a s u r e . Proceedings of t h e C a m b r i d g e P h i l o s o p h - ical Society. Vol: X X V I . P a r t 1.

L. C. Y o u x ' o considers v a r i a t i o n of a f u n c t i o n on a given set as m e a s u r e of t h i s set. T h e definition we are u s i n g is n o t so general a n d it is n o t o b v i o u s t h a t if v a r i a t i o n of s o m e f u n c t i o n s on a given set is p o s i t i v e t h e n also m e a s u r e of t h e set w i t h respect to s o m e f u n c t i o n is positive.

We s h a l l consider in w 3 B - m e a s u r a b l e s e t s of m e a s u r e zero w i t h respect to a n y function.

Concentrated and Rarified Sets of Points. 291 sequence of positive numbers. Define an increasing f u n c t i o n ~ (1) satisfying the conditions

~0 (li) :'=- -. I f o r i -~ I, 2, . . .

Obviously 9 ( + o ) - - o , a n d c o n s e q u e n t l y m,p E is defined and by h y p o t h e s i s is equal to zero. T h e r e f o r e t h e r e exists a set I of intervals (x$, x'/), x ~ ' - - x ~ = l;, d e c r e a s i n g in length, c o n t a i n i n g t h e whole of t h e set E a n d such t h a t

T h e n f o r any n

~ , ~ (1;) < , .

> ~ ~ (l;)

~ (1,'~),

i = 1

i. e.

~t~ < -

%s

n

1,, < 1,, t

a n d t h e condition is t h e r e f o r e satisfied.

The condition is also sufficient. F o r suppose a set E satisfies this c o n d i t i o n : t h e n g i v e n a f u n c t i o n T(1) s a t i s f y i n g the usual conditions and two positive c o n s t a n t s ~, ~ we can always find a sequence of positive n u m b e r s l~ < ~ such t h a t

~ (1;) <

a n d t h e n a sequence I of intervals (x'i, x~'), x~'--x~ ~-li, containing/~', which shews t h a t m e a s u r e of E w i t h respect to 99 (1) is zero.

3. W e can now solve the problem m e n t i o n e d in t h e f o o t n o t e to w 1.

T h e o r e m 2. The only B-measurable sets of measure zero with respect to any function are enumerable sets. For any perfect set P there exists a function ~ (l)

with respect to which the measure of the set is positive.

On a c c o u n t of 2 the t h e o r e m will be p r o v e d if we prove t h a t given any p e r f e c t set P t h e r e exists a sequence of positive n u m b e r s l~, n = I, 2 , . . . such t h a t no set of intervals of lengths l~, l ~ , . . , can cover t h e whole of t h e set P.

Obviously the set P c a n n o t be covered with one i n t e r v a l of a r b i t r a r i l y small

292 A . S . Besicovitch.

length. L e t ).~ be the lower b o u n d of the l e n g t h of an i n t e r v a l which covers t h e whole of P (evidently).~ is t h e distance b e t w e e n the e x t r e m e points of P).

T a k e f o r ll an a r b i t r a r y positive n u m b e r less t h a n ~l.

I t is also easy to shew t h a t t h e set E c a n n o t be covered with one i n t e r v a l of l e n g t h l~ a n d a n o t h e r i n t e r v a l of a r b i t r a r i l y small length. Otherwise let two intervals

(x,, x , + l~), (y,,, y , + 12,,)

where 4 , , - ~ o , as n - ~ o c , cover t h e whole of P for any ~. L e t ( x , y ) b e a l i m i t p o i n t of (x,,yn). T h e n f o r any ~ t h e two intervals

(x -- ~, x + ll + e), ( y - - e , y + ~)

cover P . C o n s e q u e n t l y the set P is included in t h e set consisting of t h e i n t e r v a l ( x , x + l~) a n d of the p o i n t y. B e i n g a p e r f e c t set, P has no isolated p o i n t and t h u s is c o n t a i n e d in t h e i n t e r v a l (x, x + l~), which is impossible.

L e t E.o be the l o w e r b o u n d of t h e l e n g t h of an i n t e r v a l which t o g e t h e r w i t h a n o t h e r i n t e r v a l of l e n g t h l~ covers the whole of t h e set P . D e n o t i n g by 4 a positive n u m b e r less t h a n ).~ we have t h a t no pair of intervals of l e n g t h s l~, 4 can cover the whole of the set P. Similarly we can find/3 > o such t h a t no t h r e e intervals of l e n g t h s

11, [~,l~

we arrive at a sequence of positive n u m b e r s ll, 4, 9 9 9

such t h a t no finite set of intervals whose l e n g t h s are r e p r e s e n t e d by different n u m b e r s of the sequence, can cover the set P. T h e n by the H e i n e - B o r e l t h e o r e m no infinite set of intervals of l e n g t h s

ll, 4 , 9 ..

can cover the set P , which proves t h e t h e o r e m .

w 4. T h e o r e m 3. I f the measure o f a set E is zero w i t h respect to any f u n c t i o n q~ (1), then the variation o f any monotone continuous f u n c t i o n f ( x ) on E is" zero.

D e n o t e by 0 ( f ( x ) , l ) the u p p e r b o u n d of t h e oscillation of f ( x ) on any i n t e r v a l of l e n g t h 1. Obviously

O{f(x), 1) ~ o ,

Consequently, given ~ > o we can always find a decreasing sequence of positive n u m b e r s l~, i = I, 2 , . . . such t h a t

Concentrated and Rarified Sets of Points. 293

F, o {f(x), l,} < , .

i = l

On t h e o t h e r h a n d , by T h e o r e m I the set E can be included in a set of intervals

(x~,x'/),x~'--x'~<l~.

Vf.E

f(x)

V/ E <= ~ { f (x;')

<= o { f ( . ) , l,}

which proves the theorem.

A f t e r these p r e l i m i n a r y t h e o r e m s have been p r o v e d we can pass to the main problem of this note, i.e., ~o t h e p r o o f of the existence of n o n - e n u m e r a b l e sets of m e a s u r e zero with r e s p e c t to a n y f u n c t i o n 9 ( l ) . I t can be easily shewn t h a t c o n c e n t r a t e d sets, if t h e y exist, are of m e a s u r e zero with respect to a n y f u n c t i o n 9(1). F o r let E be a set c o n c e n t r a t e d in the n e i g h b o u r h o o d of a set

~ (n==I 2 .) and {ln} an a r b i t r a r y sequence of positive numbers. By

~. X n I ~ , 9 9

I I 1,2,~)contains all points of E the definition the set of intervals ( x n - - 2 l~, x,~ + 2

except at most an e n u m e r a b l e set of them, which can be included in intervals of l e n g t h l ~ , l ~ , . . . Consequently by T h e o r e m I t h e set E is of m e a s u r e zero with respect to an 3 ' function.

T h u s our problem reduces to the p r o o f of the existence of c o n c e n t r a t e d sets.

w 5. This p r o o f will be based on t h e existence of a transfinite sequence {~i) of positive d e c r e a s i n g f u n c t i o n s which will be called a f u n d a m e n t a l sequence.

W e say t h a t a f u n c t i o n f ( n )

is ultimately greater than (greater than or equal to) another function

f ( n ) ~- g (n)

(f(n).= g (n)),

if t h e r e exists a n u m b e r % such t h a t

294 A . S . Besicovitch.

f(u) > g (n) (f(n) >_- g (.)) f o r n > no.

Definition. A sequence ~ qDi (n)i, where i takes all values less than %, % being the first transfinite number of power ~r is called a fundamental sequence i f any :positive function f ( n ) ultimately greater tha~, all ~i(n) except at most an enumerable set

of them.

T h e existence of such a sequence can be established in the following way.

W e first o r d e r t h e set of all positive d e c r e a s i n g f u n c t i o n s into a sequence off (n) i t a k i n g all values less t h a n to 1.

W e t h e n define

~, (n) -~-fl (n)

q~, (n) = min {qg~ (n),f~ (n)}

9% (n) ~-- m i n ' , q~_. (n),fa (n)}

a n d so on.

F o r any transfinite n u m b e r i + I of the f i r s t class we p u t

~0i+ 1 (1~) = m i n

(r

F o r a n y t r a n s f i n i t e n u m b e r i of the second class we first define a sequence kt (1 < ~Oo, co o being t h e first transfinite n u m b e r of p o w e r too) of increasing transfinite n u m b e r s such t h a t

lim kt = i.

T h e n we define ~i(1) f o r any l by t h e e q u a t i o n

~i (l) ---- min {%. (l), ~kl (/), 9 9 ~'~ (/)}.

I n this way t h e sequence

is defined, where i 'takes all values less t h a n r T h e sequence possesses the following properties:

(i) For any i < ~o 1

f , (.) > (.) (ii) A n y 9Di (u) is a decreasing function.

Concentrated and Ratified Sets of points. 295 (iii) I f i < k, then

q,,(,,)

The properties (i), (ii) are obvious. I n order to prove (iii) we observe t h a t

~, (n) *= 9~ (n) *__ ~ ( n ) . . .

Suppose t h a t these inequalities have been proved for all indices less t h a n i.

Then if i is a transfinite n u m b e r of the first class, we have

by the definition of ~ ( , , ) . I f i is of the second class, then i = lira kl

and again from the definition of 9~/(n) we conclude t h a t

for all l, and consequently

for a n y j < i .

(iv) To any transfinite n u m b e r i < % corresponds an i ' > i and less than to1, such t h a t

For, all decreasing functions having been e n u m e r a t e d we have

~2 r (') = ~ ' (n)

where i' must be g r e a t e r than i since ~i (n) >-f/' (n) .

~, (~) ~ ~,, (,).

Let now g (n.)be any positive function.

Define f(n) by the condition f(,~) = rain g (,n)

f(n) is a decreasing function and consequently

f(n) = j ~ (,,), ; < o,, and thus

f(n) ~- q~, (n)

and as there exists i ' > i such t h a t ~(n)~-T~, (n) we have t h a t

a s .~, (,,) ~ ~,, (,,) we have

f o r I ~ m = - < ~ .

9.96 A . S . Besicovitch.

f ( n ) > ~0j (n) f o r all j ~ i'

i. e., f o r all j e x c e p t an e n u m e r a b l e set of values. T h u s t h e sequence ( ~ , ( n ) )

is f u n d a m e n t a l .

w 6. L e m m a . Let ~x~*j, (~ < COo), be a set of poi~ts everywhere dense, ~ yn

any enumerable set of points and f ( n ) any positi~:e Ju~etion. There always exists a poi~t a &:brerent from all y~ and such that the inequality

[ a - - x . [ < f ( , , )

is satisfied j b r infiMtely ma~y values o f n.

T a k e an a r b i t r a r y positive i n t e g e r ~h a n d the i n t e r v a l ( x , ~ - - ~ t , x , , + ~), where ~a < f ( n ~ ) and is so small t h a t the i n t e r v a l does n o t c o n t a i n any of the n u m b e r s Yl, Y ~ , . . . , Y , , different f r o m x,,. T a k e now an i n t e g e r ~ > J q a n d a positive n u m b e r ).e < f ( n e ) such t h a t the i n t e r v a l ( x , . - ) . ~ , x~: + ).~) is c o n t a i n e d in t h e i n t e r v a l (x,,--).l, x,~ +)-l) and does n o t c o n t a i n any of the n m n b e r s y~, y,_, .. . . . yn: different f r o m xn~, a n d so on.

I n t h i s way we arrive at a sequence of intervals

(x, i - Xi, x,, i + ).i) i -= I, z, . . . < ~o o

each of t h e m being i n t e r i o r to t h e preceding one. A n y p o i n t a i n t e r i o r to all these intervals (if ~i--~ o t h e r e is only one such point) is different f r o m all y,~

and satisfies t h e conditions

I -x,,I < z;

< f ( n i ) f o r i = I , 2 , . . .

which proves t h e lemma.

w 7. W e shall now L e t ~i (n), (i < oq), be

c o n s t r u c t a c o n c e n t r a t e d set of points of p o w e r ~1.

a

e v e r y w h e r e dense set of points.

such t h a t t h e i n e q u a l i t y

f u n d a m e n t a l sequence of f u n c t i o n s and ~ x,~ j an T a k e a point cq different f r o m all points x,~ and

161 - I < (')

is satisfied f o r infinitely m a n y values of ~. T h e n choose a p o i n t a.~ different f r o m all xn a n d f r o m a I a n d such t h a t the i n e q u a l i t y

I ", - x~ I < (")

Concentrated and Rarified Sets of Points. 997 is satisfied for infinitely m a n y values of n. W h e n ak has been defined f o r all k < i we define a~ to be different f r o m all x,~ a n d f r o m all ak, k < i , a n d to satisfy the i n e q u a l i t y l a~--x~[ < 9~(n) f o r infinitely m a n y values of n. T h u s we arrive at a se~ of points t~ai~, (i < Wl).

L e t now {ln} be a n a r b i t r a r y sequence of positive numbers. T a k e the set of intervals (x~--l~,xn+ln), n = I , 2 , . . , a n d write f ( n ) = l n . Le~ i 0 be t h e least transfinite n u m b e r such t h a t

f(n) (n).

Take now any i > i 0. W e k n o w t h a t

< r f o r infinitely m a n y values of n and t h u s

[ a, - - x~ I < f ( n )

for infinitely m a n y values of n. T h u s a m o n g all t h e i n t e r v a l s (x~--l~, x,,+ l,) t h e r e are infinitely m a n y ones c o n t a i n i n g a~ f o r any i > i o. T h e se~ of points a ~ , . . , a,. o being e n u m e r a b l e we conclude t h a t t h e set of points ai, ( i < eo~), is c o n c e n t r a t e d in the n e i g h b o u r h o o d of t h e set {xn} and t h u s our problem is solved.

C H A P T E R I I . Rarified Sets.

I have a r r i v e d at the definition of rarified sets f r o m t h e f o l l o w i n g :

P r o b l e m . Has every plane set of infinite linear measure a subset of finite measure ?

This seems obvious and y e t it is n o t t h e case.

W e shall c o n s t r u c t a linearly measurable plane set of infinite measure, every subset of which is either of infinite measure or of measure zero.

T h e solution will be given by rarified sets. T h e y are defined as follows:

A non-enumerable plane set is said to be rarified i f any subset of this set, of plane measure zero, consists of at most an euumerable set of points.

T h e c o n s t r u c t i o n of most rarified sets will be based on the a s s u m p t i o n t h a t t h e p o w e r of the c o n t i n u u m is ~r

D e n o t e by E a n y set of plane m e a s u r e zero which can be r e p r e s e n t e d by the p r o d u c t

3 8 - - 3 3 6 1 7 . A c t a mathematica. 62. I m p r i m ~ ]e 18 a v r l l 1934.

298 A . S . Besicovitch.

E = l ~ A i

1

where each A~ is an open set. As any open set is defined by an enumerable set of points (for instance by the points with r a t i o n a l coordinates belonging to the set), t h e n so is any set E .

Now the aggregate of all enumerable sets of points on a plane is k n o w n to have the power of the continuum. Hence the aggregate of all the sets E has power of' continmlm.

L e t now two transfinite sequences El, Pi, i r u n n i n g t h r o u g h all values of power less t h a n ~ , represent all t h e sets E and all perfect sets of positive plane measure, belonging to the square (o =< x =< I, o_--< y < 1).

Denote by 21I 1 an arbitrary point of the set P, - E, P , , by M~ an arbitrary point of the set

P ~ - ( E ~ + E~) P_~

different from ~1I~, by ~]I~ an arbitrary point of

different from 31~ a n d 3[~, and so on. A f t e r we have defined 21'[i for all i < i0, 21/:.,0 will be defined as an arbitrary point of

P;o - (E, + E~ + different from all the points Mi for i < io.

For, the set

9 :, + E~ + .

9 + E , o ) Pio

Such a point exists for any i o < w 1.

+E,.o

is of plane measure zero, since it is the sum of an enumerable set of sets, each of plane measure zero. On the other h a n d /9,.o is of positive plane measure and consequently so is the set

P , o - - ( E 1 + ~ + ' + E,,,) P,o

and thus it contains more t h a n an enumerable set of points. H e n c e it m u s t c o n t a i n points different from an enumerable set of points Mi, (i < i0).

Denote by G the set of all the points Mi.

Concentrated and Rarified Sets of Points. 299 (i) The exterior plane measure of G is I.

F o r by c o n s t r u c t i o n t h e c o m p l e m e n t a r y set of G in the square (o G x ~ I, o _--< y G I) does n o t c o n t a i n any p e r f e c t subset of positive plane measure.

C o r o l l a r y . The exterior linear measure of G is i~finite.

(ii) The intersection of G with any set Eio is an enumerable set.

F o r the points of G c o n t a i n e d in Eio are all a m o n g the points Mi (i 1 , 2 , . . . < i 0 ) .

(iii) Any subset oJ G of plane measure zero is an enumerable set.

For, any such set is c o n t a i n e d in a set El.

C o r o l l a r y 1. G is a rarified set.

C o r o U a r y 9.. The set G is linearly measurable.

F o r let H be a n y set of finite positive e x t e r i o r l i n e a r m e a s u r e t t * H , a fortiori of plane m e a s u r e z e r o .

T h e set G . H b e i n g e n u m e r a b l e we have

and t h u s

~ * ( H - - G 9 H ) : ~ * H t t * G . H = o

~ * H = t** ( H - - G 9 H) + #* G . H which proves the corollary.

Theorem. The set G has no subset of finite positive exterior linear measure.

F o r let H be a subset of G of finite e x t e r i o r l i n e a r measure. T h e n by (iii) G~ H is an e n u m e r a b l e set a n d so is H , since G . H = H .

T h u s t h e set G gives a solution of t h e p r o b l e m of this chapter.

N o t e some o t h e r p r o p e r t i e s of the set G.

(iv) The set G being itself measurable has no non-measurable subset.

(v) The set G is not measurable with respect to plane measure.

F o r if it were m e a s u r a b l e its m e a s u r e would be equal to its e x t e r i o r measure, i.e., to I. T h e n by t h e F u b i n i t h e o r e m its intersection with allmost all lines of t h e square, parallel to t h e sides of t h e square, will be sets of l i n e a r m e a s u r e

I, w h i c h is impossible by (iii).

Thus G gives an example of a set which is measurable with respect to linear measure and non-measurable with respect to plane measure.

300 A . S . Besicovitch.

T h e c o n v e r s e p h e n o m e n o n is, of course, very t r i v i a l since a n y set l y i n g on a s t r a i g h t line is of p l a n e m e a s u r e zero b u t n e e d n o t be l i n e a r l y m e a s u r a b l e .

R e m a r k . W e can r e m o v e t h e a b o v e a n o m a l o u s possibilities by i n t r o d u c i n g a n e w definition of m e a s u r a b l e sets.

Definition. A set H is said to be linearly measurable ~f it ea~ be represented by the difference of the product of a fi~ite or enttmerably i~2fi~,ite set of open sets and of a set of linear measure zero.