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 the definition of concentrated sets from 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 II.
What are the linear sets on which the variation of any con-
tinuous monotone function is zero?
Concentrated sets are defined as follows:
A non-enumerable set of points E is said to be concentrated in the neighbourhood 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 increasing
function satisfying defined for 1 ~ o and such t h a t q~ ( + o) ~ o and let E be a
linear set of points.
D e n o t e by I ~
~l~ any sequence of intervals containing the
whole of the set E (li denoting both an interval and its length).
the lower bound of the sum
37--33617.
Acta mathematica.
62.
Imprim6 le 12 f@vrier 1934.
Denote by m ~ E
290
A . S . Besicovitch.
for all possible
exceed ~.
sets I ,
The
for
which the l e n g t h of the l a r g e s t i n t e r v a l does not
of m~E,
limit
as ). tends
to zero, we shall call the exterior
~0-measure of E , or the e x t e r i o r m e a s u r e of E w i t h respect to ~.
Obviously the qg-measure of an e n u m e r a b l e set and the v a r i a t i o n of a continuous m o n o t o n e f u n c t i o n with respect to an e n u m e r a b l e set are both zero.
Now
we k n o w t h a t a B - m e a s u r a b l e set E either is e n u m e r a b l e or else contains a perf e c t subset E'.
I n the first ease the v a r i a t i o n of any m o n o t o n e continuous funcI n the second case E ' m a y be t r a n s f o r m e d into an i n t e r v a l
tion on E is zero.
by m e a n s of a m o n o t o n e continuous f u n c t i o n f ( x ) .
The l e n g t h of such an inter-
val is obviously the v a r i a t i o n of f ( x ) with respect to E ' , and hence t h e v a r i a t i o n
with respect to E is positive.
Hence
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
In
the general case there are ~wn-enumerable sets whose measure is zero n, ith
respect to any fu~ction 9~,
but
the
p r o o f will
be based
on the a s s u m p t i o n t h a t the power of the conti-
n u u m is ~r
w 2.
W e shall first prove some p r e l i m i n a r y theorems.
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 .
The condition is necessary.
to
any
function
and
li
(i~
F o r let E be a set of m e a s u r e zero with respect
I, 2 , . . . )
assume t h a t t h e r e exists any decreasing
1 L. C. YOUNG. ~Note on t h e t h e o r y of measure. Proceedings of t h e Cambridge Philosophical Society. Vol: X X V I . P a r t 1.
L. C. Y o u x ' o considers variation of a function on a given set as m e a s u r e of this set. The
definition we are u s i n g is n o t so general and it is not obvious t h a t if variation of some functions
on a given set is positive t h e n also m e a s u r e of the set with respect to some function is positive.
We shall consider in w 3 B - m e a s u r a b l e sets of m e a s u r e zero w i t h respect to any function.
Concentrated and Rarified Sets of Points.
sequence of positive numbers.
291
Define an increasing function ~ (1) satisfying the
conditions
I
~0(li) :'=--.
Obviously 9 ( + o ) - - o ,
and
for i -~ I, 2, . . .
consequently m,p E is defined and by hypothesis is
T h e r e f o r e there exists a set I of intervals (x$, x'/), x ~ ' - - x ~ = l;,
equal to zero.
decreasing in length, containing the whole of the set E and such t h a t
~ , ~ (1;) < , .
T h e n for any n
> ~ ~ (l;) >= ~, ~ (1,'~),
i=1
~t~ < %
s
n
i. e.
t
1,, < 1,,
and the condition is t h e r e f o r e satisfied.
The condition is also sufficient.
then
F o r suppose a set E satisfies this condition:
given a function T(1) satisfying the
usual
conditions and two positive
constants ~, ~ we can always find a sequence of positive numbers l~ < ~ such t h a t
~ (1;) <
and t h e n a sequence I of intervals (x'i, x~'), x~'--x~ ~-li, containing/~', which shews
t h a t measure of E with respect to 99 (1) is zero.
3.
W e can now solve the problem m e n t i o n e d in the footnote to w 1.
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.
T h e o r e m 2.
On account of 2 the t h e o r e m will be proved if we prove t h a t given any
perfect set P there exists a sequence of positive numbers l~,
that
n=
I, 2 , . . .
such
no set of intervals of lengths l~, l ~ , . . , can cover the whole of the set P.
Obviously the
set P
c a n n o t be covered with one interval of arbitrarily small
292
A . S . Besicovitch.
length.
the
L e t ).~ be the lower bound of the length of an interval which covers
whole of P (evidently).~ is the distance between the extreme points of P).
Take for 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 the set E c a n n o t be covered with one interval
of length l~ and a n o t h e r interval of arbitrarily small length.
Otherwise let two
intervals
(x,, x , + l~), (y,,, y , + 12,,)
where 4 , , - ~ o ,
as n - ~ o c , cover the whole of P for any ~.
point of (x,,yn).
Then for any ~ the two intervals
(x -- ~, x + ll + e),
cover P .
L e t (x,y) b e a l i m i t
( y - - e , y + ~)
Consequently the set P is included in the set consisting of the interval
( x , x + l~) and of the point y.
Being a perfect set, P has no isolated point and
thus is contained in the interval (x, x + l~), which is impossible.
L e t E.o be the
lower bound
of the l e n g t h of an interval which t o g e t h e r
with a n o t h e r interval of l e n g t h l~ covers the whole of the set P .
4
D e n o t i n g by
a positive n u m b e r less t h a n ).~ we have t h a t no pair of intervals of lengths
l~, 4 can cover the whole of the set P.
three intervals of lengths
11, [~,l~
Similarly we can find/3 > o such t h a t no
can cover the whole of P, and so on.
Thus
we arrive at a sequence of positive numbers
ll, 4, 9 9 9
such t h a t
no finite set of intervals whose lengths are represented by different
numbers of the sequence, can cover the set P. T h e n by the Heine-Borel theorem
no infinite set of intervals of lengths
ll,
4,
9
..
can cover the set P , which proves the theorem.
w 4.
T h e o r e m 3.
If
the measure
of 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 function f ( x ) on E is" zero.
Denote
by
0 (f(x),l)
interval of length 1.
the
upper bound of the oscillation of f ( x ) on any
Obviously
O{f(x), 1) ~ o ,
Consequently,
numbers l~,
as l - ~ o .
given ~ > o we can always find a decreasing sequence of positive
i = I, 2 , . . .
such t h a t
Concentrated and Rarified Sets of Points.
293
F, o {f(x), l,} < , .
i=l
On the other hand, 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~.
Then
denoting by
Vf.E the variation of f(x) on the set E
we have
V/ E <=~ { f (x;') - - f ( x ; ) }
o{f(.), l,}
<=
which proves the theorem.
A f t e r these preliminary theorems have been proved we can pass to the main
problem
of this note, i.e., ~o the proof of the
existence of non-enumerable
sets of measure zero with respect to any function 9(l).
that
concentrated sets, if t h e y
function
~. X n I
~
9(1).
(n==I ~ 2 ,
I t can be easily shewn
exist, are of measure zero with respect to any
F o r let E be a set c o n c e n t r a t e d in the neighbourhood of a set
9 9
.) and {ln} an a r b i t r a r y
sequence of positive numbers.
the definition the set of intervals ( x n - - 2I l~, x,~ + 2I 1,2,~)contains all points
By
of E
except at
most an enumerable set of them, which can be included in intervals
of length
l~,l~,...
Consequently
by T h e o r e m I the set E is of measure zero
with respect to an 3' function.
Thus our problem reduces to the proof of the existence of c o n c e n t r a t e d sets.
w 5.
This proof will be based on the existence of a transfinite sequence
{~i) of positive decreasing functions 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 function f ( n )
is ultimately greater than (greater than or equal
to) another function g (n), and we write
f ( n ) ~- g (n)
(f(n).= g (n)),
if there exists a n u m b e r % such t h a t
294
A . S . Besicovitch.
f(u) > g (n)
(f(n) >_-g (.))
for n > no.
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.
Definition.
The existence of such a sequence can be established in the following way.
W e first order the set of all positive decreasing functions into a sequence off (n)
i t a k i n g all values less t h a n to1.
W e then define
~, (n) -~-fl (n)
q~, (n) = min {qg~(n),f~ (n)}
9% (n) ~-- m i n ' , q~_.(n),fa (n)}
and so on.
F o r any transfinite n u m b e r i + I of the first class we put
~0i+ 1 (1~) = m i n
(r
(,,),fl+l (1~)).
F o r any transfinite n u m b e r i of the second class we first define a sequence
kt
(1 < ~Oo,coo being the first transfinite n u m b e r of power too)
of increasing transfinite numbers such t h a t
lim kt = i.
T h e n we define ~i(1) for any l by the equation
~i (l) ---- min {%. (l), ~kl (/), 9 9 ~'~ (/)}.
I n this way the sequence
is defined, where i 'takes all values less t h a n r
following properties:
(i)
For any i < ~o1
f, (.) >
(ii)
(.)
A n y 9Di(u) is a decreasing function.
The sequence possesses the
Concentrated and Ratified Sets of points.
(iii)
295
I f i < k, then
q,,(,,)
The properties (i), (ii) are obvious.
*_- ~k (,,).
In order to prove (iii) we observe that
~, (n) *= 9~ (n) *__~ ( n ) . . .
Suppose that
these inequalities
have been proved for all indices less than i.
Then if i is a transfinite number of the first class, we have
by the definition of ~(,,).
If i is of the second class, then
i = lira kl
and again from the definition of 9~/(n) we conclude that
for all l, and consequently
for a n y j < i .
(iv) To any transfinite number i < % corresponds an i ' > i and less than
to1, such that
For, all decreasing functions having been enumerated we have
~2 r (') = ~ ' (n)
where i' must be greater than i since ~i (n) >-f/' (n) .
as .~, (,,) ~ ~,, (,,) we have
~, (~) ~ ~,, (,).
Let now g (n.)be any positive function.
Define f(n) by the condition
f(,~) = rain g (,n)
for I ~m=-<~.
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 that
9.96
A . S . Besicovitch.
f ( n ) > ~0j (n)
for all j ~ i'
i. e., for all j except an enumerable set of values.
Thus the sequence
(~,(n))
is fundamental.
w 6.
Lemma.
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 jbr infiMtely ma~y values of n.
Take an a r b i t r a r y positive integer ~h and the interval (x,~--~t,x,, + ~),
where ~a <f(n~) and is so small t h a t the interval does not contain any of the
Yl, Y ~ , . . . , Y , , different from x,,. Take now an integer ~ > J q and a
positive n u m b e r ).e < f ( n e ) such t h a t the interval ( x , . - ) . ~ , x~: + ).~) is contained
in the interval (x,,--).l, x,~ +)-l) and does not contain any of the nmnbers
numbers
y~, y,_,. . . . . yn: different from xn~, and 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, ... < ~oo
each of them being interior to the preceding one.
Any point a interior to all
these intervals (if ~i--~ o there is only one such point) is different f r o m all y,~
and satisfies the conditions
I
-x,,I < z;
<
f(ni)
for i =
I, 2,...
which proves the lemma.
w 7.
We
shall now construct a concentrated set of points of power ~1.
L e t ~i (n), (i < oq), be a f u n d a m e n t a l sequence of functions and ~x,~j an
everywhere dense set of points. Take a point cq different from all points x,~ and
such t h a t the inequality
161 is satisfied for infinitely m a n y
I<
values of ~.
(')
Then choose a point a.~ different
f r o m all xn and from a I and such t h a t the inequality
I ", - x~ I <
(")
Concentrated and Rarified Sets of Points.
is satisfied for infinitely many values of n.
k<i
we define a~ to
997
W h e n ak has been defined for all
be different f r o m all x,~ and from all ak,
k<i,
satisfy the inequality l a~--x~[ < 9~(n) for infinitely m a n y values of n.
and to
Thus we
arrive at a se~ of points t~ai~, (i < Wl).
L e t now {ln} be an a r b i t r a r y sequence of positive numbers.
(x~--l~,xn+ln),
of intervals
n=I,2,..,
and
write f ( n ) = l n .
Take the set
Le~ i 0 be the
least transfinite n u m b e r such t h a t
f(n)
Take now any i > i 0.
(n).
W e know t h a t
< r
for infinitely many values of n and thus
[ a, - - x~ I < f ( n )
for infinitely m a n y
there
Thus among all the i n t e r v a l s (x~--l~, x,,+ l,)
values of n.
are infinitely many ones containing a~ for any i > i o.
a ~ , . . , a,.o being enumerable
The se~ of points
we conclude t h a t the 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 the set {xn} and thus our problem is solved.
CHAPTER II.
Rarified Sets.
I have arrived at the definition of rarified sets f r o m the following:
Problem.
Has every plane set of infinite linear measure a subset of finite
measure ?
This seems obvious and yet it is not the case.
W e shall construct a linearly measurable plane set of infinite measure, every
subset of which is either of infinite measure or of measure zero.
The 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.
The construction of most rarified sets will be based on the assumption t h a t
the power of the c o n t i n u u m is ~r
Denote
by E any set of plane measure zero which can be represented by
the p r o d u c t
38--33617.
Acta mathematica.
62.
Imprim~ ]e 18 avrll 1934.
298
A . S . Besicovitch.
E=
l~Ai
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 rational 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 known
to have the power of the continuum.
Hence the aggregate of all the sets E has
power of' continmlm.
Let 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 the sets E and all perfect sets of positive plane
measure, belonging to the square (o =< x =< I, o_--< y < 1).
Denote by 21I1 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~ and 3[~, and so on.
After we have defined 21'[i for all i < i0,
21/:.,0 will be defined as an arbitrary point of
P;o - (E, + E~ +
9 + E , o ) Pio
different from all the points Mi for i < io. Such a point exists for any i o < w1.
For, the set
9 :, + E~ + .
+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 hand /9,.o is of positive plane measure and
consequently so is the set
P,o--(E1 + ~
+ '
+
E,,,) P,o
and thus it contains more t h a n an enumerable set of points.
Hence it must
contain 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.
(i)
299
The exterior plane measure of G is I.
F o r by construction the complementary set of G in the square (o G x ~ I,
o _--<y G I) does not contain any perfect subset of positive plane measure.
The exterior linear measure of G is i~finite.
Corollary.
(ii)
The intersection of G with any set Eio is an enumerable set.
F o r the points of G contained in Eio are all a m o n g the points Mi
(i
1,2,...<i0).
(iii) Any subset oJ G of plane measure zero is an enumerable set.
For, any such set is contained 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 any set of finite positive exterior linear measure t t * H , a
fortiori of plane measure z e r o .
The set G . H being enumerable we have
~*(H--
G 9H ) :
~*H
tt*G.H=o
and thus
~ * H = t** ( H - - G 9 H) + #* G . H
which proves the corollary.
Theorem.
F o r let H
The set G has no subset of finite positive exterior linear measure.
be a subset of G of finite exterior linear measure.
(iii) G~ H is an enumerable set and so is H , since G . H =
T h e n by
H.
Thus the set G gives a solution of the problem of this chapter.
N o t e some other properties 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 measurable its measure would be equal to its exterior measure,
i.e., to I.
Then
by the
Fubini t h e o r e m its intersection with allmost all lines
of the square, parallel to the sides of the square, will be sets of linear measure
I, which 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 converse p h e n o m e n o n is, of course, very trivial since any set lying on
a s t r a i g h t line is of plane m e a s u r e zero but need not be linearly m e a s u r a b l e .
Remark.
W e can remove the above a n o m a l o u s possibilities by i n t r o d u c i n g
a new 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.