NUMBER 0FDIMENSIONS. THE THEORY OF INTEGRATION IN h SPACE OF AN INFINITE

THE THEORY OF INTEGRATION IN h SPACE OF AN INFINITE NUMBER 0FDIMENSIONS.

B. JESSEN ill CoPENIIAG I~.N'.

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I O . t~ I I , I 2 .

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"32--34198.

Contents.

Introduction.

The torus-space in an infinite number of dimensions.

Limit points. Covering theorems.

Continuous and semi-continuous functions.

The Lebesgue measure.

The construction of nets.

The transferring principle.

The Jordan measure.

The definite and indefinite integrals.

The Riemann integral.

An important lemma.

Application Of Fubini's theorem.

Infinitely multiple integrals.

Representation of a function as the limit of an integral.

Strong convergence.

Majorised convergence.

Fourier series.

A special orthogonal system.

A case of Birkhoff's ergodic theorem.

Application to almost periodic functions.

0rthogonal series whose coefficients are analytic functions.

General exponential series.

Analytic almost periodic functions.

Distribution functions.

Acta mathematica. 63. I m p r i m 6 le 6 jllillet 1 9 3 4 .

250 B. Jessen.

w 9

w 26.

w 27.

Weyl's theorem on equal distribution.

Application to a class of almost periodic functions.

Distribution of the values of certain classes of analytic functions.

A class of analytic almost periodic functions.

w I. I n t r o d u c t i o n .

T h e object of the p r e s e n t p a p e r is to s t u d y in g r e a t e r detail t h a n has been done before a c e r t a i n space of an infinite n u m b e r of dimensions, in which a t h e o r y of i n t e g r a t i o n can be developed. T h e 'space in question was first con- sidered by Daniell i n c o n n e c t i o n with his studies on i n f e g r a t i o n in a b s t r a c t spaces~; since t h e n it has been i n v e s t i g a t e d by Wiener, Steinhaus, P a l e y and Z y g m u n d , Carlson and myself. ~ T h e t h e o r y has applications to a n a l y t i c a l problems and to problems in t h e calculus of probabilities.

L e t us consider a real or complex f u n c t i o n f ( x l , x~, x3, . . . ) d e p e n d i n g on a sequence of real variables; such a f u n c t i o n is called p e r i o d i c w i t h the periods i, I, I , . . . if f o r a r b i t r a r y i n t e g e r s nl, ~ , ~ , . . . we have always

f ( x L , x~., xa, . . . ) = f ( x , + nl, x~ + ~ , x a + ~a, . . .).

W e r e s t r i c t ourselves to the c o n s i d e r a t i o n of such functions. As always when dealing with periodic functions, i t is c o n v e n i e n t to consider t h e f u n c t i o n s as defined n o t in the usual >>open* space b u t in t h e closed space which we obtain by replacing the c o o r d i n a t e axes by circles. Thus t h e space with which we have to deal in the p r e s e n t p a p e r is t h a t closed space or torus-space which we obtain f r o m the space of all real sequences x l , x~, xa, . . . by r e d u c t i o n of the coordinates rood. I. This r e d u c e d space is d e n o t e d t h r o u g h o u t by Q~.

I t is not usual to speak a b o u t a space before relations between its points h a v e been defined. W e give these relations in t h e n e x t few sections by i n t r o d u c i n g the notions of intervals, limit points and closed a n d open sets a n d by p r o v i n g the classical covering theorems. These sections c o n t a i n w h a t may be called the topology of the space Qr upon which the whole t h e o r y is based. I t is also possible to in- t r o d u c e a distance b e t w e e n the points of Q~ b u t n o t w i t h o u t violating the sym- m e t r y of the variables, a n d we do not need t h e notion. The f a c t t h a t we con- sider the variables as a r r a n g e d in a fixed o r d e r is only a m a t t e r of convenience.

' P. J. Daniell Ill, [2].

2 N. Wiener [I], H. Steinhaus II], [2], R. E. A. C. Paley-A. Zygmund Ill, I2], [3], F. Carlson Ill, ]',. Jessen [[], [2], I3].

The Theory o[ Integration ill a Space of an Infinite Number of Dimensions. 251 O t h e r spaces t h a n the space (2~ can be t r e a t e d in a similar way; such spaces have been considered by Daniell, Feller a n d Tornier, and K o l m o g o r o f f l ; a quite general result has been a n n o u n c e d by Ulam. 2 T h e t h e o r y is also r e l a t e d to a t y p e of i n t e g r a t i o n in f u n c t i o n a l space i n t r o d u c e d b y Wiener. 3

T h e idea of the p r e s e n t exposition is to develop the t h e o r y of f u n c t i o n s in the space Q~,, in close a n a l o g y to the t h e o r y of t h e n-dimensional torus-space Qn, obtained f r o m the o r d i n a r y ~,-dimensional space by r e d u c t i o n of the coordinates mod. I. This is possible; in fact, as soon as intervals and the measure of intervals have been defined, the o r d i n a r y definitions of e x t e r i o r and i n t e r i o r measure, and so also the o r d i n a r y definition of m e a s u r a b l e and i n t e g r a b l e func- tions, m a y be applied. T h e proof t h a t t h e m e a s u r e and so the i n t e g r a l have t h e o r d i n a r y p r o p e r t i e s is obtained, w i t h o u t the t r o u b l e of r e p e a t i n g all a r g u m e n t s , by using a simple

transferring principle;

the c o r r e s p o n d i n g principle for n dimensions has been used by Lebesgue, F. Riesz and de la Vall6e-Poussin. 4 I t would be easy to prove t h a t m y definition leads to t h e same n o t i o n as Daniell's which was based on Y o u n g ' s definition of the Lebesgue integral. - - The mos~ i n t e r e s t i n g p a r t of the t h e o r y is t h a t which deals with such problems as have no analogue f o r f u n c t i o n s of a finite n u m b e r of variables; f o r instance, the problem of w h a t m e a n i n g can be a t t a c h e d to an infinitely multiple integral. - - A m a i n p o i n t in t h e t h e o r y is the e s t a b l i s h m e n t of a t h e o r y of F o u r i e r series f o r f u n c t i o n s in

Qo~.

A solution of this problem in the case of c o n t i n u o u s f u n c t i o n s was given by B o h r in his second p a p e r on almost periodic f u n c t i o n s ; in the p r e s e n t p a p e r we a t t a c h to any i n t e g r a b l e f u n c t i o n in

Qo~ a

F o u r i e r series of the form

E CPl, P'z . . . Pn e ~ z i ( P l X ~ + p ~ x 2 + 9 9 9 + P n Xn)

(where the s u m m a t i o n is b o t h over the p ' s and over n) a n d we prove t h a t this series d e t e r m i n e s the f u n c t i o n uniquely; we also prove the P a r s e v a l a n d Riesz- Fischer t h e o r e m s for these series.

Daniell and W i e n e r used the t h e o r y of the space Q~ as an example a n d special p r o p e r t i e s of t h e space were not given. S t e i n h a u s gave two applications of the t h e o r y ; he p o i n t e d out t h a t a t h e o r y of m e a s u r e in Qo~ would m a k e it

1 p. j. Daniell [211 W. Fcller-E. Tornier [I], A. Kolmogoroff [2] 24--30.

2 S. Ulam [I]. It is of interest to remark that the main results of the present paper, in particular the theorems of w167 13 and 14, hold also in the case considered by Ulam. The proofs must be rearranged.

8 See R. E. A. C. Paley ~--3L Wiener-A. Zygmund I I ] . . . . H. Lebesgue [I] 365; F. Riesz [I 1 497; C. de la Vall6e-Poussin [I].

252 B. Jessen.

possible to generalise the theory of probabilities by a sequence of choices of real signs + I (where the two signs are supposed to be equally probable) to the case of sequences of complex signs. This leads in our notation to considerations concerning the orthogonal system e2~ixk. Translating a beautiful result of Kol- mogoroff from the language of the calculus of probabilities into the language of real functions, Steinhaus proved an interesting convergence theorem for series of the form

Z ake2Z~iXk;

k = l

he also proved a theorem on the analytical continuation of power series. Paley and Zygmund developed the theory for both real and complex signs in a syste- matic way; they obtained results not only for power series, but also for Fourier series and Dirichlet series; Carlson added certain results concerning Dirichlet series.

The present author was led to the theory in connection with some investi- gations by Bohr concerning the distribution of the values of the Riemann zeta- function, which were carried out in collaboration with the author. ~ These in- vestigations have a certain connection with those just mentioned, but here it is the case of complex signs t h a t occurs. I t proved advantageous to consider the zeta-function

(8) = Z I I ( i - n = l k = l

in relation to the general class of functions

H - 9

k ~ l

In the present paper I apply the theory of the space Q~ not to the zeta-function itself but to other almost periodic cases f o r which the details are simpler. The results of Steinhaus and some of the results of Paley and Zygmund are of importance for these applications; I give a slightly simplified exposition of these results before I give my own applications.

w 2. The Torus-spaee of all Infinite Number of Dimensions.

We start from the space of all sequences xl, x2, x3, . . . of real numbers.

Reducing the coordinates of this space mod. I we obtain a certain closed space;

H. Bohr-B. Jessen [I], [2], [J].

The Theory of Integration in a Space of an Infinite Number of Dimensions. 253 we call it the t o r u s - s p a c e Q~. W e shall h a v e in t h e sequel to consider f u n c t i o n s defined in this space. T h r o u g h t h e r e d u c t i o n of t h e c o o r d i n a t e s mod. I t h e c o o r d i n a t e axes in t h e o r i g i n a l space b e c o m e circles; we call these circles t h e

coordinate circles

of Qo~ a n d d e n o t e t h e m by cl, ca, cs, . . . ; t h e y all h a v e t h e peri- m e t e r I. T h e l e t t e r xk is in w h a t follows used in t w o different senses: b o t h to d e n o t e a p o i n t of t h e c o o r d i n a t e circle ck a n d to d e n o t e t h e abscissa of this p o i n t on ek; i n t h e l a t t e r case xk is only d e t e r m i n e d rood. I. T h e p o i n t of Qo~

w h i c h is d e t e r m i n e d by t h e c o o r d i n a t e points, or s i m p l y coordinates, xj, x~, x s , . , . is d e n o t e d by

x = ( x . . . . . ).

I t will be c o n v e n i e n t f o r us to h a v e a fixed n o t a t i o n f o r w h a t is usually called t h e

product

of a finite or infinite n u m b e r of (arbitrary) sets A1, A~, A s . . . . T h e p r o d u c t

A = ( A D A~, A s , . . . )

is defined as t h e set of all s y m b o l s x ~ : ( x 1, x~, x 3 , . . . ) , w h e r e xk b e l o n g s to Ak. I n t h i s sense t h e t o r u s - s p a c e Q~ is t h e p r o d u c t of t h e c o o r d i n a t e circles e~, ca, c3, . . . or

= ( e . %,

Of t h e g r e a t e s t i m p o r t a n c e f o r t h e p r e s e n t p a p e r is n o w t h e definition of a n

interval

in t h e space Q~. D e n o t e as an arc b on a circle c e i t h e r a n ordi- n a r y o p e n arc (perhaps t h e circle wi~h e x c e p t i o n of one p o i n t ) o r t h e circle itself. T h e obvious t h i n g to do would be to d e n o t e as a n i n t e r v a l in Q~ a n y set of p o i n t s o b t a i n e d by c h o o s i n g on a n y c o o r d i n a t e circle ck a n arc bk a n d t h e n f o r m i n g the p r o d u c t

z = . . . )

of t h e s e arcs. T h i s definition, however, t u r n s o u t to be very u n s a t i s f a c t o r y . O u r t h e o r y d e p e n d s e n t i r e l y on t h e fact, t h a t we a d m i t as i n t e r v a l s only those sets of t h e f o r m (2.I) f o r w h i c h o n l y a

finite

n u m b e r of t h e arcs hi, b~, bs, . . . a r e o r d i n a r y arcs, t h e r e s t of t h e m being t h e c o o r d i n a t e circles themselves. T h e l e n g t h s of t h e arcs a r e called t h e

edge-lengths

of t h e i n t e r v a l ; so f o r an i n t e r v a l in o u r sense ~he e d g e - l e n g t h s are all = I w i t h t h e e x c e p t i o n of a finite n u m b e r of t h e m which are < I. T h e space itself is a n i n t e r v a l a n d its e d g e - l e n g t h s are all = I.

1 In the applications we shall also consider spaces of the form

(tt'n, Qr

where /~n is, as usual, a Euclidean space,. It was found convenicn~ not to complicate the general theory by the consideration of this case.

254 B. Jessen.

I t is o f t e n c o n v e n i e n t to consider Q~ as t h e p r o d u c t of the ~-dimensional torus-space

and the infinite-dimensional torus-spaee

~n,r

⁼

(Cn+l, Cn§

W e write t h e n in accordance with our general n o t a t i o n Q~ = (Q,, Qn,~). I f x ' = (x~, x 2 , . . . , x~) a n d x " = (x~+~, x,~+2, . . . ) are two points of Q~ a n d Q,, o,, we d e n o t e the c o r r e s p o n d i n g point of Qo~ by x = (x', x"). The points x' and x " are said to be the projections of x on the spaces Q~ and Q ... I f we p r o j e c t all the points of a set in Q~ on Q,~ or Q~;~ we get the p r o j e c t i o n of t h e set itself.

T h e projections of t h e i n t e r v a l (z.I) on Q~ and Q~,r are t h e intervals

a n d

I ' = (bl, b~ . . . . , b.)

I " - -

(bn+l, bn+2, ...);

so we have I = (I', I"). I f n is large e n o u g h we have I " = Q,,,~ and so I - -

= (I', Q~, ~)). F o r an a r b i t r a r y set A in Q~ the r e l a t i o n A = (A', A") will gener- ally n o t be true. I f we have A = ( A ' , Q~,o,) or A = ( Q = , A " ) the set A i s c a l l e d a cylinder; in t h e first case its base is t h e set A' in Qn, in the second case the set A " in Q~, ~.

O u r t h e o r y of f u n c t i o n s in the space Q~ will be developed in close a n a l o g y to t h e t h e o r y of the space Q,~, b u t it should be observed t h a t t h e t h e o r y of the space Q~, contains t h a t of the space Q,~. I n f a c t a n y f u n c t i o n f ( x l , x2, . . . , x,~) in Q,~ m a y as well be c o n s i d e r e d as a f u n c t i o n in Q~, which does n o t depend on the variables x~+~, x,+2 . . . . and to any set A' in Q,~ corresponds t h e cylinder A--~

= (A', Q~, ~) in Q~. F u n c t i o n s which only depend e i t h e r on the variables xl, x,~,..., xn or on t h e variables x,+~, x~+~, . . . play an i m p o r t a n t rble in the t h e o r y .

w 3. Limit Points. Covering Theorems.

A sequence of points x ('~) = (x~ '~), x~"), x~"),...) in Q~ is said to be convergent and to have the limit poifit x = (xj, x~, x 3 .. . . ) if x~ ~) -~ xk as n -~ ~ for any fixed k (but not necessarily u n i f o r m l y in k); we t h e n write x ('~) -o x as n--~ ~ . L e t t h e sequence x (1), x (~), x(S), . . . be c o n v e r g e n t to t h e limit p o i n t x and let ~ b e an interval s u r r o u n d i n g x; t h e n it follows i m m e d i a t e l y t h a t x(") must lie in T~

The T h e o r y of Integration in a Space of an Infinite Number of Dimensions. 255 for all sufficiently large n. T h e converse of this is also t r u e : Suppose t h a t t h e sequence x (1), x (2), x(3), . . . and the p o i n t x have the p r o p e r t y t h a t f o r any inter- val /~ c o n t a i n i n g x the points x (n) of the sequence lie u l t i m a t e l y in T.~, t h e n t h e sequence m u s t be c o n v e r g e n t a n d m u s t have t h e limit p o i n t x. T h e Weierstrass- Bolzano t h e o r e m is t r u e f o r t h e space Q~, t h a t is: A n y sequence x (1), x C2), x(8), . . . of points in Q~j contains a convergent subsequence. This is proved in the o r d i n a r y way by means of the d i a g o n a l m e t h o d .

I f the limit p o i n t of every c o n v e r g e n t sequence of points of a set A belongs to A t h e n we say t h a t A is a closed set; a set A is said to be open if its c o m p l e m e n t a r y set Q ~ - A is closed. An i n t e r v a l is evidently an open set, b u t the same is n o t t r u e for a set of the form (2. J) w h e r e an infinite n u m b e r of t h e arcs bk m a y be o r d i n a r y arcs. W e shall now p r o v e t h e following more g e n e r a l t h e o r e m : A set A of points in Qo~ is open i f and only ~f corresponding to any of its point~ ~ x it contains an interval I~ surrounding x.

T h e one p a r t of this t h e o r e m is obvious: I f the set A contains c o r r e s p o n d i n g to any of its points x an i n t e r v a l L: s u r r o u n d i n g x, t h e n no point x of A can be a limit p o i n t f o r the c o m p l e m e n t a r y set Q ~ - A ; so this set m u s t be closed a n d h e n c e A open. I n o r d e r to prove the converse suppose A to be open a n d x = (xl, x,2, x ~ , . . . ) to be a p o i n t of A, and consider f o r any n t h e i n t e r v a l I (~) = (b~, b~, . . . , b,~, Cn+i, en+2 . . . . ) where bk for I _--<_ k ~ n is the arc on ck w h i c h has its m i d p o i n t in xk and t h e l e n g t h ~; t h e n it is clear t h a t I (~) m u s t belong to A for all sufficiently large n; for if n o t we should have f o r any n a p o i n t x (~) of Q o ~ - A lying in I (n) and Q ~ - A would n o t be closed since evidently

X(~)---~X a s n ---~ o r

I f we add to an a r b i t r a r y set A all points outside A which are limit points for sequences of points of A we obtain t h e closure A of A, which is t h e smallest closed set c o n t a l n i n g A. I f A : ( e ~ , e~, c a , . . . ) where ek is an a r b i t r a r y set o n the c o o r d i n a t e circle c~ we have .4--- (el, e~, ~3, 9 9 .). I f I is an i n t e r v a l we call the c o r r e s p o n d i n g closed interval.

Finally we call to mind the two classical covering theorems:

I. T h e c o v e r i n g t h e o r e m of Lindel5f. I f to any point x of a set A in Q~

there corresponds an interval I~ surrounding x then we can find a finite or churner- able number of these inter~'als which will cover A.

I n o r d e r to prove this we d e n o t e as a r a t i o n a l p o i n t on each c o o r d i n a t e circle a point with r a t i o n a l abscissa and as a r a t i o n a l arc an arc whose e n d p o i n t s

256 B. Jessen.

are r a t i o n a l (the circle itself shall also be considered as a r a t i o n a l are). An in- t e r v a l I in Q~o is now called a r a t i o n a l interval, w h e n the defining arcs are all r a t i o n a l ; evidently t h e r e is only an e n u m e r a b l e n u m b e r of r a t i o n a l intervals in Q~. N o w a n y i n t e r v a l Is s u r r o u n d i n g a point x in Q~ c e r t a i n l y contains a r a t i o n a l i n t e r v a l s u r r o u n d i n g x. This proves the theorem.

2. T h e covering t h e o r e m of B o r e l .

I f the set A co~zsidered is a closed set, then a finite number of the intervals will always be enough to cover A.

A simple way of p r o v i n g this t h e o r e m is to deduce it f r o m Lindelhf's t h e o r e m . Suppose t h a t A is covered by a sequence of intervals I(~), i(2), /<3), . . . a n d d e n o t e by A ('~) t h e p a r t of A which lies outside the first n intervals i ( 1 ) , p ) , . . . , i ( n ) ; t h e n A (~) is evidently closed. W e have t o prove t h a t A (n) vanishes for all sufficiently large n; this, however, is clear; for f r o m the Weier- strass-Bolzano t h e o r e m it e a s i l y follows t h a t a sequence of closed sets A (~)==_

>= A(2)~A(3)>--_ ...,

n o n e of which vanishes, m u s t have a p o i n t in c o m m o n ; and such a point does n o t exist in o u r case.

W e could of course carry this study of the space Qo, f u r t h e r ; we shall only add the following t h e o r e m , which is i n itself of no g r e a t i n t e r e s t b u t which we shall require l a t e r on: L e t A' d e n o t e an a r b i t r a r y set in (~)~ and let the set

A ~ (A', Q~,~)

in

Q,,~

be enclosed in an open set 0 in

Q,.~;

t h e n ttmre exists an open set

U'

in Q~ w h i c h contains

A'

and is such t h a t U = ( U ' ,

Q,,.~j)

is con- t a i n e d in 0. T h e p r o o f is as follows: F o r a fixed point x' of

A'

we consider t h e set

(x', Q~,,~)

in Q~; this set is c o n t a i n e d in 0, so to any of its points x t h e r e corresponds an i n t e r v a l /.~ s u r r o u n d i n g x and c o n t a i n e d in 0 ; now the set

(x', Q,,, ~)

is closed, so by Borel's t h e o r e m a finite n u m b e r of t h e intervals Is will cover the set a n d we may conclude t h a t t h e r e exists an i n t e r v a l

I'

in

Qn

c o n t a i n i n g

x '

such t h a t (I', Q,~,~) is c o n t a i n e d in O. This proves the t h e o r e m . W e have also t h e c o r r e s p o n d i n g theorem, t h a t if a set of t h e f o r m

(Q,, A"),

where

A"

is a set in Q,, .... is c o n t a i n e d in an o p e n set 0 in Q .... t h e n A " is c o n t a i n e d in an open set U" such t h a t

U-~ (Qn, U")

lies in 0.

w 4. C o n t i n u o u s and S e m i - c o n t i n u o u s F u n c t i o n s .

A f u n c t i o n

f ( x ) = f ( x t ,

x~, x 3 , . . . ) defined in Q~,, is said to be

eo~ti~ous

(v01!stetig) in QoJ, if

f ( x (n))---*f(x)

w h e n e v e r x(")-~ x. I t follows easily: A func- tion

f ( x )

is c o n t i n u o u s if and only if for any x in Q~,, and for any e > o there exists an i n t e r v a l [~ s u r r o u n d i n g x such t h a t tile inequality

I f ( z ) - f ( x ' ) l <

The Theory of integration in a Space of an Infinite Number of Dimensions. 257 holds for all points x' in L,. I f we keep ~ fixed and v a r y x, it follows at once f r o m the covering t h e o r e m of B o r e l (since Q~o is c l o s e d ) t h a t a n y c o n t i n u o u s f u n c t i o n f ( x ) must be b o u n d e d in Qo,. A real f u n c t i o n f ( x ) is c o n t i n u o u s if, a n d only if, f o r a n y real a the points where f ( x ) > a or f ( x ) ~ a both f o r m a closed set. T h e f u n c t i o n attains b o t h its u p p e r and its lower bound.

We shall give one more p r o p e r t y of c o n t i n u o u s f u n c t i o n s which also follows imuiediately f r o m Borei's covering t h e o r e m : I f f ( x ) : f ( x ~ , x2, x 3 , . . . ) i s c o n t i n u o u s in Q. t h e n t h e r e exists for a n y e > o a n u m b e r n and a n u m b e r d > o, such t h a t the inequality

t

I f ( x , x,~, x~, . . .) - f ( x ; , x'~, x ~ , . . . ) 1 <

holds w h e n e v e r the n inequalities Ixk - - x; I < d f o r I < ]c ~< n are fulfilled. This t h e o r e m shows t h a t any c o n t i n u o u s f u n c t i o n in (r m a y be u n i f o r m l y a p p r o x i m a t e d by c o n t i n u o u s functions, each of which depends only on a finite n u m b e r of t h e variables.

A real f u n c t i o n f(x) defined in Q. is called semi-confinuous f r o m above in Q~ if lira f ( x ('~)) <=f(x) w h e n e v e r x (hI --~ x; a f u n c t i o n is semi-continuous f r o m above if, and only if, for any real a the points where f(x)>~ a f o r m a closed set. A f u n c t i o n f(x) is semi-continuous f r o m below if - - f ( x ) is semi-continuous f r o m above.

w 5. T h e L e b e s g u e Measure.

T h e way to a theory of measure in Q~ is now r a t h e r obvious; we need only a t t a c h to any (open) i n t e r v a l 1 (in the sense defined a b o v e ) a s m e a s u r e the p r o d u c t of its edge-lengths a n d t h e n apply the o r d i n a r y Lebesgue definitions.

T h e s e definitions are the following:

L e t A be any set in Qo~ and consider all coverings of A with a (finite or) e n u m e r a b l e n u m b e r of intervals I ; d e t e r m i n e f o r each such c o v e r i n g the sum of the measures of the covering intervals; the set of n u m b e r s obtained c e r t a i n l y contains the n u m b e r I since Q~o itself is a covering interval. W e call its lower b o u n d m~A t h e exterior Lebesgue measure of t h e set A; the interior Lebesgue measure n~A of A is defined by the relation m i A - ~ I me(Qo,- A).

W e have o < m e A < I a n d h e n c e also o_--<miAN I. F u r t h e r m ~ A < m e A or meA + m e ( Q o ~ A ) > I; for two coverings of A and @ , - - A form t o g e t h e r a covering of Qo. T h e n , however, it follows f r o m Borel's covering t h e o r e m t h a t a finite n u m b e r of the intervals considered will cover Qo~; now these intervals

3:3--34198. Acta mathematica. 63. Imprim6 le 6 j u i l l e t 1934.

258 B. Jessen.

m a y f o r some fixed n all be w r i t t e n in the f o r m I ~ (1', Q,,~) where I ' denotes an i n t e r v a l in Q~, whose measure is equal to t h a t of L ]Now as these inter- vals Y m u s t cover Q~ the sum of t h e i r measures m u s t be >= I and so the t h e o r e m is proved.

I f the interior measure is equal to the exterior measure, the set A is said to be measurable in the Lebesgue sense with the infinite-dimensional measure

m A = m~A ~ meA. "

N o w it m u s t of course be proved, t h a t inter~'als are measurable sets and t h a t t h e i r m e a s u r e is equal to t h a t already defined. This could easily be proved directly but follows also f r o m the m o r e general r e m a r k t h a t if A' denotes any set in Qn t h e n the e x t e r i o r and i n t e r i o r measure of t h e cylinder A--~ (A', Qn,(o) will be equal to the e x t e r i o r and i n t e r i o r n-dimensional measure of A'. I n par- t i c u l a r the two sets A a n d A' will be m e a s u r a b l e t o g e t h e r and w i t h the same measure, i n o r d e r to prove this we suppose first t h a t A' is m e a s u r a b l e ; t h e n the r e l a t i o n m e A + me(Q~ - A) >= I in c o n n e c t i o n With the relations

give

m ~ A = ~ n A ' , m~(Q(o-- A ) = m ( Q n - A ' )

and so m~A -- m~A - m A ' . Suppose now t h a t A' is n o t measurable; it is evidently e n o u g h to prove t h a t m e A ~ m~A' and since n~eA G m~A' it is e n o u g h to prove t h a t me A + ~ > me A ' f o r a n y ~ > o. Consider n o w a (finite o r ) e n u m e r a b l e n u m b e r of intervals I covering A such t h a t the sum of t h e i r measures is

< m~A + ~; d e n o t e t h e open set composed of these i n t e r v a l s by O; t h e n we have clearly m e A + ~ > meO; now we have proved before t h a t t h e r e exists an open set U' in Qn c o n t a i n i n g A' so t h a t U = (U', Q,,,,,,) is c o n t a i n e d in 0 ; but this proves the t h e o r e m since t h e n by t h e case already c o n s i d e r e d

~ A + e > m ~ O > m U = m U ' > m , ~ A ' .

T h e t h e o r e m is to be considered as showing t h a t n-dimensional mea.~,ure (in the space Q,~) is a particular case o f the infinite-dimensional measure. I t follows especially t h a t any closed i n t e r v a l _1 in Q,~ is also measurable and t h a t its measure is equal to the p r o d u c t of its edge-lengths.

I n exactly the same way it can be proved t h a t if A " denotes an a r b i t r a r y set in Q ... and A = (Qn, A") is the c o r r e s p o n d i n g cylinder in Q,.,, t h e n the two

The Theory of Integration in a Space of an Infinite Number of Dimensions. 259 sets A" and A have the same exterior and interior measure (now both measures are infinite-dimensional). The proof depends on the theorem (not yet established) t h a t any open infinite-dimensional set is measurable. Observe also the following theorem which is very easy to prove: I f A' and A" are arbitrary measurable sets in Qn and Q .... , then the set A ~ (A', A") in Q~ is also measurable and we have m A ~ m A ' 9 m A " .

A special r61e is played by the null-sets in Q~, t h a t is the measurable sets of measure o. The sum of an enumerable number of null-sets is again a null- set. I f Q* is a subset of Q~ which differs from Q~o only by a null-set and if A* denotes the common part of Q~ and an arbitrary set A in Q~, then it is easily seen tha~ m~A ~ - m e A * and m~A ~ m i A ~. As an example of a null-set we mention the set of all points in Q~ whose coordinates are not all irrational.

w 6. The Construction of Nets.

After the above discussion it will not be surprising t h a t the measure in- troduced has all the properties of the v-dimensional measure in the space Qn.

This could be proved b y repeating all the ordinary proofs. I t is, however, much easier to use a simple transferring principle, w h i c h gives everything without the trouble of repeating all arguments. This transferring principle depends entirely on the concept of a net, introduced with such great success in the theory of real functions by de la Vall~e-Poussin. I n the case of functions of a finite number of variables the importance of the nets is rather t h a t they give a simple technique for proving theorems which have themselves nothing to do with the nets and could have been proved without them. I n the theory of the space Q~ it seems (as we shall see later on) that the use of nets is the only way of ob- taining the deeper results. We shall have to use nets not only for the proof of the transferring principle but also later on; we therefore postpone the proof of the transferring principle and deal in this section only with the construc- tion of nets.

We shall speak of a dissection of a circle c into arcs b, when we leave out of c a finite number of points; the extreme cases where no point, or only one point, is left out are also to be considered as dissections. W h e n we denote the arcs by b t h e y are supposed to be open; the corresponding closed arcs are denoted by b. Now consider the first ~ coordinate circles ck which form the torus-space Q~ and take a dissection of each of them into arcs bk; t h e n we ob-

260 B. Jessen.

rain a dissection D ' of Q,~ into i n t e r v a l s , which we shall d e n o t e by I ' , by t a k i n g all i n t e r v a l s of t h e f o r m

I ' = (bl, b ~ , . . . , b~)

w h e r e e a c h bk is an are f r o m t h e dissection of ek considered. W e o b t a i n a subdisseetion of a circle e by l e a v i n g out m o r e points of e; a c c o r d i n g l y a sub- dissection of Q,, is o b t a i n e d by t a k i n g subdissections of t h e c o o r d i n a t e circles ek or a t l e a s t of some of t h e m . W e n o w define a net in Q~ as a sequence of

t v t r P

dissections D , , D ~ , D,, . . . w h e r e Dm+l is a l w a y s a subdisseeti0n of D,~ a n d w h e r e f u r t h e r t h e f u n d a m e n t a l c o n d i t i o n is fulfilled t h a t t h e m a x i m u m of t h e edge-lengths of all i n t e r v a l s in D ~ tends to zero as m - ~ ~ . t

T h i s definition is easily g e n e r a l i s e d to t h e case of t h e space Q~. I n order to o b t a i n a dissection D of Q~, we t a k e a dissection of each of t h e c o o r d i n a t e circles ek into ares bk b u t so t h a t only a finite n u m b e r of these dissections are real dissections no points b e i n g l e f t ou~ f r o m t h e r e s t of t h e circles. T h e n a g a i n we m a y f o r m t h e set of all i n t e r v a l s

I = (bl, b2, b ~ , . . . )

a n d these sets I are i n d e e d i n t e r v a l s in Q~ a c c o r d i n g to o u r definition. I t follows at once t h a t a n y dissection D of Qo~ m a y also be considered as g e n e r a t e d by a dissection of t h e space Q,~ where n is sufficiently large, in t h e sense t h a t t h e i n t e r v a l s I of t h e dissection D are all of t h e f o r m I = (r, Q~, ~,), t h e inter- vals I ' f o r m i n g a dissection of Q~. N o w consider a sequence of dissections D,, D2, D ~ , . . . of Qo~ w h e r e D,~+I is always a subdissection of D~; we t h e n say t h a t this sequence f o r m s a ~et i,n Q,., if f o r a n y fixed k t h e m a x i m a l l e n g t h of t h e arcs of t h a t dissection of ck which c o r r e s p o n d s to D,~ t e n d s to zero as n --*

(but evidently n o t u n i f o r m l y in k). W i t h this definition of a n e t in Q~ t h e f u n d a m e n t a l p r o p e r t y holds t h a t if ~ > I~ > ~ > . - d e n o t e s a n y sequence of closed i n t e r v a l s such t h a t L~ f o r a n y n b e l o n g s to D~,, t h e n these i n t e r v a l s will h a v e e x a c t l y one p o i n t in c o m m o n . Obviously t h e r e exist n e t s in Qo, T h e set of all points w h i c h lie on t h e b o u n d a r y of s o m e i n t e r v a l in a n e t f o r m a null-set.

w 7. The Transferring Principle. 1

By m e a n s of t h e c o n s t r u c t i o n of n e t s it is n o w easy to p r o v e tile trans- f e r r i n g principle r e f e r r e d to above. L e t Q a n d q be t w o t o r u s - s p a c e s of t h e k i n d

1 Cf. H. Lebesgue [I] 367; F. Riesz [I] 497; C. de la Vallde-Poussin [I],

The Theory of Integration in a Space of an Infinite Number of Dimensions. 261 considered; t h e y need n o t be of an infinite n u m b e r of dimensions; suppose f o r instance t h a t Q is Q,,, and q a circle of l e n g t h I. W e shall p r o v e t h a t t h e r e exists an application of Q on q which conserves the measure in t h e f o l l o w i n g precise sense:

There exists a o~e-one application of the points of Q with the exception o f a null-set on the points of q with the exception of a ~ull-set, with the property that eorresfgondiug sets in Q and q have always the same exterior a~d interior measure.

F o r o u r p r e s e n t object (the p r o o f t h a t the m e a s u r e in Q~ has all t h e pro- perties of ~he measure on a circle) t h e existence of one such application in t h e special case m e n t i o n e d above would be sufficient; the way in which it is con- s t r u c t e d is, however, of i m p o r t a n c e f o r t h e applications, and it is also of im:

p o r t a n c e t h a t we do n o t r e s t r i c t ourselves to this case alone.

W e consider a sequence of dissections D~, D~, D~, . . . which f o r m a n e t in Q and a sequence of dissections all, d~, d3, . . . which f o r m a n e t in q. T h e inter- vals of Dn will be d e n o t e d by L~, ~he i n t e r v a l s of d~ by i~. N o w we suppose t h a t the two nets in Q and q correspond in t h e sense t h a t f o r a n y n we have a one-one correspondence b e t w e e n t h e intervals L~ of Dn and the intervals i,~ of d~ with the following two p r o p e r t i e s : C o r r e s p o n d i n g intervals L~ and i~ have always the same measure; to an i n t e r v a l I.+1 c o n t a i n e d in an i n t e r v a l In always corresponds an i n t e r v a l i~+1 c o n t a i n e d in the c o r r e s p o n d i n g i n t e r v a l / ~ . E v i d e n t l y t h e r e exist c o r r e s p o n d i n g nets in Q a n d q.

W e now consider any sequence

(7. i) ]I => L _-> L _->

of closed intervals in Q, one f r o m each of the dissections Dn; this s e q u e n c e de- termines u n i q u e l y a p o i n t x of Q b u t t h e r e are points of Q which are deter- m i n e d by more t h a n one sequence (7. I); those are the points x which lie on t h e b o u n d a r y of some i n t e r v a l /,~. N o w to a n y sequence (7. I) by the corres- pondence of t h e two n e t s t h e r e corresponds a sequence 9

(7. ~) ~ => ~-~ => ~3 ->- "

of closed intervals in q, w h i c h d e t e r m i n e s u n i q u e l y a certain p o i n t t in q. So if we let two points x and t of Q and q c o r r e s p o n d if t h e y are d e t e r m i n e d by c o r r e s p o n d i n g sequences (7. I) and (7.2), we obtain an a p p l i c a t i o n of Q on q. T h i s application is not a one-one a p p l i c a t i o n except in trivial c a s e s - the:possiSilities

262 B. Jcssen.

are illustrated in fig. I - - b u t we shall show t h a t if we only consider the applica- tion f o r pairs of c o r r e s p o n d i n g points x and t, each of which has only one c o r r e s p o n d i n g point, the e x c e p t i o n a l sets are null-sets and the application has the desired properties. T h e p r o o f of this is simple, but it requires a certain care a n d we m a y of course d u r i n g the p r o o f only apply the few properties of the m e a s u r e which we have a l r e a d y established in ,~ 5.

I t is c o n v e n i e n t to delay f o r a m o m e n t t h e omission of t h e exceptional sets a n d first consider ~he application as it is defined by t h e correspondence b e t w e e n t h e sequences (7. I) a n d (7.2). F o r any set S in Q we may consider the set s of all points in q which c o r r e s p o n d to some point of S (this does not imply t h a t S contains all points in Q which correspond to some point of s);

Fig. L

we can now prove very easily t h a t m~s ~ rueS. Consider f o r a given ~ > o a (finite or) e n u m e r a b l e n u m b e r of intervals I covering S, so t h a t the sum of the measures of these intervals is < me S + e . Take now first all the closed intervals I l of D 1 which belong to one of these covering intervals I , t h e n all the closed intervals i~ of D ~ which belong to one of the intervals I w i t h o u t being c o n t a i n e d in one of the intervals I1 already selected, and c o n t i n u e this process; the sum of the measures of all the closed intervals T, obtained in this way is clearly

< m e S + ~ ; f u r t h e r m o r e f o r each sequence (7. I) which defines a p o i n t x of S, t h e . first interval /~, of the sequence which belongs to one of the intervals I is a m o n g the selected intervals i~. Now consider the set in q composed of all the c o r r e s p o n d i n g intervals i,; this set must c o n t a i n s; now an)" i n t e r v a l ~,, can be enclosed in an open interval whose measure is only slightly g r e a t e r ; so we get m e s < m e S + e and since ~ was arbitrary,

mesdameS.

W e have of course the

The Theory of Integration in a Space of an Infinite Number of Dimensions. 263- corresponding result t h a t if r is a set in q and R the set of all p o i n t s in Q which correspond to the points of r, then ~ e R __~ met.

Now consider first the set $1 of all points in Q which have more than one corresponding point in q, and denote by s I the set of all corresponding points in q; the set S t is known to be a null-set; since rues 1 <= rues 1 we see t h a t sl must also be a null-set: Similarly if rl denotes the set of points in q with more than one corresponding point in Q, and if R~ is the corresponding set in Q, we have t h a t r~, and hence also R1, is a null-set. Now denote by Q* the set obtainecI from Q by leaving out the two null-sets $1 and R~ and by q* the set obtained from q by leaving out the sets s~ and r~; then Q* and q* are formed by all the pairs of points x and t each of which has the other point as its only corresponding point. Q* and q* differ from Q and q by null-sets and the one- one application o f Q* on q* preserves the exterior and interior measure; in fact, if S and s are corresponding sets in Q* and q*, we have both rnes ~ rues and rueS <= rues and hence m ~ S = m~s, and the corresponding result for the interior measure follows by considering the complementary sets with respect to Q~ and q*, using the remark that the omission of a null-set does not alter either the exterior or the interior measure.

Thus the theorem is proved; together with the above remark on null-sets the transferring principle proves that the measure in Q~ has all the properties o f the measure on a circle; we emphasize especially the main theorem: I f A1, A2, A~, . . . denotes a finite or enumerable sequence of measurable sets in Q~, then the common part and. the sum of these sets are again measurable. I f no two of the sets have common points we have f u r t h e r

m(AI + A2 + A~ + ...) = m A l + mA,~ + m A s + . . . .

From this theorem it follows t h a t any open set A in Q~ must be measurable; in fact, if we choose for each point x of A an interval /.~ surrounding x and con- tained in A, it follows from LindelSf's theorem t h a t A is the sum of a finite or enumerable number of these intervals. From this it follows t h a t also any closed set, as complementary set to an open set, must be measurable.

I n the measure; in

w 8. T h e J o r d a n Measure.

general theory we use exclusively the infinite-dimensional Lebesgue the applications, however, the corresponding J o r d a n measure

264 B. Jessen.

also plays an essential r61e. W e define this measure i m i t a t i n g the ordinary definitions.

L e t A d e n o t e an a r b i t r a r y set in Q .... W e consider all coverings of A hy a finite n u m b e r of intervals I and d e t e r m i n e for each such covering the sum of the measures of the c o v e r i n g intervals. W e define the exterior J o r d a n measure of the set A to be t h e lower bound u~A of the n u m b e r s obtained; the i~terior J o r d a n m e a s u r e is defined by t h e relation t t i A - - I--#~,(Q,.~--A). W e have evi- dently o = < # ~ A = < I and hence also o < # i A = < I; the relation tt~A:<=tt~A is for the J o r d a n measure an e l e m e n t a r y relation.

I f the interior Jordan measure is equal to the exterior, we say that the set A is measurable in the Jordan se~se with the i~finite-dimensional Jordan measure

t t A -- # i A = #~A.

T h e properties of the J o r d a n measure are, exactly as in the n-dimensional case, most easily t r e a t e d by means of the Lebesgue measure. L e t A be an a r b i t r a r y set in Q~ and let ~-1 d e n o t e the closure of A in the sense defined above;

since A is closed, it is measurable in the Lebesgue sense and we shall now p r o v e t h a t

# c A = m-A.

T h a t #~A ~ r n A follows at once f r o m Borel's covering t h e o r e m ; in fact, if we have a covering of A by a (finite or) e n u m e r a b l e set of intervals 1, t h e n a finite n u m b e r of these intervals will cover A and so A. On the o t h e r hand, if A is covered by a f n i t e n u m b e r of intervals I, t h e n the c o r r e s p o n d i n g closed intervals f will cover A; so #~A>= m A and the two inequalities t o g e t h e r give the result.

By consideration of the c o m p l e m e n t a r y sets we get i m m e d i a t e l y the correspond- ing result f o r the i n t e r i o r J o r d a n measure t h a t

,aiA - m A

where A = Q , , , - ( Q ~ , - A ) denotes the open kernel of A (that is, the open set composed of all i n t e r i o r points of A). T h e two relations t o g e t h e r show t h a t a set A is measurable in the J o r d a n sense if and only if the closed set A - - A , which we may d e n o t e as the bom~dary of A, has the Leheso.ue measure, and so also the J o r d a n measure, zero. F r o m this result it follows at once t h a t the stun a n d common p a r t of a finite n u m b e r of sets measurable in the J o r d a n sense are agMn m e a s u r a b l e in the J o r d a n sense.

The Theory of Integration in a Space of an Infinite Number of Dimensions. 265 T h e infinite-dimensional J o r d a n m e a s u r e is m o r e n e a r l y r e l a t e d to the n- d i m e n s i o n a l m e a s u r e t h a n we should p e r h a p s t h i n k a t first sight. L e t A d e n o t e an a r b i t r a r y set in Qo, a n d let

An

f o r each n d e n o t e t h e p r o j e c t i o n of A on t h e n - d i m e n s i o n a l torus-space Q,,; t h e n t h e sequence of n u m b e r s

tteAi, tteA2, # e A a , . . .

is evidently d e c r e a s i n g (in t h e wide sense); we shall p r o v e t h a t lim #~ A n ~ - tte A .

n ~ o r

As t h e i n t e r i o r J o r d a n m e a s u r e is defined by m e a n s of t h e exterior, this gives us a definition of t h e infinite-dimensional J o r d a n m e a s u r e by m e a n s of t h e n-dimensional. 1 T h e p r o o f is i m m e d i a t e ; f o r in t h e first place we h a v e

t,~A,~>=tt~A

f o r all n; a n d on t h e o t h e r h a n d if e > o is given, t h e n t h e r e exists a c o v e r i n g of A by a finite n u m b e r of i n t e r v a l s such t h a t t h e s u m of t h e i r m e a s u r e s is

< #~ A + e ; f o r all sufficiently large n these i n t e r v a l s h a v e t h e s a m e m e a s u r e as t h e i r p r o j e c t i o n s on Q~; so f o r l a r g e n we h a v e tt~ A~ <

#eA+~.

A c o r r e s p o n d i n g d e t e r m i n a t i o n of t h e e x t e r i o r L e b e s g u e m e a s u r e in Qo, is n o t possible.

w O. T h e D e f i n i t e a n d I n d e f i n i t e I n t e g r a l s .

On t h e basis of t h e L e b e s g u e m e a s u r e t h e n o t i o n s of

measurable

a n d

inte- grable

f u n c t i o n s

f ( x ) ~ f ( x l , x2, x~ . . . . )

can n o w i m m e d i a t e l y be i n t r o d u c e d .

S u p p o s e first t h a t f ( x ) is a

real

f u n c t i o n in Q~. I f n o w f o r a n y a t h e set of points in which f(x)_--> a is a m e a s u r a b l e set, t h e n we call f ( x ) a m e a s u r a b l e f u n c t i o n . Suppose this to be t r u e a n d consider a n a r b i t r a r y scale

' " < y - 2 < y--1 ~ Y0 ~ Yl ~ Y~ ~ " "

of i n c r e a s i n g n u m b e r s f o r which y-,~-~ - - av a n d y~--~ ac as n - ~ ac a n d f u r t h e r t h e n u m b e r

(9. I) b o u n d (Y,~+i - - Y~)

is finite; we denote by

m,~

t h e m e a s u r e of t h a t p a r t of

Qo,

in which we h a v e

y,~ ~ f ( x ) <

y~+i a n d f o r m t h e series

(9 Y

T h i s d e f i n i t i o n w a s u s e d i n H. B o h r - - B . dessert [~] 6 5 - - 6 9 . 3 4 - - 3 4 [ 9 8 . Aeta matheraatlea. 63. Imprim4 le 6 ]uille~ 1984.

266 B. Jessen.

I f n o w f o r some scale this series is a b s o l u t e l y c o n v e r g e n t , t h e n it is absolutely c o n v e r g e n t f o r a n y scale of t h e k i n d considered; t h e f u n c t i o n

f(x)

is t h e n called i n t e g r a b l e a n d its i n t e g r a l is defined as t h e l i m i t of the stun of t h e series (9. z) w h e n t h e n u m b e r (9. I) tends to zero in an a r b i t r a r y way. A

complex

f u n c t i o n

f(x)=u(x) 4-iv(x)

is called m e a s u r a b l e or i n t e g r a b l e if the t w o f u n c t i o n s

u(x)and

v(x) are b o t h m e a s u r a b l e or i n t e g r a b l e respectively. I n t h e l a t t e r case we define t h e

infinite-dime~sional integral

(9.3) f

/ ( x ) d

Qco

of f ( z ) over Q~ by i n t e g r a t i n g the r e a l a n d i m a g i n a r y p a r t s separately.

A n y

conti~mo~ts

f u n c t i o n

f ( x )

is i n t e g r a b l e ; f o r a c o n t i n u o u s a n d real func- t i o n is always b o u n d e d a n d t h e set of points in which

f ( x ) ~ a

is f o r a n y a closed a n d so m e a s u r a b l e . I t should also be o b s e r v e d t h a t if a f u n c t i o n f ( x ) in (2(,, d e p e n d s only on t h e variables xl, x2, . . . , x~, t h e n it is m e a s u r a b l e considered as a f u n c t i o n in t h e n-dimensional torus-space @~. if and only if it is m e a s u r a b l e considered as a f u n c t i o n in (2~,; the f u n c t i o n s are also i n t e g r a b l e t o g e t h e r a n d t h e i n t e g r a l over

(2,,,

is equal to the i n t e g r a l over (2.,~. T h i s r e m a r k m a y be considered as an

expressio~ of the r~-dime~sional integral

(for f u n c t i o n s in (2~~)

as a particular ease of the infinite-dimeJ~sio~al.

Similarly if a f u n c t i o n depends only on t h e variables

x,~+~,

xn+2, 9 9 t h e n it is i n t e g r a b l e c o n s i d e r e d as a f u n c t i o n in (2 ... if a n d only if it is i n t e g r a b l e over Qo~ a n d t h e i n t e g r a l s are equal. W e shall s o m e t i m e s allow ourselves to write

f ^f(x)dw,~

^or

Qn %,

i n s t e a d of (9.3) in these cases.

E x a c t l y as in the n - d i m e n s i o n a l case, t h e c i r c u m s t a n c e s in a set of m e a s u r e zero are of no i m p o r t a n c e in t h e i n t e g r a t i o n . T w o f u n c t i o n s which differ only in a null-set will t h e r e f o r e in the followirLg n o t be considered as different func- tions a n d a f u n c t i o n will also be considered as defined in (2~ if it is only defined outside a set of m e a s u r e zero.

T h e i n t e g r a l h a s e v i d e n t l y

the same ftt.ndamental properties as the ordina~'y

Lebesgue integral.

This follows i m m e d i a t e l y f r o m the t r a n s f e r r i n g principle; in f a c t if we denote by

x=x(t)

the a p p l i c a t i o n of a circle q of l e n g t h I on Qo~

The Theory of Integration in a Space of an Infinite Number of Dimensions. 267 constructed by means of corresponding nets in Q~ and q, then for any real function f ( x ) in Q. the corresponding function

qD(t)=f(x(t))

on q will have the same distribution of its values as f(a:) in the sense t h a t for any real a the sets of points where we have

f ( x ) ~ a

and ~(t)~= a will have the same exterior and the same interior measures. This, however, implies t h a t the two functions

f(x)

and q~(t) will always be measurable or integrable together and in the latter case always with the same integral.

In addition to the definite integral (9.3), for the considerations of the present paper the

indefinite

integral of an integrable function f @ ) in Qr is also very important. Let E denote an arbitrary measurable set in Qr we consider t h a t function defined in Qo~ which in E is equal to f(~') and is o elsewhere;

this function is again integrable; we denote its integral over Q~ as the integral

F ( E ) = d

E

of

f(x)

over the set E. The function F ( E ) of the variable set .E is called the indefinite integral of

f(x).

From the transferring principle follows at once the main theorem of Lebesgue:

A f~tnetion F(E) defined for all measurable sets E in (2o~ is the indefinite integral of an iutegrable function f(x) in Q~ i f and only i f it is additive and absolutely continuous,

t h a t is if, firstly, for any two sets E j . a n d E2 without com- mon points we h~ve

and, secondly, to any e > o there corresponds an '0 > o, so t h a t ]/~'(E) I < e when m . E < ' q .

The indefinite integral determines uniquely the integrated function f(x);

but how shall we obtai~

f(x)

when F ( E ) is given? The Lebesgue theory of symmetric derivatives certainly cannot be generalised to our case (at least there is no obvious generalisation); this follows from the fact that there are no sym- metric neighbourhoods for the points of Q~, all intervals in Q~ (except Q, itself) being highly unsymmetric. We may, however, always differentiate F ( E ) o n any

net

in Q~; so the construction of a net turns out to be of greater importance

968 "B. Jessen.

f o r t h e t h e o r y t h a n merely to supply a simple t e c h n i q u e of proofs. T h e diffe- r e n t i a t i o n t h e o r e m which w e obtain is as follows:

Suppose that the sequence of dissections D~, D~, D 3 .. . . . of Q~ form a net and denote by H~(v), for any value of n, the >>stepfunction>> which in any interval L~ of the n-th dissection D,~ is equal to the corresponding quotient

m ~

then the sequence of functions J , ( x ) will tend to f ( x ) as n ~ ~ almost everywhere in Q~o.

The p r o o f of this t h e o r e m follows immediately. On a circle q of l e n g t h i we c o n s t r u c t a net dl, de, d s . . . . which corresponds to the n e t Dl, De, D 3 . . . . in Qr W e d e n o t e by x = x ( t ) t h e c o r r e s p o n d i n g application of q on Q,,, a n d by

@(e) t h e indefinite i n t e g r a l of the f u n c t i o n ~ ( t ) : f ( x ( t ) ) on q; t h e n we have e v i d e n t l y for any i n t e r v a l I,, in Q~ and the c o r r e s p o n d i n g i n t e r v a l i~ on q the r e l a t i o n

=

(i,,)

h e n c e to the sequence of f u n c t i o n s Hn(x) in Qo, t h e r e corresponds by the applica- t i o n of Qo, on q a sequence of f u n c t i o n s which tends to ~(t) a l m o s t e v e r y w h e r e in q; b u t t h e n the sequence H,~(x) must t e n d to f ( x ) almost e v e r y w h e r e in Qo~.

I n the last t h e o r e m we have the f u n d a m e n t a l s t a r t i n g point for a deeper s t u d y of i n t e g r a b l e functions of an infinite n u m b e r of variables. I t shows t h a t the relationship between these functions and i n t e g r a b l e f u n c t i o n s of a finite n u m b e r of variables is n o t so d i s t a n t as m i g h t have been e x p e c t e d f r o m the beginning, each of the functions H,~(x) being in fact a f u n c t i o n d e p e n d i n g only on a finite n u m b e r of the variables x~, x.~, x~ . . . This state of affairs is finally only a characteristic reflection of the definition of an i n t e r v a l u p o n w h i c h the t h e o r y is based.

w io. The Riemann Integral.

I n the applications we shall also make use of an infinite-dimensional Rie- m a n n integral. W e i n t r o d u c e this integral, i m i t a t i n g the o r d i n a r y definitions.

L e t f ( x ) be an a r b i t r a r y f u n c t i o n in Q~ which is real and bounded, and let us form, for any dissection D of Q~, t h e two sums

The Theory of Integration in a Space of arl Infinite Number of Dimensions. 269

s(D) := Z g ( I ) m I

and

S ( D ) : Z G ( I ) m I

where the summations are over all intervals I of D , and

g(I)

a n d G ( I ) denote the lower and upper bounds of

f(x)

in I. I t is easily seen tllat

b o u n d {s(D)} ~ bound {S(D)}

where the upper and lower bounds are with respect to all dissections D of Q~.

These two numbers define the lower and upper R i e m a n n integrals of

f(x)

over Q,o. I f t h e y are equal, we call

f(x)

integrable

in the Biemann sense;

in this case it is easily seen t h a t

f(x)

is measurable and t h a t the integral is equal to the Lebesgue integral. W h e n we say t h a t a function is integrable in the R i e m a n n sense, it is always understood t h a t

f(x)

is real and bounded.

The simplest way of d e a l i n g with the R i e m a n n integral is to reduce it to the Lebesgue integral. I f we introduce to a give n function

f(x),

which we suppose to be real a n d bounded, the two functions

~o (x) ~ bound {g (Ix)} and tO (x) = bound { G (Ix)}

where the upper a n d lower bounds are with respect to all intervals Ix s u r r o u n d i n g x, it is seen t h a t these functions ~ (x) and tO(x), are semi-continuous from below a n d above respectively. The integrals of ~ (x) and tO (x) are simply the lower a n d upper R i e m a n n integrals of

f(x).

W e therefore obtain the usual criterion for the integrability in the Rieinann sense t h a t t O ( x ) - q~(x)must be a null- function.

W e shall also use the following remark: A f u n c t i o n

f(x)

is integrable in the Riemann sense if, a n d only if, there exists corresponding to a n y e > o two

continuous

functions a (x) and A (x) such t h a t

and

a (x) f(x) A (x) f o r a l l x

f (A (x) -- a d o, <

Q~

I t is always possible to choose for a (x) and A (x) functions depending only on a finite n u m b e r of the variables xl, x2, x 3 , . . . ; if we wish we m a y take a(x) and A (x) as finite trigonometrical polynomials.

270 B. Jessen.

A f u n c t i o n f ( x ) which takes only a finite n u m b e r of values is integrable in the R i e m a n n sense if, a n d only if, t h e sets of points where it takes its values are all measurable in the J o r d a n sense.

I I . An I m p o r t a n t L e m m a .

W e shall sometimes meet, in t h e following considerations, f u n c t i o n s f(x) in Q~o which have the p r o p e r t y t h a t for any two points x in Q~) which differ only in a finite n u m b e r of coordinates the f u n c t i o n is e i t h e r n o t defined in the two points or is defined and has the same values in b o t h points. W e shall call this p r o p e r t y t h e property S. A set of points in Q,,, will be said to have the p r o p e r t y S if the characteristic f u n c t i o n of the set has the p r o p e r t y , t h a t is if any two points in Q~ which differ only in a finite n u m b e r of c o o r d i n a t e s either both belong to the set o r b o t h belong to the c o m p l e m e n t a r y set. A n o t h e r way of expressing tlmt a f u n c t i o n f ( x ) has the p r o p e r t y S is by saying t h a t for any n the f u n c t i o n does n o t depend on the variables x~, x~ . . . . , x,~ b u t may be con- sidered as a f u n c t i o n in (2~,, ~; f r o m this f o r m u l a t i o n it is n a t u r a l to conclude t h a t the f u n c t i o n c a n n o t depend on a n y t h i n g but m u s t be a constant. I f the f u n c t i o n f ( x ) is measurable, this is actually true in the sense t h a t the function m u s t be a c o n s t a n t almost everywhere. This t h e o r e m , and the c o r r e s p o n d i n g t h e o r e m f o r measurable sets, is a very useful l e m m a f o r m a n y considerations:

A measurable function f(x) in Q,, which has the property S must be a constant almost everywhere. A measurable set with the property S has either the measure o or the measure I.

I t is clearly sufficient to prove the t h e o r e m for sets. The p r o o f follows at once from the d i f f e r e n t i a t i o n t h e o r e m given above. L e t A d e n o t e a measur- able set in Q~) with the p r o p e r t y S and let f ( x ) be the characteristic f u n c t i o n of A; t h e n for a n y value of n the c o r r e s p o n d i n g function//:~(x) m u s t be c o n s t a n t and - - m A . This follows at once f r o m the fact t h a t for any value of n we have A=(Q~, A"), d e n o t i n g by A" the p r o j e c t i o n of A on Q ... ; now A is measurable;

it follows t h a t also A " must be measurable and t h a t m A - - m A " . On the o t h e r hand, if I ~ ( I ' , Q ... ) denotes an a r b i t r a r y i n t e r v a l in Q,.) with its base I ' in Q~, t h e n we have A I - - ( I ' , A ' ) and consequently m A I = m I ' m A " - - m l m A . But this proves t h a t we have for any i n t e r v a l L~ of tile dissection to which Jn(X) belongs

The Theory of Integration in a Space of an Infinite Number of Dimensions.

f f ( x ) dw~ = m A L = m l;, m A

271

and consequently J ~ , ( x ) = m A in all points of Q,.). H e n c e since zl,~(x)--~f(x) almost e v e r y w h e r e and since f ( x ) only takes the two values o and I, we m u s t have e i t h e r m A = o or m A = I .

T h e r e is a c o r r e s p o n d i n g t h e o r e m t h a t if a measurable f u n c t i o n of a single real variable has a r b i t r a r i l y small periods t h e n it m u s t be a c o n s t a n t almost everywhere. This t h e o r e m is familiar. T h e application of t h e same idea to the space Q,,, is due to S t e i n h a u s 1 who used t h e above l e m m a to prove an i n t e r e s t i n g t h e o r e m on anMytie c o n t i n u a t i o n of power series. I shall q u o t e this t h e o r e m which is a s t a n d a r d example f o r the application of the lemma.

L e t us consider all power series of the f o r m

k ~ l

where the ak are g i v e n numbers; all these series have the same radius of convergence; suppose this radius r to be > o and finite. T h e n to a n y point x = ( x l , x,2, x 3 , . . . ) i n Q~ t h e r e corresponds by ( ~ . I) an analytic f u n c t i o n f ( z , x) i n < ,~ Now consider those of these f u n c t i o n s which are not contiuuable (for which the circle ]z I = / " is a n a t u r a l boundary)i The c o r r e s p o n d i n g set of points x in Q~.~ has clearly the p r o p e r t y S ~nd it is also measurable; t h e l a t t e r assertion follows from the r e m a r k t h a t it is the set of points x in Q~ f o r which

1

for all points z - = a + i b in [z[ < r w h e r e a and b are rational. Consequently we may conclude t h a t t h e measure of the set is always e i t h e r o or I. S t e i n h a u s proved t h a t it is always the l a t t e r case t h a t occurs and so gave a very n a t u r a l i n t e r p r e t a t i o n of the t h e o r e m t h a t almost all power series are not continuable. A more general (and more difficult) t h e o r e m f o r D i r i c h l e t series was p r o v e d by P a l e y and Z y g m u n d ; I shall deal with a generMisation of this t h e o r e m in w 22.

1 H* S t e i n h a u s [21.