Integration in locally compact spaces by means of uniformly distributed sequences

Integration in locally compact spaces by means of uniformly

distributed sequences

Citation for published version (APA):

Post, K. A. (1967). Integration in locally compact spaces by means of uniformly distributed sequences. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR75301

DOI:

10.6100/IR75301

Document status and date: Published: 01/01/1967 Document Version:

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INTEGRATION IN LOCALLY COMPACT SPACES BY MEANS OF UNIFORMLY DISTRIBUTED SEQUENCES

INTEGRATION IN LOCALLY COMPACT SPACES

BY MEANS OF

UNIFORMLY DISTRIBUTED SEQUENCES

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL TE EINDHOVEN OP GEZAG VAN DE RECTOR MAGNIFICUS DR. K. POSTHUMUS, HOOGLERAAR IN DE AFDELING DER SCHEIKUNDIGE TECHNO LOGIE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN

OP DINSDAG 20 JUNI 1967 TE 16 UUR

DOOR

KAREL ALBERTUS POST

GEBOREN TE HAARLEM

DIT PROEFSCHRIFT IS GOEDGEKEUHD DOOR DE PROMOTOR

CONTENTS

Chapter 0

7

0.1. Introductory remarks

7

0.2. Notations 10

Chapter I 12

1 • 1. Preliminary remarks - Auxiliary proposi tiona 12 1.2. Construction of a function with prescribed

mean and integral values 16

1.3. Uniformly distributed sequences with repetitions 29

Chapter II

33

2.1. (t;;,dJ.L)-summability - Introduction

33

2.2. Topological and measure-theoretical preliminaries

34

2.3. Sufficient conditions for (t,dJ.L)-summability 48

2.4. Discussion of best-possible results 52

Chapter III

59

3.1. Introduction

59

3.2.

Applications of (t,dJ.L)-summability

69

References

77

Samenvatting

79

Curriculum Vitae 81

CHAPTER 0

0.1. INTRODUCTORY REMARKS

The first reference to uniformly distributed sequences of real numbers on the segment [ 0,1] is to be found in a paper by Weyl

[14].

One may consider uniformly distributed sequences from two dif-ferent points of view.

00

On the one hand, a. uniformly distributed sequence !;. = {x } 1 n n= (0 ~ x "' 1 for all n) is characterised by the relation

n

(1.1) lim _N1 I: N =1 X[ b] (x ) = b - a.

N-oo n a, n

for all a,b (0 ~a< b.;;; 1), where X[a,b] denotes the charac-teristic function of the interval [a., b] •

On the other hand, a. uniformly distributed sequence 1;. "' {xn};. 1 (0 < x .;;; 1 for all n) may be characterised by the fact that

n 1 lim

~I:

N= 1 f(x ) =

J

f(x)dx N-oo n n 0 ( 1. 2)

for all continuous functions f on the interval [0,1].

The studies made of the aspects of uniformly distributed sequenc-es may be roughly divided into quantitative and qualitative onsequenc-es. Quantitative questions are concerned with the order of conver-gence of the mean values in ( 1.1) and (1.2) to their respective limits, a.nd have been investigated e.g. by Van der Corput

[3]

and Koksma [ 12] (for

mor~

information on the literature about this and the following subjects see Cigler and Helmberg [2] ). The 7

study of the qualitative aspects of uniform distribution was further developed particularly by Hlawka

[a], [9],

who, starting from a paper by Eckmann

[4],

introduced the concept of uniformly distributed sequences in a compact topological space with a nor-malised Borel-measure ll• Generalising the situation on the unit

interval, it appears that every Riemann-integrable function on a compact space may be integrated by taking its asymptotic mean

value on the points of a uniformly distributed sequence (of.

(1.2) ).

All these investigations concerned bounded functions. Helmberg

[7]

studied the case of non- bounded continuous functions on a

locally compact normalised measure-space. Such a function may

also be regarded as a function on a compact space (the one-point

compactification of the original space), which is continuous

everywhere, one single point excepted, and may be approximated (pointwise) by a sequence of bounded continuous functions. Helm-berg, amongst others, proved that for any uniformly distributed

sequence

~

=

{x } 00 and for any non- negative continuous func-n n==1

tion f on a locally compact space X we have the following in-equality:

(1.))

Jx

f(x)dJ.!.(x)

<

lim inf

~

I: N f(x )

N-00 N n=t n

He also gave an example of a situation where fxf(x)d!l(x)< co and

lim sup

~

I: N f(x )

=

oo •

N-co n=1 n

In accordance with Helmberg [

7]

we shall define a function f

to be (;,dJ.I.)-summable, i f f is integrable and

(1.4)

lim

~

I: N f(x )

=

J

f(x)diJ. (x)

N-oo n=t n X

about the behaviour of unbounded functions with respect to a uniformly distributed sequence.

In Chapter I we shall prove that, given any uniformly distributed sequence I;

=

{x } 00 without repeti tiona and any three numbers

n n=1

a,

j3, y (0 <a~ (3 ~ y ~

oo),

there exists a non-negative contin-uous function f such that

Jfd.J!

=

a,

lim inf

~

EN

1 f(x ) =

p

and

N-oo n

lim sup

~ E~

f(xn) = y. In Chapter IT some sufficient conditions N-co

for (!;,d~)-summability are discussed (Section 2.3 ). We shall sketch here some background considerations, which gave rise

to

the investigation of these particular conditions:

(i) The (!;,d~)-summability of a continuous function f on a lo-cally compact space depends on the behaviour of f near the point

00 _{, i.,e. the point which has to be added in order to compactify}

our space.

(ii)

The topological and measure-theoretical situation near tile point 00 _{is related to the system of neighbourhoods of oo.}

(iii) We are able to construct a special sequence of neighbour-hoods of 00 , related to I; (Section 2.2 ), such that the global behaviour of a function f with respect to this particular se-quence of neighbourhoods is decisive for the (!;,dJ.l)- summability of f o These arguments are found to apply also to a larger class of functions than continuous functions only (Section 2.3 ). In Section 2.4 we shall exhibit some examples which show that the results of Section 2.3 are in a certain sense best-possibl~

In Chapter III we shall illustrate the results of Chapter II, mainly for the special case of a sequence introduced by von Neu-mann [13]. One of the arguments used here also applies to almost all sequences {np} 00 mod 1 (where

p

is an irrational number), as

n==1

0.2. NOT AT IONS

In order to avoid repetition of definitions ofsome notions which are used in all Chapters of this thesis we here give a survey of some concepts and notations used.

X is given to be a locally compact, non-compact Hausdorff-space with a countable base ~of open neighbourhoods E. Its one-point compaotification X U

{~}

is denoted by x*. If A is a subset of X we write A't

A,

A0, and oA, respectively, for the complement,

closure, interior, and boundary of A.

1.1. is defined to be a normalised non-atomic Borel -measure on X with a non-compact support S • This measure 1.1. induces a

norma-1.1 * * *

lised non-atomic Borel-measure 1.1 on X if we define 1.1 (A) =

= ll *(A U

{~})

= IJ.(A) for all Borel-sets A c

x.

Without loss of generality we may assume that all neighbourhoods

E

€W

have compact closures and zero-boundaries, i.e. j.!(oE) = 0

for all E €'\-() (of. Helmberg

[7]).

~

t

= {xn}n=

1 denotes a diJ.-uniformly distributed sequence of points

x € X (n,.,1,2,3, ••• ), in other words we have n

(2.1) _lim

*

_{EN f(x )}

₌

J

_{f(x)d!J. (x)} N-~ n=t n X

for all continuous complex-valued functions f on X with compact support. It is well known that for all Borel- sets F with zero-boundaries we have

(2.2) lim

~

A(t,F;N)

=

!J.(F)

N-~

where A(!; ,F;N) "" , ••• ,N for which

!:: ..

1 XF(xn) denotes the number t>f indices n

=

1, x € F (of. Helmberg [7], p. 172 Hilfssatz 2).

n

Fbr every finite collection of sets

{E.}~

1 the following

inclu-J J=

o (

U

~

E . ) c U

~

1 ( oEj ) ,

o (

n

~

Ej ) c U kj ( oEj )

J=1 J J= J=1 =1 .

(of. Kelley [10]). These relations imply that the union and the intersection of a finite number of sets Ej with zero-boundaries are sets with zero-boundaries themselve·s.

12

CHAPTER I

1.1. PRELIMINARY REMARkS • AUXILIARY PROPOSITIONS

In the present Section we shall derive some lemmas concerning dfl-uniformly distributed sequences and set-theoretical topology.

Lemma 1 .1.1 • I f F is an open set with compact closure and zero-boundary, such that

~(F)

> 0, if

~

=

{~}~

1

is

d~-uniformly

dis-tributed in

x,

and if

{ki}~

1

is the sequence of all indices k

k. 1

such that

~

€ F, then . :!"moo

~~

= 1 •

l. l.

Proof. Obviously A(t,F;k

1+1) = A(t,F;ki) + 1 (i=1,2,3, ••• ) and

this expression tends to infinity i f i ~oo. Now we have

1 A(~,F;k.) k. ki

1 = v.(F) • "(F) == lim 1 • l.+l

=

lim_..±!.

.. i -oo ki A(E;.,F;ki+1) i -oo ki •

Lemma 1.1.2. If x € S , and i f

0 ~ E;.

=

{~}~

1

is dv.-uniformly

dis-of indices

{ti}~

1

such

tributed, then there exists a sequence

,e.

that 1 . d 1. l.+1 1

• llll xt. • x₀ an . l.lll

-::e-

= • J.-00 l. J.-00 i

Proof. Let {F'kj:₁ be a countable base of open neighbourhoods of x

0, ~

EW

(k=1,2,3, ••• ). Then \: is compact', fl.(a~) = 0 (k

=

= 1,2,3, ••• ) and

n~

..

1 \ : = {x0} (cf. Ch.O). I f Fk is defined by

Fk = n

~

..

1 Ej (k = 1 ,2, 3, ••• ), then Fk is compact, and f.1

(oFk) .,;;:

or;;

1::~

..

1 (l(oEj) "' 0 (k= 1 ,2,3, ••• ), F1 ::> F2 ::> ••• ::> Fk ::> •••, and

n

~

...

₁ Fk = {x

we may find for any neighbourhood U of x

0 a set Fk, such that

Fk c U. As ll is non-atomic we obtain lim ll (Fk)

=

0 , and x

0

e:

s

₁₁

implies I.L(Fk)

>

0 (k=1,2,3, ••• ). k .. oo

For each of the sets Fk lemma 1.1.1 applies and we obtain a chain of successive sub-sequences of indices {.tk . } ~

1

(k = 1, 2,

3, ••• )

tJ

J-auch that for every fixed k we have x₁

e:

Fk for all j and k,j

lim

~,j+

1

=

1. Let

.t~,j+t

<

1 + 2-k for all j satisfying

j-oo k,j k,j

.tk, j > \:. Without loss of genera.li ty we may assume that \: = lk, j for some index j, and that 1\+

1 > ~ (k ... 1,2,3, ••• ). If~ is the

(non-empty) finite sequence of all lk . for which N. <.tk . <N. ,

tJ k tJ k+1

and {.t j} ; , is the sequence formed by writing down all blocks ~

J,. -k

in succession, we see that 1

<

~7

< 1 + 2 for all lj

e:

~·

J,. +1

Hence l i m +

=

1. The construction of {Fk};

1 implies that

j-co j

Corollary 1.1.2.1. There that .lim ~.

=

00 and

J . - 0 0 l. exists a sub-sequence k. +I lim - 1- .. 1 • i -00 ki Lemma 1 .1. 3. If {k. } :"'

1 and {m.} ~ 1 are two monotonically

increas-J increas-J"" J J= k. m.

ing sequences of integers such that .lim

'~

1

=

lim

~~

1 = 1 ,

J .. oo J j .. oo J then there such that exists a sub-sequence {.t.};'

=

{k.

}~

1

of {k.}~ l. •1 Ji l.• J J=1 .t.+1 .lim

~.

= 1 J.-00 l.

and J,j > mj for all j .

Proof. We define J 1 =Min {kjlkj > m1} and by induction .ti+1 = =Min{kjlkj>mi+t and kj>.ti} (1=1,2,3, ••• ).

and for all m. ~ N J

k. 1

1

<

_ . L

<

(1 + e)2' , k. _J-1

Let m. :iiJ N. Then we distinguish two cases :

l.

(1 )

In this case .ti+1 ~ is the quotient of two

l.

.ti+1 consecutive k., both of which are greater than N. Hence 1 <~ <

1 J .ti < (1 + e)2' < 1 + e • (2) :;;.m i+1 ' so that

m₁ .,.; .ti < mi+₁ .,.; .ti+₁ • Then .ti+₁ is the smallest kj :;;. say ti+₁

=

kt :;;. mi+₁ > kt_₁ • Hence

*

I f

V

is a closed subset of X with zero-boundary,

*

x

0

r/.

V,

and lt and A. are real numbers such that 0<lt<A.<1-fl

(V),

*

then there exists an open neighbourhood U of :x:

0 in X suoh that

*

*

u

n

v

==

rj

and x < fl (u) = fl

(U)

< A. •

~·

The normality of

x*

(cf. Kelley [10]) implies that there

exist disjoint open sets W and W such that :x: € W and

V

c W •

*

1 2 0 1 2

fl is a regular measure (of. Ha.lmos [

5 ] ),

hence W

2 contains an

open set W~ with zero-boundary,

V

c W c W , satisfying

* .., 3 2 *

*

1 - fl (w

3) > A.. Again by the regularity of fl for any point :x:€ X

*

*

there exists an open neighbourhood F such that fl (F )

=

fl ~ ) <

X X X

< A. - x. By Heine-Borel1_{s lemma a finite number of these}

neigh-*

n

*

bourhoods already covers X , say, U k=

1 Fk = X • Let x0 E F1• The

intersections Fk

\W

3 = Fk

n

fi)'

(k==1, ••• ,n) cover

x\

W

3 and

all sets Fk \

i

3 are open and have zero- boundaries. Since

14 1.1.

*

(Fk \

i'

m such that the open set U = U ~ ...

1 (Fk \

W

3) has the desired

prop-erties.

Remark. In Chapter II we shall prove a lemma (lemma 2.2.1 ), which is both a sharpening and a generalisation of lemma 1.1.4. Since the proof of this lemma is fairly more complicated and in this

Section only weaker results of lemma 1.1.4 are needed, we do

not give the stronger results here.

Lemma 1.1.

5.

'If {xi} : :₁ is a sequence of distinct points of X converging to the point oo in

x*,

and i f

{a):,

and

{PJ : ,

are sequences of real numbers, 0<

a.<

f3.

(i=1,2,3, ••• ), :E~

f3.

<1,

J. J. 1=1 l

then there exist open neighbourhoods U. of x. (i=1,2,3, ••• ) with

J. l

compact closures and zero-boundaries in X such that

n

U.

=

¢

J (ifj)and a.<tJ.(U.)<f3. (i=1,2,3, ••• ).

J. J. l

Proof. The construction may be performed by induction.

( 1 ) Let

V~

) be an open neighbourhood of oo in X* with

zero-boundary and

tJ.\vi

1)) <t(1-

f3

₁), such that x₁

¢

vi

1). Since

vl

1

) covers {xi};:₂ except for a finite number of points, say x;1), •••

,~

1

),

we may construct open neighbourhoods

v;

1), •••

,v~

1

)

1 1

of

x~

1

),

•••

,~

1

)

respectively, with zero-boundaries, such that

d k,

::r0

1 ( (1) ( 1) 1

x₁ y:. U ,e..₁ VJ, and ll V₁ U ••• U Vk ) < 2"(1

-13

₁ ). I f we choose

v(

1

) =

vi

1)

uv;

1

)u ... uv~:)

then

0D,

x

1, a1 and

f3

1 may be sub·

etituted for

v,

x

0, ~and A in order to make lemma 1.1.4

appli-cable, so that there exists an open neighbourhood U

1 of x1 with

the property that U is compact, a < t.t

(u )

= ll (u ) <

13

and

- - 1 1 1 1 1

U

1

n

V( 1 ) =

¢.

As a consequence, we

have~

i

U

1 (k= 2,3,4, ••• ).

(2) Theinductivestep. Let

u,, ••.

,ur be open neighbourhoods

of x , ••• ,x respectively, such that

U.

n

U.

=

¢

(if

j ), UJ.. is_ 15

compact, a:. < ll (u. ) .. tJ.

ru:. )

< ~.

(i -

1, ••• 'r) and let X.

¢

ui

~ ~ ~ ~ K

(i

~k).

Since l!

~

1

~.

< 1 and

~

+t

>

0 there exists an open

~= ~ r

(r+t)

* .

(

(r+1))

neighbourhood V of oo in X w~ th zero boundary and 1.1 V

<

00 - - - 00

1 ( r+1 ) d (r+t )

<

2 1 - i::i=t

1\ ,

xr+1 y.. V"" • Beoa.use xi .. oo as i - oo , this

neighbourhood contains all x. (i ~ r + 1 ) except for a finite

num-~

(r+1) (r+t) . .

ber, say x₁ , ••• ,~ , wh~ch in the~r turn have open neigh-r+1

(r+1) (r+1) . .

bourhoods v

1 , ••• , Vk , respect~vely, w~ th zero- boundaries,

r+1

k - - k

d r+1 (r+1 ) ( r+1 ) 1 ( r+1 )

such that xr+t y.. U .£=t V .£ and 11 U .l=t v~,

<

2 1 -I: i•t 13_{1 •}

u- d f. v<r+t ) -

u u

u u u

v<r+t )

u

u

v.<r+t ) d 1'1\JW we e ~ne -1 • • • r 00 • • • k an _ _ _ r+1 obtain I!(V(r+t ))

=

!k(V(r+t))

<

z:7

1

~·

+

2•i(1-E:+

1 1

J3.)

= ~= ~ ~= ~

= 1 - ~ +1 • Again lemma. 1.1.4 applies when we substitute

V

=

~

.

= V , x = x + , x = o:

1, A = ~ 1, so that there ronsts an

o r 1 r+ r+

open neighbourhood U +1 of x with compact closure,

r r+1

a:

<

"(u

) "'

"'U

)

< (.!

u

n

v(r+1 ) ..

¢

As a

con-r+1 ,... r+1 .-\ur+t "'r+t ' r+1 •

sequence, we have x.

r/.

U

(k ~ r + 1 ), so that lemma 1.1.5 is

k r+1

proved.

1.2. CONSTRUCTION OF A FUNCTION WITH PRESCRIBED MEAN AND INTEGRAL VALUES

Helmberg [

7 ]

proved that for any continuous non- negative

func-oo

tion f on X and any dl!- uniformly distributed sequence ~

=

{~}k=t we have

(2.1)

He also gave an example of an unbounded non- negative function f such that

J

f(:x:)d!J. < oo and lim sup

if

I:

:=

1

f(~)

= oo •

X N-oo

con-tinuous function f with prescribed values of

J

fdjl ,

lim inf

i

1::

~~ f(~),

and lim

sup~

Z

~ f(~).

The proof shows that we must impose an additional condition on~. viz. that all points

~ € ~ are distinct, in other words, that

t

has no repetitions. A discussion of the consequences which may arise when we delete this condition will follow at the end of this Chapter.

Definition 1. 2.1. A continuous function g is called a Urysohn-function if' 0 ~ g(x) ~ 1 for all x € X, g(x) == 1 for some point x

= x

0, the support of g is compact, and if the sets

A =

{xlg(x) >

0}

and B • {xlg(x)

=

1}

both have

zero-bound-g g

aries.

Obviously, if g is a Urysohn-function and if a is a positive

num-a

ber, then h = g is also a Urysohn-funotion, and Ag = ~' Bg = ~·

Lemma 1. 2.1 • I f g is a Urysohn-function then I ==

J

ga df.L is a

a X

non-increasing continuous function of a ,

lim Icx

=

J.L (B ) •

a-oo g

lim I

=

!l(A ) and a~ 0 a g

P.roof. This lemma f'ollows from an application of Lebesgue 1 _s

theorem on monotone convergence (cf. Zaanen

[15]).

Lemma

1.2.2. Let U be an open neighbourhood of x

0 with compact

closure

U

and zero-boundary, and let 0 < b < J.L(U). Then there exists a Urysohn-function h with support Sh = U such that ~ == {x

0}

and Jxhdf.L =

o •

P.roof. From

a

result mentioned by Halmos ( cf. [

5 ] ,

p. 21'() it follows that there exists a non-negative continuous function f ,

1

which vanishes at the point x if and only i f x = x , and a

non-e

negative continu6us function £₂, vanishing at the point x i r and only if x

r/.

u.

We define f by { exp (-f f-1

(x))

1 2 f(x) = 0 if X € U i f xr/.U 17

so that f is a Urysohn-function with support

U

and Bf • {x

0}.

Ac-cording to lemma 1. 2.1 the integral I =

J

.f'Xdfl is a continuous

ex X

function of ex and lim rex = ll (U) = ll

(U),

lim ~ • 0 so that there

ex lo ex-oo

exists a positive number ~ satisfying ~ = b. The corresponding Urysohn-function h =

f~

has the desired properties.

Notations. I f !; =

{xk}:,

is dfl- uniformly distributed and f is a continuous non-negative function we write by way of abbreviation f[N]

=

i

~:=, f(~)

and ll(f) =

J

f(x)dfl. I f f is bounded, then

def def X

we have (of. Helmberg [ 7 ] )

(2.2) lim f[N] = ll(f)

N-oo

In general we do not know anything about the behaviour of f [N] for "small" N, i.e. those N for which f[N] is not yet "close" to ll(f) in some specified sense. This behaviour depends on the one hand on the topological properties of X, which have a connection with the values of f since f is continuous; on the other hand, it depends on the measure-theoretical and topological structure of X, which relate to the properties of !;. The idea of the following

investigations is to construct functions for which we have this behaviour under control. Under the assumption that !; has no

repe-titions we shall construct a continuous function f with

pre-scrilied integral for which the support is contained in a given open set and for which the value in one single given point xK E !; al-ready determines the behaviour of f[N] up to a given index and for "small" N. In fact, f[N] =

9

for all N < K and f[N] =

% (

C being constant) for all N~ K up to a given index and until f[N] is "close" to ll(f). More precisely, we shall prove the following lemma:

Lemma 1.2.3. Suppose!;=

{~}:

1

is a dfl-uniformly distributed

an open neighbourhOod of ~ with compact closure, ~ (U)

=

~

(u)

>

o,

and c > 0 and a> p > 0 are given, such that .Ka~(U) > p .

Then there exists a non-negative continuous function f with com-pact support Sf c

u,

~(asf)

= o,

and an index M

>

L such that

• (a)

f(N] 0 (1 < N < K) (b) (c) f[N]

=

K; (K

<

N < M), in particular

f

[K]

=

a Ka < f[N] < p + c (N > M) N (d) ~(f) = p •

Proof. According to lemma 1.2.2 there exists a Urysohn-function h such that ~ ==

u,

~ = {~} and 11 (h) has a given value smaller than !1(U), for example

fo.<~(h)<Min(PK:c, ~(U)

). Since .Kah[N] tends to Ka~-t(h) (of. (2.2)) as N-00 _and Ka~-t(h) _{< p} + e, _there exists an index M

0 such that Kah[N] < p + e for all N > M0• Take

M = max(L,M ). Now let V be open neighbourhoods of x (q = 1, ••• ,

0 q q

M-1; q~K) with compact closures and zero-boundaries, and let V

be an open neighbourhood of

:XX:

with compact closure and

zero-bound-ary suchthat

~(V)<Tf

and X

f/.V

(q=1, ••• ,M-1;qfK).

r.O' q

This is possible since ~ has no repetitions. The set

M-1

W = [ U U ( U

1 V ) ] \

V

is open.

i

is compact and i.l. (oW)

=

o.

More-q= q

q,lK

over, we observe that

f

Koh(x)d~(x)

=

J_Kah(x)d~(x)<

Kaj.t.(V) <

Jx\

w

v

< p • Lemma 1.2.2 enables us to construct Urysohn- functions h q

with respect to

i

as~ and {xq} as~ (q•1, ••• , M-1; q,iK),

q q M-1

their integrals 'being immaterial. If we define f = Kah.IT q=l (1-hq)a.

qfK

(a> 0), then f(x)

=

0 (q=1, ••• , M-1; q,iK) and f(xK) = Ka so that

r[n]

has th; properties (a), (b) and (c), no matter what value of

a.

is given. Since lim

J

fd~

=

J

KahdtJ. =

Ka~(h)

>

p and 19

20

lim

J

fdiJ. =

J

KahdiJ. < p there exists all exponent ex for which

ex-oo X X\W

J

X fdl! = p ( cf. Lebesgue 1 _{s theorem on monotone convergence (Zaanen}

(15])). Q.E.D.

no repetitions, and if ex, ~, y are generalised real numbers, 0 < < ex ..; ~ .,.; y .,.; oo , then there exists an unbounded, non- negative ,

continuous function f such that IJ.(f) = ex, lim inf

f

[N]

= ~ alld

N-oo

lim sup

f[N]

=y.

N-oo

Proof:

I.

The case

0

<ex

=

~ y

=

oo is trivial : we take a non-negative continuous function f such that IJ.(f)

=

oo.

II. We shall now prove the four cases 0 < ex .,.; ~ .,.; y < oo • By co-rollary 1.1.2.1 there exists a sub-sequence

{XJc.} ;:

1 of t

tend-J ing to oo as j - oo Blld such that

k.

lim~= 1. k.

j -00 J

lemma

1.1.3

this sequence contains a sub-sequence J,.

which still lim J.+1 = 1 alld moreover

i -00 J,i 00 _{-1 1} ~.

.e.

<

2 • J.=1 J. According to for

(in lemma

1.1.3

we choose e.g. m.

J

(j +1 )P (j

=

1 ,2,3, ••• )

for some suitable exponent p >

0).

In

view of

(2.3),

using lemma 1.1.5, we may construct disjoint neighbourhoods wJ,. of xJ,. with compact closures such that

J. J.

(i=1,2,3, ••• ) .

For the construction of the required function f we shall make use of a sub-sequence {n } 00

1 •

{.t }

00

r r= i r=1 of { }

00

.ei

i=1 by means of a

repeated application of lemma 1.2.3 in the following inductive process:

(a) We start by defining n₁

=

K₁

= .t

₁, L₁ = .t 1 + 1,

u

1 = Wt , p1

=

-ba, a1

= y,

e:1 1 1 =2· Notice that K₁a 11J.(U1) > .t1

r.e;

1

;;. a> p₁, therefore lemma 1.2.3

is indeed applicable. Let £

1 be the corresponding function, the

existence of which is proved by this lemma.

( ) b Now let nt' pt' et' ft t={ }r-1 ₁ be determined by lemma 1.2.3 so ·

-t -t

that pt .. 2 a,e:t = 2 , Sft c Wnt (t=1, ••• , r-1). Let gs

=

=

z

~=1ft

if

1

""s"" r -

1.

Then gr-1

[N]

= E

~=~

ft [N] tends to tt(g 1)

= a:(1 - 2

1 -r) as N .. 00 • Obviously, a(1 - 21-r) < p(1 - 2-r)<

r-< p, which makes it possible to define the data for application of lemma 1.2.3 for the construction of f , viz.

r n

= K

=

Min {.t. l.t.>n ₁, VN;p. •

g

(N] < p(1 - 2-r)} r r ~ ~ r- "'i-1 r-1 1 = Max (K , M 1 ) + 1 r r

r-u ..

w

r

n

r

o.

_r

=

Y - ; _-r-1

[n ]

_r

Notice that K o IJ.(U )

>

n a n _,

>

y - p(1 .. 2-r) ;;. p2-r ;;. a2-r = p

r r r r r r r

(cr.

the hypothesis of lemma 1.2.3). Finally,

£

is defined as

Some of the properties of f are immediately clear:

(i)

f is continuous since all f are continuous and have dis-r

joint supports; (ii) ~(f)

=

l: ; :

1 ~(fr)

=

a by Lebesgue 1s theorem are positive;

(iii) lim sup f [N];;., y since for all r we obtain N-oo f [n ]

=

:E rt 1 ft [n ]

=

g

1 [n ] +

f

[n ] = r = r r- r r r

= g

[n ]

+a

=

y r-1 r r

( i v) lim inf f [N] "'" ~ , because for all r we have N-oo

f

[n

-1]

= g

[n

-1] < ~ (1 - 2-r) <

p •

r r-1 r

since all f

r

As lim inf f[N];;. J.l(f) ('of. Helmberg

[7 ])

we may conclude N- oo

that in the case a=~ we already have proved lim inf

f

[N] = f3

N-oo

To prove the required equality signs in the remaining cases let us assume that ~ is an arbitrary positive real number.

Consider the following arguments (a) and (b)

(a) If r

1 ;;. r0(n) then for all s > r1 we have

(2.6) :E s (p + £ ) < E oo (p + £ ) .. (a+ 1 )2 -r1 <

~

r=r.;+1 r r r=r₁+1 r r

(b) If r

1 is fixed and N> N

0

(r

1

,~) then for all r"'" r1 we obtain

l'f

[N]- 2-ra

I< ~2-r

so that r

(2~

7) E

:~

1

!'fr [N]- 2-r

aj

<

~

We take r

1 = r0 (~) in (a). Then the integer N0 (r1 ,~) is a constant

which only depends on

n,

say, N

0(r1

,TJ)

= N

0

(~). Suppose that s

a. n > N (rJ) 8 0

b. P.2 -8-1 <TJ

(2.8) _{..ek-1 - 2 1J}

c. 1 -

_..ek

< - -_{y +1)} for all ..ek > ns d. 8 ;;;,. r

1

•

.t

The condition (2.8c) may be satisfied since lim

~~

=

1. In the k -00 _k

interval ns ~ f'T < n

8+1 the functions

f[

N]

and gs

[N]

are identi-cal. Restricting ourselves to the interval n ~ N< n we may

S S+1

therefore consider g _s

[N]

instead of

f

[N ].

'::'his is useful since the behaviour of

g

[N]

for N;;,.

s determines the value of n S+1

(cr.

(2.4)).

We obtain gs [N]= Z r,

f

[N] + Z r2

f

[N] r=1 r r=r +1 r 1 +I: s

f

(N

J

r=r 2 +1 r (N ;;a.n ) s where r

2 is defined as the maximum of r1 and the greatest index r

for which

M

~n (it may occur that some of these sums are

r s

empty). We consider the sums in

(2.9)

separately, calling them

(1)

_s1

= l: r_{r=1 r} 1

1

[N] differs from a ( 1 - 2 -r1 ) by an amount less than 1J

(cr.

(2.7), (2.8a)).

r2 - r2

( 2)

o

~

s

2 =

z

r=r f

[NJ

< I:

(p

+ e: )

1 +t r r=r1 +1 r r

since for these values of r we have M ~ N (cf. lemma 1.2.3 (c)).

r

(3)

S

=

I: 8

f

[N] • For these values of r we have n < M so

3 r=r +1 r s r

2 K a

that by lemma 1.2.3 (b), (c) we may write

f

[N] = rNr + bN ,

r

,r

Ka

where 0 ~ bN < p + e; and therefore

s2

=

z

s ....!:....£ +

s

I

, r r r' r=r 2 +1 N 3 ' where 0 ~ s~ <

z

s 1 (p + e ) • r=r 2 + r r

Combining these results we obtain estimates for

g

[N] in the in- 23 s

24

terval N ~ n :

B

-r

<

a (1 - 2 1 )

In other words we have

(2.1

0)

a (1 -2 -r, ) + 11:6 K a - 11

<

g

[N]

<

N r=r₂ +1 r r s

-r, 1 s

< a: (1 - 2 ) + -N l: K a + 21') (N;;. n ) ,

r=r₂+1 r r s

which may symbolically be written in the form

L _-N,s

<

g

_s [N)

<

RN _,s (N ;;. n )

s

•

s

I t should be remarked that the sum l: ₁K cr depends only on s r=r₂+ r r

and 1') (cf. the definition of r

2 in (2.9)). Therefore we may

con-clude

(2.10b) R is a non-increasing function of N (N ;;. n )

N,s s

Moreover, R - IL. has the constant value 31'Jt so that N,s -N,s

- [N] > R._ _--N,s - 311 (N ;;. n ) •

s

An explicit estimate for 1:8 +₁K

o

follows from (2.10) if we r=r₂ r ~

substitute in it N = n and consequently

g

[n ] = y • Then we obtain

s s s

(2.11) y-a:(1-2-r1)-21')<-1 l:s Kcr <y -a(1-2-r1)+1'J• n r=r +1 r r

s 2

Combination of the right hand inequalities of (2.10) and (2.11) yields for

N ;;.

n

s

g

[N]

< a:(1 -2-r1)

s

n Bearing L"l mind that : <EO 1

g

_s

[N]

< y + 3'1l n +~ N

-r,

(y-a:(1 -2 ) +1')) + 2'1) we conclude (N ;;. n ) s

•

or

f

(N]

<

Y

+ 3TJ

(n :;;;

N <

n

)

s s+1

'I'his holds for all sufficiently large values of s ( cf. ( 2. !Ia)). Hence lim sup

f[N]

:;;;y so that in view of (iii) lim sup

f[N] "'Y•

N-w N-®

Jror the proof that lim inf

f[N] ;;;.

p

we distinguish two, cases: N-oo

(a) RN _,s attains no value _{· ·} < ~- TJ (n _s

<

lkn _s+1 ). 'l'hen, by (?.10°)

f[N

]=

g

_s

[N]

> ~ - 4TJ (n _s

< N

< n ).

8+1

(b) R-..

< p

~ TJ for some N (n < N< n ). Let N be the greatest

~.,a s s+1 o

value of

N

in this interval such that

~

;;;,.

p -

TJ• By (2.10b'c) o's

i t follows that for all N such that n < N < N

s 0

(2.12)

g

_s

[N]

==

f(N]

> ~ - 3T) ;;;..

p -

4TJ 0

,s

b .

From (2.8 ) we deduce that for all N

>

N

0

g

[N] < RN <

p -

TJ <

p

(1 - 2-s-1 )

a ,s

so that by (2.4) we obtain

ns+1 <Min {.ei

I

.ei;;;.. No + 2} • Suppose

.ek

== Min {.e

1

l.e

1;;;.. N0 + 2} • Then we have the inequality ns

<

.ek-1 < No

+

2

<

.tk ' and

R = R - (R - R ) ;;;. t:! - TJ

-(J_ -

j_ ) E 6 K a •

.t s N s N s J, s ~"' N L r=~ +1 r r

--k'

o'

o'

k'

o

·1c G

We may write for the last term

<__!!_ (y+1)) = T)

Therefore Ril.. _{__ k,s} > ~ - 21) and consequently by the monotonicity

property (2.1 ob) we obtain

R

1 >

~

- 21) , which implies (cf. 2.10b' 0 ) n₆+1 - ,s (2.13)

Combination of (2.12) and (2.13) yields

•

f

(N]

> j3 - 5lJ (n .;;;;; _s N < n _s + )

1

so that both in case (a) and (b) we have, under the hypotheses (2.8a,b,o ,d), the inequality

flN]

>

~-

5'1) (n .;;;;; N < n ), which

s s+1

proves lim inf

f[N];;a.

p.

Therefore (of. (iv)) lim inf

f[N]==j3,

N-co N .... co

which completes the proof of theorem 1 • 2.1

O<cx.,.;pc;;;y<oo.

in the oases

III. The remaining three oases ( 0 < ex < oo, y • cot a .;;;;; j3 < y) may

be proved by slight modifications of the foregoing proof. We

shall restrict ourselves to the most important definitions and

consequences:

00

A. I f j3 < oo we define a sequence of functions {f } ₁ as in II

rr==

by an inductive method, using lemma 1.2.3 as follows:

(a) We start by defining n

1 -K1 ==.t1, 11 =.t1+1, U1 =W.e/ p1

==ia,

a1 ==j3+1, e1

=i

and construct the corresponding function f

1 by lemma 1.2.3.

( ) _{b If} { _{nt' Pt'}

_et,

_{ft t=1} }r-1 are determined by lemma 1.2.3 so that

-t -t - ( )

Pt•2 a,et'"'2 , s f

cw

t=1, ••• ,r-1 andg

t

~ s is defined

b ~ 8 f J..f

(2.14) n = K = Min{.t.!.t. >n 1 , VN._ 11 1

g

1 [N] <

13(1-

2-r)} r r 1 1 r- ...-N.- r-L .., Max {K , M } + 1 r r r-1 U =W r n _r -r p = 2 a r a = ~ +r -

g

1 [n ] r r- r -r E: = 2 r 1 Now we put f = l! 00

1 f and immediately conclude that JL(f) = a and r= r

lim. sup

I

[N] ;;~> lim.

I[

n ] = lim.

(13

+ r) = oo • The proof that

N-oo r-oo r r-oo

lim. inf

I

[N]

= 13 may be given in a manner quite analogous to N-co

the foregoing proof, as follows: obviously, lim. inf

I[N]

c;

13.

N-co

Now

let us assume 1') > 0 and consider the same arguments (a),

(b),

(2.6) and (2.7), taking r

1 = r0

(1'J),

N0(r1

,1'J)

= N0

(1J).

Supposethat

s has the following properties (of. (2.8))

(2.15) a. ns > N 0

(TJ)

A -s-1 b. I"' • 2 <

TJ

c.

l -2 ( k-1 ) -r1 2 1 -

:t

(13 -

a (1 - 2 ) -

3TJ)

< TJ k

We may apply exactly the same argumentation as in the lines as far as (2.10°). In particular, (2.16),(2.10a),(2.10b) and (2.10°) remain valid (the reasoning then following, which leads to the statement lim sup f[N] = y is irrelevant to the present case,

N-oo

since we have already shown lim. sup f[N] = oo ). The discussion_ 27 N-oo

28

of the cases (a) and (b) remains also valid without change up to the point where in case (b) we find

Since ~ +

1,6 <

f3 -

1'), the last factor in the right hand side of

0

this inequality may be majorised by (2.10)

Hence .ek 1 - 2 -r1

-

-

~

)(f3 -

IX (1 - 2 ) - 31')) > R, >

f3 -

T) - 2(1 "'k's >

f3 -

21') (cf. (2.15c)) • Now the monotonicity of

RN

implies

,s

g

[N]

=

f

[N]

> j3 -

51l

s (n .;;; s N

<

n s+1 )

and lim inf

f

[N] ;;..

f3.

Since the converse inequality has already N ... oo

been shown we obtain lim inf

f

[N·]

= j3 • N ... co

B. The case IX < oo , j3 ==

r

= oo may be treated as follows:

-r We take n = K = J, , L

=

J, + 1, U

=

W ~ , p = 2 IX,

r r r r r r "'r r

and a > p and so large that for all r

r r

z::

₁

f.[N]

> r J• J

-r

Then we define f

=

.E 00 f and immediately verify that ~J.(f) • a,

r=t r lim

f[N)

""oo • Q.E.D.

N-oo

1.3. UNIFORMLY DISTRIBUTED SEQUENCES WITH REPETITIONS

In Section 1.2 we proved theorem 1. 2.1 under the assumption that the sequence

t

contained no repetitions. It remains an open ques-tion whether a similar statement holds for uniformly distributed sequences with repetitions. This Section will show that the case of sequences with repetitions cannot simply be reduced to the case of sequences without repetitions:

Let 1J = {yk}:

1 be a dj.l-uniformly distributed sequence. Consider

the sub-sequence 1J 1

= {z .}

~

1 = {yk } ~ of all points yk

(j

=

J J= j J-1 j

= 1,2,3, ••• ) which do not coincide with any of the preceding points y (r

<

k.) of the sequence TJ•

(If

1J has no repetitions

r J

then TJ I = TJ; for every 1J the sequence TJ' has no repetitions).

One might expect that 1]1 _{is d~J.-uniformly distributed too, and then}

try to apply theorem 1.2.1. This conjecture, however, turns out to be false,

as

may be shown by the following example.

EXAMPLE: Let X be the unit interval ] O, 1 ] and let 11 be the Lebesgue-measure on the Borel-subsets of

x.

The sequence

{ }

00

t; = z j j=1

wh ere z _1' Z = 2.e -1

(p

=

-'D)

1

=

_#+.e

_2p+1 0, 1 , 2, ••• ;

.e

= 1 , ••• , ;c- , is a

sequence without repetitions. It is easily seen that

C

is not dfl.-A(

C,

]o,-k]'

2P +2p-1)

=

_g_ for all uniformly distributed, since

??

+~-1

3

p

=

1,2,;, •••• We shall construct a dj.l-uniformly distributed se-quence 1] such that T)1

=

t.

_{To this end we define 1]}

=

{yk}~

1

,

30

(mod

1 )

(p =

o,

1 , 2, ••• ;

.t

=

1 , ••• , 4P)

(mod 1) (p=0,1,2, ••• ; .t=1, ••• ,2.4P)

The structure of ~ becomes clear, when we arrange its elements into a scheme of rows which has to be read in the onier of its rows from top to bottom to generate ~; the total number k of elements in ~ up to and including each row is placed at the end of each row, and the elements of ~· are placed in parentheses:

k

(1)

1

Ct)

2 1 1 2

4

(:})

(t)

1 _l. 4 4 8 1 1 3 ₁

4

2 4 1 1 3 ₁

4

2 4 16 (i-)

Ct)

(f)

Cf)

•

•

•

1 3 5 _1.. 8 8

s

8

32

1 ₁ 3 1 5 3

t

a

4

a

2 8 4

•

•

•

1 1 _.! 1

t

3 _1..

a

4 8 2 4 8 1

64

•

.

.

• • •

.

•

. .

• •

It should be remarked that among the first 4P (rasp. 2.4P) ele-ments of TJ each of the rational numbers np (n = 1, ••• , 2P) (resp •

2

~~

₁

(m=1, ••• ,2P+1)) occurs

exactly~

times (p=0,1,2, ••• ), so that, i f ~ is a real number, 0 < 13 .s;; 1, we obtain

{

A(TJ,]O,~] ;4p) • [~13]2P

A( TJ, ]0, p]; 2.4P) = [~+1 13]2P

and

•

The total number of elements in ]0,13] occurring in each row between k = 4P and k

=

2.4P is equal to [2P13 + ~], and if 2.4P < k .s;; 4p+1 this row-score attains the value [ 2p+1 13] • In view of (3. 3) this implies for all p = 1, 2,

3, •••

{

[~13]~

+ (.e-1

)[2P~ +~

]<A(TJ,]O,p]

;k)<[~p]2P +.t(~l3 +~]

. p ( ) p p _n ( _ P) l.f 4 + .t-1 2 <k<4

+.t.;c-

.t-1, ••• ,2 ;

{

(~+1

J3](2P

+.t

-1) < A(1),]0,p] ;k) < [2P+1J3](2P

+.t)

i f 2.4P +

(.t

-1 )~+1 < k < 2.4P +.t.~+1

(.e

= 1 ' ••• '

~)

•

From the inequalities (3.4),

(3.5)

it follows from elementary

estimates that

(3.6)

f3 -

21-p <

A(TJ,]~,p];k)

<

f3

+ 21-p 1

provided that 4P•< k .s;; 4p+1 •

Hence lim A(TJ,

]~,~];k)

= p, and, since f3 is chosen arbitrarily k-00

between 0 and 1, we obtain for each interval I= ]a,f3]

(3. 7) lim

A(!),]~,@]

;N) = 11{]a,J3]) ,

32

. which implies that TJ is d~t-uniformly distributed. It is easily seen that TJ 1

=

_{C , so that the assertions in the example are} proved.

CHAPTER II

2.1. ( lf,d 1d. SUMMABILITY • IH-rRODUCTIOH

Definition 2.1.1. A Borel-measurable function f is called (~,d~)­ -summable, if f is integrable and lim

i

l: : .. ₁ f (:x:n)

~(f)

•

N-oo

It is well known that every bounded continuous function ( in fact every bounded R-integrable function on X; cf. e.g. llelmberg [

7])

is (~,d~)-summable.

Our intention is to investigate the (~,d~) for a

larger class of Borel-measurable functions. Chapter I shows that there are limitations to the possibility of extending the range of functions, for which (t: ,d~ )-summabili ty can be established, even when we consider continuous functions only.

I t turns out that the concept of local R- integrability which

will be defined now, is suitable for our purpose.

Definition 2.1.2. A real-valued Borel-measurable function f is

·called locally R-integrable, if for every compact set V with zero-boundary and for every &

>

0 there exist two continuous functions f₁ ~nd f

2 with compact supports, such that f1 Xy IE> fXv IE> f2Xy and

~((f2 - f1

)xv)

< & •

Remark. Every continuous function is locally R- integrable. Section 2. 2 gives the topological and. measure -theoretical back-ground for some theorems, which will be proved in Section 2.

3

In Section 2.4 we shall discuss that the results of Section 2.3 are best possible in a certain sense.

from now on that the support S(~) coincides with X itself.

Besides, we make the notational convention to write 1.1. both for

*

the measure defined on

X

and for its natural extension on

X •

2.2. TOPOLOGICAL AMD MEASURE.THEORETICAL PRELIMINARIES

The following lemma both sharpens and generalises lemma

1.1.4

(cf. the remark at the end of lemma

1.1.4).

*

Lemma 2.2.1. If A and Bare disjoint closed subsets of X ,

I.I.(A) +!!(B)< 1, IJ.(oA)

=

IJ.(oB) =

o,

and i f vis areal number such that ll(A) < v

<

1 - IJ.(B), then there exists an open set C => A such that C 11' B = ~ and

JJ.(C)

= !l

(c')

= v •

Proof. We shall use the notation A=

u<

1 ) , B =

u<

2) • Since X

*

is a compact Hausdorff-space, it is normal, and therefore there exist open neighbourhoods

z(

1) and

z(

2) of

u<

1) and U(2) respectively,

such that Z ( 1 )

n

Z (2 ) =

¢.

As 1.1. is a regular measure, there exist

open sets

v<

1) and W(2) with zero-boundaries, such that

u<

1 ) c c

v(t)

₀ c

z(~) and~

c

w<

₀ 2 ) c

z(

2) _, and that moreover _{, ,}

(2.1)

Jl(V~

1

))

<

v < 1 -

Jl(W~

2

))

•

Suppose that

9.>(1)"' {V.(1

_k

)}co

_k=1 is an open covering of

w<

2)

'

=

0

=

X* \

w~

2

),

consisting of

V~

1) and a sequence of open sets

{v~

1

)};₁, each of which has measure < 2-1 and zero- bounda.zy• According to Heine-Borel's lemma a finite subcollection

{v~

1

)}~

0

'

already covers

W~

2

)

and we consider the expanding rstem of open

(1) Nt

sets

{Uk

}k=o defined by

For exactly one index k1 we have

(2.3)

11(u(

1 ))

~

v

<

11(T1(

1) ) <

!!(H(

1 )) +

2-1

-k1 -k, + 1 -k,

and the corresponding set

u~

1

)

has zero-boundary (see Fig. 1). I f

11(~

1

_{)) = v, the proof is}

_c~mpleted

_{by taking C =}

_~

1

₎

_{, since}

1 I 1

rr<1 ) c

w<

2) , which is a closed set disjoint with

U(

2) , so that

-k1 0

CnB=¢.

Fig. 1

Therefore, let us assume that

!1(~

1

J)

< v. We define u(3) =

~

1

>.

1 1 _ _ 1

Because

u<

3) c

W~

2

)

we obtain

:;;:T2'Jn u(

3) =¢and !l(U(2))<1- v<1-- !l(U(3 )). Following the same method as before, we may state the existence of open sets

V~

2

)

and

W~

3

)

with zero -boundaries, such that

u(

2) c

v(

2)

u<

3 ) c

w(

3)

v(

2)

n

w<

3) =

¢

and

0 ,, 0 ' 0 0 ,

•

Now let

~

(2) •

{v.(

2 )} co be an open covering of

w<

3)

1

=

x*\w(

3)

k k=1

o

o '

consisting of

V~

2

)

and a sequence of open sets {

V~

2

)h:

which has measure < 2-2 and zero-boundary. By Beine-Borel's lemma N

a finite subcollection

{v~

2

)}k~o

already covers

w<

3) 1

and from

0 '

{ (2 )}N2

the expanding system uk k=o' where

(2.5)

(o ,.; k ,.; N )

2

exactly one set, say

~

2

),

satisfies the inequalities

2 (2.6) and has 11. (u (2)) ,.; 1 - v < 11. (u (2 ) ) < ll (u<2)) k . k +1 k 2 2 2

zero-boundary (see Fig. 2).

-2

+ 2

Fig. 2

1 - v then the proof is completed by taking C

=

(u~

2

0

'.

2

(Obviously C is Open and has zero-boundary. 0 I : :

~

2 ):::>

~

2 ):::>

V~

2

):>

( )' 2 2

:> B, so that C is contained in a closed set V

0 2

disjoint with B.

Hence

C

n

B

=

¢.

On the other hand C :>

W~

3

)

:>A, so that all con-ditions for Care satisfied). Assuming that

!l(U~

2

))

< 1 - v, we define U( 4)

=

~

2) and observe that we are in th! same position

2

by 2. In the same way we continue this process by induction, choosing in the jth step an open covering

~(j)

=

{v~j)}~

0

of

'

W~j+

1

₎

_{consisting of}

_V~j)

_{and a sequence of open sets}

_{v~j)}~

1

each of which has measure < 2-j and zero-boundary. If this process results (in the manner just described) in a set U(j+2) of measure v (if j is odd), or in a set U(j+2) of measure 1 - " (if j is even), then the proof of the lemma is completed by taking

I

(•+2)

(•+2)

C

=

U J and C = U J respectively. Otherwise, we obtain two expanding sequences of open sets

{

(A

c

)u(

3) c

u<

5) c

u<

7 ) c

(B

c)u(

4 ) c

u<

6 )

c u<

8 ) c

...

,

and

...

'

such that

u(k) n

u(j) =

¢

(k

+ j odd),

1~-t

(u(2 i+t)) - v 1

<

2-2i+1 (i=1,2,3, ••• ) and

l~-t(U(2i+

2

))-

1 +vI

<

2-2i (i

=

1,2,3, ••• ). Now U00 u(2i+1) and U00 U(2i+2) are open disJ·oint sets of

i=1 i=1

measure \1 and 1 - v respectively, and neighbourhoods of A and B respectively. Hence the set

c,

defined by C = U

.'>0

U(2i+t) has

l.=1 ,

the desired properties, which proves lemma 2.2.1.

Lemma 2.2.2. (Interpolation lemma). If A and D are subsets of X with zero-boundaries, A is compact, D is open, and D :::> A, and i f v is a real number, such that !l(A)

<

v < !J.(D), then there exists an open set C with zero-boundary and compact closure, such .that A c C c

C

c D and !J.(C) = v •

*

Proof. Consider A and D as subsets of X • We distinguish two cases:

*

D is compact in X). In this case lemma 2.2.1 applies i f we choose

*

Il = X \ D, and the lemma is proved.

*

(2) D'

=

X \ D contains the point co as a boundary- point. Now

*

we construct an open neighbourhood E of oo in X with zero-bound-ary, such that

En

A

=

¢and

ll (E)

<

IJ.(D) -

v. Then the set

B

de-*

*

fined by B = (X \ D) U E is closed in X and has zero- boundary, B

n

A=

¢,

and

*

IJ.(A)

< v <

IJ.(D) - IJ.(E)

=

1 - {J.L(X \D)

+

J.L(E)} .;:; 1 - J.L(B) ,

so that lemma 2.2.1 may be applied.

Q.E.D.

Lemma 2.2.3. If

{a }

00 is a sequence of real numbers satisfying n n=1

0 < a < a < • • • and lim a = 1 , then there exists a sequence

1 2 n

n-oo

of open sets { C } 00 in X with compact closures and zero-bound-n zero-bound-n=1

aries, such that

(ii)

U 00 C =X

n=1 n = a

n (n= 1 ,2,3, ••• )

•

*

Proof. The normality of X implies that there exists a strictly contracting sequence of open neighbourhoods

{F.}':'

of oo , such

l. :!.=1

that

x*

=

F ;:)

F ;:)

F ;:)

F ;:) ••• , J.LcF. \F.)=

o

(i=1,2,3, ••• ),

1 1 2 2 l. l.

n ':'

F.

=

{co} ,

and lim 11

(F. ) =

o.

Consider the sequence

{A.}':' ,

l.=1 l. . l. l: l.=1

l . -

co

*

defined by

A.=

X

\F.

(i=1,2,3, ••• ) and

l. l.

*

-defined by

D.=

X

\F.

(i=1,2,3, ••• ). l. l.+1 the sequence

{n

1} 1.=1 ~ ,

Then the pair

{A. ,D.}

]. ].

is defined as ~i = ~(Ai) (i=1,2,3, ••• ), then ~(D.)=~- (i=

~ ~-11

= 1 ,2,3, ••• ), and

o ..

~ <P <~ < ••• ,

1 2 3 . lim ~. 1 = 1.

1 - 0 0

It follows that for all <X (n= 1 ,2,3, ••. ) there exists exactly n

one ~ i' such that ~i .,.; an< ~i+

₁

, and that for fixed ~ i there are at most finitely many ex satisfying~ . .,.; <X <

p.

(such a "block"

n 1 n 1+1

of elements ex .between two consecutive elements

p

may pe empty). Let us consider a non-empty block

We define C i f <X = ~-, otherwise C is defined

n. 1 1, =A. 1 n. 1 l. n. J., 1, 1

as a set of measure ex , obtained by interpolation n. 1

1,

and D. in lemma 2.2.2. In either case C and D;

1 n. 1 ... 1, between A. 1 satisfy the hypotheses of the interpolation lemma, so that the sets

with measure <X

(.t=

2, ••• ,k(i)) may be constructed by induction

n. b

1,.., •

as interpolatory sets between C and D. (lemma 2.2.2).

n. b J.

1,..,-1

Finally the sequence {cn}~

1

is obtained by ordering the blocks

{c

}k(i) •

n . . .

1, J J=1

The required properties (i) and (iii) of the sequence {cn}~

1

are clear. (ii) follows from the fact that for all ~. (j = 1, 2·,

3, ••• )

J there exists an element <X >

p . •

In other words for all

n J

exists a set C ::>A .•

n J

*

00 = X \ (

n

_{j=1 F j) = X,} 00 00

*

)

Since _{U j=1 Aj = U j=1 (X \ Fj} 00 we obtain U C = X • n=1 n A. there J

The following lemma is concerned with finite expanding sequences of open sets with compact cl~sures and zero-boundaries. The nota- 39

40

tion used in this lemma is chosen in order to facilitate its ap-plication in theorem

:":.2.1.

Lelllllla 2. 2.

4•

Let { Gi} i:m+₁ be a sequence of open sets with compact closures and zero-boundaries, satisfying

Gm+t c G _m+1 c G _m+2 c

G

_m+2 c ••• c G, and ll(G.) = y. (i=m+1, ••

.£ ~ J.

•• ,.,e).

Let H be a closed subsRt of G.£ with zero-boundary and sup-pose H + = H _m

n

G + , = H

n

(G. \ G. )

(i

=

m + 2, ••• , l),

1 m 1 J. ~-1

!l(Hi) = T)i (i m+1, ••• ,,e). Let {ei}f..m+t be a sequence of real numbers, satisfying

(a) 'T)i<9i<yi-yi-1 (i=m+2, ••• ,,e)

(b) '!Jm+1 < 9m+1 < Ym+1 •

Then there exists an open set E with zero-boundary, HcEcEc G,e' such that for the sets

F.,

_~ defined by

F

_m+t

=En G

_m+1 1

F. =En

_J.

n (G.\ G.

₁

)

(i=m+2, •••

,.£),

we have

!J.(F.)=e.

(i=m+1, •••

,,e).

l. J.- l. l.

(see Fig.

3).

Proof. Let b be defined as b =

t

min{e. -TJ·Im+1.;;;; i.;;;; ,e}.

Ob-l. l.

viously we have !l (II) + b < 1J

,e'

so that in view of the interpola-tion lemma there exists an open set L with zero-boundary, H c L c c L c Gn, and

fl.(L)

=

[l(H) + b • Since [J.(L

n

G ) .;;;; [J.(H ) +b <

N m+1' m+1

< 8_m+1 < Ym+1 and ll(L

n

(Gi \ Gi-1 )) .;;;; ll(Hi) + 0 < 8i < Yi - Yi-1 (i=m+2, •••

,.t),

neither of the sets G \Land (G.\ G. ) \ L

m+ 1 J. 1.-1

(i

=m +2, ••• ,.£) is empty, and because all these sets have zero-boundaries and positive measures, the interpolation lemma applies when we take

A=

¢,

D

=

((G. \G. ) \ L)"

and v

=e.-

p.(Ln

(G. \

l. l.-1 l. l.

\G.

1)) (i=m+1, ••• ,,e; we put G =¢).Let M. (i=m+1, •••

,,e)

J.- m J.

be the interpolatory sets. Then the set E = L U ( U . .t M.) has J.=m+1 J.

the required properties.

We shall now deduce a theorem, which will serve as a basis for the theorems on (~,d!l)-summability of unbounded locally R- inte-grable functions as formulated and proved in the next Section.

It will state the existence of an infinite expanding sequence of open sets with compact closures and zero-boundaries, which behave in a specified manner with respect to a given d~-t -uniformly dis-tributed sequence. IJ.he significance of the measure -theoretical requirements, stated in theorem 2.2.1 will become clear in the sequel (of. Section

2.3).

'I'HEOREM 2. 2.1. Let ~ =

{x }

00 be a d!l-uniformly distributed

se-n n=1

quence in X, and let e be any real positive number. Then there exists a sequence {Ek}~

0

of open sets with compact closures and zero-boundaries, such that

(a) (b) (c)

E =¢,!E

0 1 c

-

E 1 c E 2 c E 2 c ••• (k ..

o,

1 '2' ••• ; N=1,2,3, ••• ) 41

42

(d) !l(~ \ ]\_

1} /!l(]\+1 \ ]\) = c, where cis a constant

1

(as a consequence we have c =

_{1 -p.(E ) ) •}

1

Proof. Let

o

and p be fixed real numbers such that

(2.8) 1 +

t

0 < p

(1+o) ·_;e__

₁ <1

+e •

p-It is not difficult to find numbers

o

and p with these properties. We choose for example

o =

t

e and p sufficiently large.

lemma 2.2.3 there exists an expanding sequence of open sets {~}=

₁

with compact closures and zero-boundaries

(k=

1 ,2,3, ••• ), such that

c1

cc

₁ c

c

c:c

c ••• 2 2

(2.9)

_{uk=1 ck}

00 =

x

p.(Ck)

=

1-

~(1

o)

(k=1,2,3, ••• )

_•

p

Our intention is to extend each

Ck

in an app~priate way to an open set

Ek

with the required properties, and satisfying p.(~)

=

=

1 - 1k (k=1,2,3, ••• ). We use an inductive method:

p

(1 ) The construction of E 1

Since

t

is d!J.-uniformly distributed we have (cf.

(2.9))

A(!;,,C

1 ;N)

lim N

=

p.(C )

=

1 -

1

(1 +to)

N .... "" 1 p

from which it follows that A(t,c

1 ;N)

(2.10) N >1

1

(1

+

0)