A UNIQUENESS PROBLEM FOR FINITE MEASURES IN EUCLIDEAN SPACES UDC 519.2

N. A. Sapogov

O. Introduction i. We shall consider finite (completely additive) measures ~ sets

~

of a Euclidean space

of probability measures

~"

~= P

of dimension n.

Let ~ = ~

z=(x,,

for any

) , related to a class

IP}

of measures P.

~,

. ., z ~ ) ~

R~

, ~

,

and

$~.=={~:

t~<

rl

, where

The important well-known theorem of Cramer--Wold [i] asserts that if

~(St,~)= ~ (St.=) for any t ~ ~ n, ~ Qz~)

~, IP}

~=IPl will denote the class of al~ such ~ a s u r e s Q

t =(t,, t~,..., t~)e~ ~, t~ ~

Special attention is devoted to the case

(probabilities, probability distributions) and to the class

of corresponding probability spaces ( ~ , In particular,

on a ~ - a l g e b r a of Borel

S G ~

).

~4,

where P e ~

, J=~,Z , then

~=PL

(i.e., ~ ( ~ ) =

Its proof uses the properties of characteristic functions, i.e.,

of the Fourier transforms of probability distributions P. Let Us note that this theorem admits an almost obvious extension, i.e., instead of the probabilities P we can refer to any finite measures ~ [3, w 4], and the proof can be based on the theorem on single-valued reconstruction of a measure

~ in

R~

on the basis

of its Fourier transform. The Cramer--Wold theorem does not mention Fourier transforms. Therefore, it was natural to try and prove direc=ly the Cramer--Wold theorem without using the technique of Fourier transforms. This, however, was found to be unexpectedly difficult, and it was Dosed as a special problem by Kolmogorov in 1938 [2] (the section on "Mathematical problems,"}Problem No. 16, p. 233). A complete solution of K o l m o g o r o v ' s problem was published many years later [in 1954) in two versions: I) By Kostelyanets and Reshetnyak [3], and 2) by Khachaturov [4]. Le~ us note tha~ closely related to the Cramez--q4old theorem is the so-called Radon transform whose definition and properties can be found in the book of Ge!'fand, Graev, and Vilenkin [5]~ The formulas and results contained in this book give in fact a direct proof of the Cramer--Wold theorem, but only under additional assumptions with regard to the existence of an infinitely differentiable and fast decreasing [at infinity) density of the measures under consideration. The Cramer-Wold theorem (with its extension to finite m e a s u r e s ) can be interpreted as some sort of uniqueness theorem for finite measures ~ in ~ . More precisely, in accordance with the foregoing, a measure ~ the half-spaces $~.= systems of sets

is unique.

S = I ~ c ~,

For the case that the system from

M

by motions in ~

wi=h assigned values o~ all

Thus there at once arises the question of possible other

on which the values of

~(S)

correspond to a unique measure ~

$ is formed by a certain B-set ~

.

and by all the sets obtained

, this question has been posed already in [3, w

In many cases

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute imV. A. Steklova AN SSSR, Vol. 41, pp. 3-13, 1974. This material is protected b y copyright registered in the name o f Plenum Publishing Corporation, 227 West 1 7th Street, N e w York, N. I,. 10011. N o part o f this publication m a y be reproduced, stored in a retrieval system, or transmitted, m any f o r m or by any means, electromc, mechanical, photocopying, microfilming, recording or otherwise, w i t h o u t w ~ t t e n permtssion o f the publisher. A copy o f this article is avadable f r o m the publisher for $7.50.

this question is of interest when certain limitations are imposed on the class of measures under consideration.

Hence this problem can be formulated as follows.

given a class of finite measures

~}

and a system of sets

$ c ~ .

In

R~

we are

In which cases does

the condition

imply that for any

~ ~ ~

If (0.i) implies

un{quene88 system

we have

(0.2), we shall say that the corresponding system of B-sets

for the class of measures ~ }

Together with the system complements C a E $',

CB =~\5

S=IS}

of all the sets ~ e $ .

values o f ~ ( 5 ) = ~ ( ~ ) -

~(CS)for

tem for the given class of measures =~G81 {~

{~},

is also a uniqueness system.

for which

~ ( R ~)

.

for

~s

formed by the

From the fact that $ = {51

This applies, for example,

is a uniqueness ~I

to classes of measures

of all probability measures

2. In this paper it is assumed that the system of 8 -sets 5~-{=:

z-~E~o}

of a set

8o G ~

with u taking almost all the values in R~ .

S = ~

in R ~ .

Among such

in R ~.

~ in ~

is formed

of positive Lebesgue measure

Such a construction of the system

a sufficiently large store of sets in $ , and makes also the system choice of the origin of coordinates

The set

sys-

the system

retains the same known constant value within a class. 9

~(C~),

, then we shall know also the

it follows that for this class

classes of measures let us mention the class

by the shifts

S'=ICSl,

If together with the values of

~(~)

any 5 e ~

is a

.

let us consider a system

we are given also the values of

$

$

IS=I> 0 , S

ensures

independent of the

~o will be called the generating set

of the system $ = ~B~ Let us note that the sets obtained from this distinguishes

the systems

invariant under all motions

S

B~

by rotation in

under consideration

(including rotations)

R~

are not included in $ ;

from the system of ha!f-planes

$

that is considered in the Cramer-4?old

theorem~ In Sec. I of this paper we shall present some results obtained for this uniqueness problem for the case that the generating set 8o
of the system

g

has a finite measure: @
In Sec. II we shall study the case of a set B~ of infinite measure:

15oi= 0 0

Final-

ly, in Sec. III we shall examine the stability of one of the results obtained in Sec. I. I. The C a s e 0~18.1 ~

3.

Suppose that the system of sets $ = 15 ~ }

all ~ e R ~) interest.

of a given

B-set

~o

of measures

i ~ ,

~

~

, and let O < IBoI~oo

.

is formed by shifts The case

15oI=0

(for almost

is not of great

0

, and

$

would not be a uniqueness

~

we

system for a class

containing absolutely continuous measures,

(m)

be the indicator of the set B: ~B (m) = ~

It is evident that ~ ( ~ )

measures in R~

2

R ~

For any absolutely continuous (with respect to a Lebesgue measure) measure

would have in this case ~ ( ~ ) =

Let

in

.

By

~ . = ~.(t)

= ;So ( ~ - ~ ) . and

~ ' = ~(t)

for ~ g

Let ~// = ~

~

, and

~

(m)= 0

for

be the class of all finite

we shall denote the Fourier transforms

(characteristic functions, in abbreviated form denoted by c.f,) of ~s.(o=) and ~(5):

of the convolution

Then the c.f.

~(~3= ~ ~ . ( ~ - z ) ~ C c t ~ )

will be equal to the pro-

duct

vCt)=/ct) ~8.(-t), t ~R~ On the other h nd, !.

(l.l)

9 Therefore, the c.f.

convolution will be completely determined by the values of the measure 8m

of the system $ . It suffices if the values of ~(8~ )

e R~.

Thus the question of the uniqueness of the measure

the uniqueness of the solution ~(t) 4.

of this

~($~)

on the sets

are assigned for almost all ~

reduces to the question of

of Eq. (i.i).

For a (complex-valued) function

~(m)

in R~

we shall denote by

~ [~] the support

of this function, i.e., the closure of the set ~ m ' ~ ( m ) # 0 I in R ~. THEOREM i~

If 0~IBol < ~

and ~ [^ ~5.J~ - R

~

, then $ ={ 8 u~

will be a uniqueness system

for the class of all finite measures .~/ in R ~ . Indeed,

~(t)

can be uniquely determined from (I.i) for any t such that

from the conditions ~ for any

~o] ~-~

of continuity of

~(t)

it follows that

~,,(-t)4 0

;

#
t e R ~ , which completes =he proof. A

In the case ~[~,.]~ ~" for ~he class

2

, the system

8={5~}

of probability measures.

could not be a uniqueness system even A

In this case ~so(-t)= 0 , t e a

,

where

AW

is an open set in R ~ . This would signify the existence of an infinite set of distinct probabilities teA.

P=~

whose c.f. P ' ( t ) = ~ (t)

For n = i, the existence of

satisfy Eq.

distinct c.f.

(i.!), and which differ for values of

~(t)

that coincide for all t ~ (~', t") ,

t'< t", is a consequence, for example, of the well-known theorem of Polya [6, pp. 44-45]. For any

~ 74

it is possible to apply this same theorem of Po!ya if ~he measures P

in ~

are taken in the form of a direct product of n one-dimensional probabilities. The condition in

~

t~ ~ ' ~~

will be satisfied if ~So(t), tg ~ ,

, or a limi~ value of an analytic function.

(Z,,... , z~) A

~[~,] =

,

is analytic in an (open) region

S~ e ~ ,

Q

~= ], . . . , ~, whose closure D

. . . . , t~) = ~5~ ~,. (Z,, .... Z~),

is an analytic function

The latter signifies that ~^o < Z ) = z~. ^ in

in

if (Z ......

~ ~

of complex variables ~ = ~ +gs~, contains ~ :

Z~)e D

~ c ~

and ~ -- 0,

; here

~ ~o(t)

j= ~, Z, . . . , ~.

As a direct consequence we hence obtain the following general result: THEOREM 2.

If a set ~o s ~

then the system of its shifts

in ~

$ = ~S a ~

is bounded and it has a nonzero measure (for almost all ~ g R~ )

l~=I> 0

,

will be a uniqueness sys-

tem for ~he class of all finite measures J/ in ~ For the proof it suffices to note that under the conditions of the theorem we have 0
Q=={z:

in R ~

can be complete-

I ~ - ~ I e ~I

of

ussigned

finite

radius

z>O

(here

I ~ I ~ = :e~ § . . .

§ ~c~ ) .

The w e a k e r a s s e r t i o n

that

a mea-

sure

can be reconstructed on the basis of' its values on all the spheres with any centers

and ~ Z

~os8~b~e radii

%>0

is

trivial.

In confining ourselves for simplicity to the one-dimenslonal case ~ = I , let us note that

will be a limit value

5oC"

t,o(~)C~CC

e

is bounded on one side, i.e.,

~cb
of an analytic function if the ~ - s e t

for any

ZEbo

(this is also true for aG

zc ~ 6 ~ - ~ condition

h,:,O

is replaced by the integral

if the integral is

also

j e ~= ~,.(=)~:=~o~

sufficient:

J~

).

The following weaker

for a positive h (or .~ e"~" ~ o ( ~ ) ~

~ ~,

). A

5.

If O~ 18~= ~

, then ~,.{t) ~ 0

i.e., for I t~I~ ~ , ~=l,Z,...,~, O.

Therefore,

~[t)

5-0

in a neighborhood of the zero point , since

~6. Ct)

is continuous and

(0, .... O ) ,

~,.(0,...,0)=15oI ~

will be determined by Eq. (i.i) in the same neighborhood

It~Isg

If

A

belongs to a class of finite measures tinua=ion from a neighborhood It~l~ ~ be uniquely determined in ~ set, for which 0 ~ l ~ o l ~ o

,

J~ = ~I

, ~=~ ..... ~,

for which

/(t)

admits a unique con-

to the entire space

; hence in this case the system

~

$ = I B ~}

, then

contains all the finite measures ~

will

of shifts of a ~ -

will be a uniqueness system in the class of measures

us note that the class ~

~(t)

for which

~(t)

~bb~. in

Let ~

is

either a limit value of an analytic function, or an analytic function. Let

~"

IP~ be a class of probability measures in ~

tion from a neighborhood It~l~ ~ , ~ = ship of

P

in the class

~=

tion in this neighborhood

Y,..., ~ , to the entire ~ .

will be ensured if ~ (t)

ItI~zz-.

plete description of the class

~

that admit a unique continua-

For distributions

In this case the member-

is a limit value of an analytic funcP

on the straight line ~' , a com-

has been given by Krein [7].

Let ~ = ~ ~= be the class of all probability distributions for which the c,f, ~(t) admit a single-valued continuation from any neighborhood I~I m ~ , ~ > 0 , Hence we obtain the following theorem: THEOREM 3.

If

0~15oI ~ o ~

, then ~he system of shifts

system for the class of probability distributions II.

The Case l~ol=ou 6.

in ~

will be a uniqueness

n

with a condition l~ol= ~ 1 7 6 , we shall show at first

that sometimes it can be reduced to the case for all ~ q

{~

15oI< ~

15oi=00

If

, where C5o = R ~ \ B o

and IC~ol = ~ 1 7 6

But there can be also other cases of

Let us present an example.

Suppose that the boundary of a ~ -set on the plane Qz that originate at a point 0.

0 < ! C ~ = I < oc

, it will be possible to carry out such a reduction in ac-

cordance with the remark made at the end of Section I. reduction for which

I

,

In the case of a 5 - s e t

and ~ ( ~ ) = C 0 ~ $ t

$-~5~

is formed by two rays 0A and 0B

The broken line AOB decomposes ~

into two parts, in one of

which the angle AOB has,

for example, the value ~ ; then its value in the other part will

be ( Z ~ - ~).

5o

As the set

(the angle) we shall take the set of points of the part of the

plane for which the angle a= =he vertex 0 is smaller than ~ ; thus let O~ ~ < ~

4

.

The case

=Z~-~-~

corresponds to degeneration of the angle Bo

we exclude.

In the angle

Bo

we shall include also the points of the rays OA and OB.

shall now assert that =he system gle

~=

~B~

of shifts (for almost all ~ e ~z)

is a uniqueness system for the class of all finite measures ~

For the proof we shall take any points

A~CtlOB

struct the rays a,q)

The angles

and B~IIOA.

AA,C, ~B~B

A4

Let ~

, and ~ C

denoted, for example, by B~,, B ~

are not included in

, and

B~

and 84

B~,

We

of such an an-

on the plane.

on the rays OA and OB, and con-

be the point of intersection of

(with vertices at , of the angle

of points that belong to the parallelogram 0,8~

into a half-plane, a case which

A,,B~

B~ .

A4C

and

, and O~ ) will be shifts By

~

we denote a B -set

O,B, OA~ ; the points of the segments

~ A,

and

We have

If instead of 8o we consider its shift a ~ , then all the elements of the above construction will undergo the same shift and Eq. (2ol) can be rewritten in such a way that the sets are replaced by their corresponding shifts. Thus the values of the measure ~ will be defined @

on a system of shifts

~B~ I

of a bounded set

af

that has a measure IS$J> 0

of Theorem 2 the system IS ~ I , and hence also =he system of shifts will be a uniqueness system for the class of finite m e a s u r e s ~ R L will also be an angle, and ~ R e R z. 8~

z)

.

la~l

By virtue

of the angle 8~

The completion of

CBo

in

-is defined if we know all the value of ~ ( C 8 ~ ) ,

This follows from the fact that with an appropriate (sufficiently large) shift,

will lie outside a given arbitrarily large circle (centered at the origin), and there-

fore ~ ( B ~ ) - - O

, ~(CS~)--~(R:)

If the values of ~ ( C a ~ )

.

Since they are sets in R ~ , the angles

for the closures

C8~

By virtue of the remark made

at the end of Section i, it follows from =he foregoing that the system uniqueness system for the class of finite measures angles~ but also for open angles 8o .

jg~.

IB~I

of the angle ~o

But if ~ = ~

, then the angel

An obstacle to using Eq.

of a system of shifts

~B~I

I~

S =ICS=~ will be a

This applies not only for closed

Thus for any value of. ~ ,

a>O,

the system of

will be a uniqueness system for the class of finite measures

shifts of a half-plane in R ~ ~ 7.

are open.

for such angles are kn6wn for almost all ~ e R: , then we shall

know also the values of ~ ( C B ~ )

shifts

Ca=

~o ~

~e a uniqueness system for probability measures~

(i.I) for solving the above uniqueness problem in the case

of a set

Fourier transform of the indicator

will degenerate into a half-plane, and the system of

8o of infinite measure is the absence of a classical

~o(m)

for such sets.

However it is poss'ible to use a

Fourier transform in the sense of the theory of generalized functions [8, Chap. 7] and [9, cb

Vol. i, Chap. 2; Vol. 2, Chaps. 3and 4]. The space of (fundamental) functions

We shall use the following notation. @(m)

with all their derivatives faster than any power will be denoted by ~ .

The generalized functions

mo~e~

If ~

E increasing.

= !~ }(m;~ m # ~ m ,

~ ~ ~

the indicator ~.(m)

in R ~

that decrease for Iml--o= together

ImI"~

(i.e., the space of L. Schwartz)

<~, ~ ,

~

are said to be

is an ordinary (complex-valued) function in R ~ , then <~, ~>

(if the integral is convergent). of any

defined on

~ -set in R ~ .

Among such functions let us mention

The Fourier transform <~, @>

of a moderately

increasing generalized function If we define the convolution

<9, q >

~

9z

is defined by the formula

< ~, q > = < ~ , q >

, q~.

of two moderately increasing generalized functions,

and if the Fourier transform of one of them, say

~(~)

, is an infinitely differentiable

function each derivative of which has the largest exponential order of growth for

Itl--oo ,

we obtain the relation A

^

A

<9, ~{~, ~>--<9~, V>,

qe ~.

For the problem under consideration, this relation assumes the form ^

where ~

A

(2,2)

denotes the Fourier transform (as a generalized function) of the convolution

If the finite measure ~

in

tives will be bounded in R ~ holds.

A

~

has a compact support, then

The indicator

~8o

Moreover, the values of ? ( ~ ) = ~ ( b z )

belongs to ~ .

closed paral!elepiped in R ~ . functions ~

and all its deriva-

Therefore formula (2.2)

are assigned for almost all z e ~ ~ , and there-

fore the left-hand side of (2.2) is also assinged, the (generalized) function ~a~-x)

~q(t)

Now let us assume =hat the support of

contains an open set -[~~

The functional

<~s~-~

q>

@ ~ led ~ , where I is a

Let

on the set 0~ X of fundamental

with supports belonging to I is the derivative of order

~=(~,...,~)

of the

following function that is continuous on I: <~,.(-~), "~> = <- 4

{ I

Here

~ is a multiindex, and we are dealing with partial derivatives,

function f coincides with I.

It hence follows that <>8o~J'~-aL,9> ~ 0

that is dense (in the sense of a uniform metric) in ~z .

The support of the

for a set of functions

Therefore <~(t), ~ > '

~(t)dt function

=

I ~A' i t ~ I

can be determined from (2.2) for this same set of functions q ; hence the analytic i~ (t)

is defined for

t ~ ~~

~

, and hence everywhere in ~

.

Therfore we ob-

tain the following theorem: THEOREM 4. B=E ~

If the support of the Fourier t r a n s f o r m ~ o O f

contains a set

~=#

~

the indicator of the set

which is open in R ~ , then =he system of shifts $ = I ~ I

will be a uniqueness system for the class of all finite measures ~

with compact supports.

Let us note that there can be slight modifications of the conditions of Theorem 4.

In

particular, instead of requiring the compactness of the support of the measure ~ , we can require that the function

~(t)

be analytic; but these modifications are not discussed in

detail here. I!I.

Stability of Theorem 2 8.

In connection with each of the above theorems on the uniqueness

that have assigned values of these theorems.

of finite measures

~ ( ~ =), there could arise the question of the stability of

As an example, we shall examine here the stability of Theorem 2, and con-

fine ourselves to the class of probability measures

THEOREM 5. a sequence ~, for )__oo

Let the set

in R ~ be 5ounded, and let ~ ~

5o r ~

~= 4, %,... , of probability distributions in ~

. Let us consider

, and for almost all

~ e R~

we have ~ (5=)-- P (B~).

Then the sequence of distributions

Proof.

~

(3.1)

will (weakly) converge to the distribution

First of all it is necessary to ascertain the compactness of the sequence of

distributions ~

. Compactness, i.e., the possibility of selection of a convergent subse-

quence from any infinite part of ~

, can be ensured if the probabilities

~

infinity uniformly within the sequence of distributions under consideration. can be established as follows. Let O < T < o o have ix ; = T H.

By ~

we shall denote a cube in ~

~ = I, s . . . . . ~.

Let us write

a=

Since 8o

~15oi ' where I---$1--~l-

for whose points

6>0

is assigned.

decrease at This uniformity

x=(~ .....~ )

is bounded, we shall have ~o < ~

P
P when

we

for a positive

There exists an g > H

such that

I- e'

(3,2)

0 ~- P -~ I , it is possible to integrate the limit relation (3.1) over D ~ :

For some N , with

i> N , we then have

c

(3.3) Let us consider regions Ao. A~ , and Az

in the space

R ~•

~,

whose points


satisfy the following conditions: !)

~,~)e

2)

( ~ 9 ) e A~

signifies that ~ e ~~,

~

~.

3)

(~]~,A z

signifies that ~ ,

~

~.

Here ~

Ao

signifies that ~

=Ira: ~ - V ~ ~ i

and the conditions find that Ao c A~

, and m~ ~

~ and

~e ~.

~

is defined similarly. 9e O~

are equivalent.

If (a, J l e A ~

, then m e ~

Therefore A, c Ao

cO"

,

Similarly we

Hence

(3.4) where, as above,

#So(m)

is the indicator of the set 8~

But

Similarly,

J ~.,,. ~-~)P~'d.9od.= .,.t6,,Ip(D""~)

Next,

Therefore, we can rewrite (3.4) as follows:

IBoIP(G").~ J,o" P(B..)d.=-lSolP (D""). Such inequalities hold also for all the distributions

(3.5)

P, :

ISolP~CO"')- D,,,,~ m=)d.= -- IBolP,CD"'"). Hence for

~>N

(3.6)

, we obtain from (3.5)-(3.6) with the use of (3.2)-(3.3) the relations

Ia.l~ (D~'")~

~o~P(B~)a=-e'~

IB.IP (Dr") - ~'-IBol(~- s )- ~'= ISol-~'( ~+IBol). f

Therefore, ~(D~*")~-s ''~+IB~ = ~-a Thus we have proved =he compactness of the set of disIBoJ trlbutions % . Should the sequence % not converge to the distribution P , it would be possible to select two subsequences

~,

and

Pa,

that are (weakly) convergent to

different

limits :

P~ --P' and for almost all

~a ~

p. -.. /

it would nevertheless follow from (3ol) that

p~.(8=)-- P(BD,

4, (s=) -- P(5~)

Hence for almost all ~ we must have P'(~=)= P"(B=) and from Theorem 2 it would hence follow that

p'm P" , in contradiction to the assumption that P' and P"

are distinct.

This

completes the proof of Theorem 5.

LITERATURE CITED I. 2. 3.

4.

5. 6. 7. 8. 9.

H. Cramer and H. Wold, "Some theorems on distribution functions," J. London Math. Soe,, ii, 290-294 (1936). A . N . Kolmogorov, "Problem No. 16," Usp. Mat, Nauk, ~, 233 (1938). P . O . Kostelyanets and Yu. G. Reshetnyak, "Determination of completely additive function on the basis of its values on half-spaces," Usp. Mat. Nauk, ~, No. 3(61), 135140 (1954). A . A . Khachaturov, "Determination of value of measure for region of n-dimensional Euclidean space on the basis of its values for all the half-spaces," Usp. Mat. Nauk, ~, No. 3(61), 205-212 (1954). I . M . Gelfand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions, Vol. 5, Academic Press (1966). Yu. V. Linnik, Expansions of Probability Laws [in Russian], Leningrad Univ. Press, Leningrad (1960). M . G . Krein, "On the continuation of Hermitian position continuous functions," Dokl. Akad. Nauk SSSR (New Series), 26, No. I, 17-21 (1940). L, Schwartz, Theorie des Distributions, Vol. i, Paris (1956); Vol, 2, Paris (1951). I . M . Gelfand and G. E. Shilov, Generalized Functions, Vols. i and 2, Academic Press (1964).