(Warszawa)On the extension oî measure with values in a topological groupLet Ъе a ring of sets and

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X IX (1976) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE X IX (1976)

W o j c i e c h H e r e r

(Warszawa)

O n the extension oî measure with values in a topological group Let Ъе a ring of sets and X a topological abelian group.

A set function [ a : & - > X is called a measure if and only if for any two sets A, B e f with A n B = 0 , ^ ( A u B ) = / а (A) /л{В) and for every decreasing sequence (A n) of sets in ^ with C]An — 0 , lim/л{Ап) — 0.

П

The problem of extending [ a to a measure on some u-ring of sets containing was considered by many authors under various additional, assumptions on X and ( a (see [2] and [4] for references).

Finally, the problem of extension was solved by M. Sion in [4]. He proved there that, if X is complete, T

topological abelian group and measure [ a is exhaustive, i.e. for every sequence (A n) of pairwise disjoint sets in lim /

( А п ) = 0, then ^ can be extended to a measure on some u-ring of sets containing

t (exhaustivity of / a is, of course, necessary for an extension). Sion’s proof of this theorem depends on the general­

ization of the classical method of Carathéodory outer measure.

In the first paragraph of the paper it is shown th at this theorem may be obtained with the help of some topological extension arguments (extension by “ continuity” was used also by L. Drewnowski in [2] but his method is less direct than ours).

Second paragraph of the paper contains some remarks on the nature- of our extension.

In the third paragraph we prove th a t our and Sion’s method of extension gives the same results.

1. By

&a (£%0) we denote the class of all unions (intersections) of countable subclasses of sets in By H{M) we denote the class of all subsets of sets in 3%a (if ^ is an algebra of subsets of some set Ü then Н(Щ is simply the class of all subsets of Q). We shall consider H(&) as an abelian group with the operation of symmetric difference of sets as the addition.

This operation will be w ritten as “ + ” and this will denote the symmetric difference of sets only for sets in Н{Щ) if U*, V* c H(0ê), then U* + V*'

= {A +B\ A e Z7*, В e V*}. Let Ш be a ring of sets and [ a a measure defined,

on M with values in topological abelian group X. Let H be a fixed basis,

of closed, symmetric neighbourhoods of zero of the group X.

1.1.

Let for every Ue U the set Ü* c H(t%) will be defined as follows:

Te Ü* if and only if there exists a set E e M a such th a t T a E and ior every A e 0 t with А с E, we have p{A)e U.

By XI* we shall denote the set {U*\ Ue XI}.

1.2.

XI* is a base of neighbourhoods of 0 for some group topology in H(Ûê).

P ro o f. It follows immediately from Definition 1.1 th at the set XI*

has the following properties:

(i) for every U* e XI*, 0 e U*,

(ii) for every U*, V* e XI* there is 17*e XI* such th at W* <= U*r\V*.

Thus it suffices to prove ([1]; Chapter 3, § 1, n.2) th at XI* possesses the property:

(iii) for every U* e U* there exists such a V* e XI* th at F* + V* c: U*.

We shall prove this by showing th at if V + 7 + 7 c U, then V* + + V* c U*.

Let TeV* and ZeV*, what means th a t there exist sets E e M a and Qe Ma such th a t T a E, Z a Q and p ( A ) e V for any set A e M with A a E or A c Q. Since T + Z c E u Q and E^jQeMa it suffices to show th at ior any set A e M with A c= Eu Q , /u(A)e U.

Let E = U A n and Q = [J E n, where (An) and (Bn) are increasing sequences of sets in Thus for any with A cz E kj Q we have p ( A ) = lim [y (A n (An u B j)) = Hm (p (A n A n) + y {A n B j ) - y(A n (An n B j )

е У + F + F c U, what completes the proof.

The group topology in H(Ûë) determined by XI* obviously does not depend on the choice the basis XI of neighbourhoods of zero in X. This

topology will be called ju-topology of H(M).

1.3.

Operation of union of sets is uniformly continuous in y-topology of H{âê).

P ro o f. Analogically as in the proof of (iii) of Theorem 1.2, we can show th a t for any U*e XI* and V e XX, if V + F + V <= U, then for every T, Ze H{M) with T, Ze F*, we have TvjZe U*. This means th a t the operation (T, Z) -> T kj Z is continuos at the zero of the group H(ûê).

Since for every U*e XI* and Ae XI* if В c A, then Be U*, we obtain our thesis from the formula (T u Z) -i -(T u Z ') c (T -\-T”) v ( Z -\-Z'), holding for any sets T, T', Z, Z ’eH{0t).

1.4.

Operations of difference and intersection of sets are uniformly continuous in y-topology of H(M).

P ro o f. T \ Z = { T u Z ) + Z and T n Z = { T v Z ) + {T + Z) for any T,

,ZeE{@).

Extension of measure 75

1.5.

Closure of 22 in y-iopology, 222, is a ring of sets.

P ro o f. The family 22 is closed with respect to the operations of union and difference of sets and the thesis follows from the fact that this operations are uniformly continuous in ^-topology.

Now we are going to prove the main result on /^-topology of I I (22) which will allow us to prove our extension theorem.

1.6.

I f the measure p is exhausitive, then for any increasing sequence (Mn) of sets in 22, ^-lim Mn = { J M n (^-lim denotes the limit in

П /г-topology).

P r o o f of the theorem will be produced in four steps:

1° If (An) is an increasing sequence of sets in

, then y-\imAn = U A n.

П

We must show th a t for an arbitrary neighbourhood TJ* e U*, there is an index n

such th a t for all indices Jc > щ , (L U „) + A k€ lJ*. What means th a t there exists a set E t

a such th at ( I J i J + i j . с E for all k ^ n

and p(A)e TJ for any A e 22 and A <=. E.

First we shall prove th at there is an index w0such th a t for all n = 1,2,...

if A e

and A с= А п\ А Пй, then p(A)e TJ. Suppose the contrary. Then for any index n x there is an index n

and the set B

t

such th a t B x a A n

\ A ni and p(BJ

TJ. Iterating this procedure we obtain a sequence (Bn) of pairwise disjoint sets in 22 with p(B

TJ for n = 1, 2, ... what contradicts exhaustivity of p.

Taking E = ( U Hn) \ H no we obtain a set in

a such th a t for all k'jz n0,

(UiJ + А к c ( I J i J M с E and for any set A t 22 with A cz E, we have p(A) = lim ^ fJ .n ( A n\ J . no)))« U.

П

2

° If (Gn) is an increasing sequence of sets in 22a,then/bi-lim.Gn = U Gn.

П

Let Gn = U A k, where (A^) are for n = 1,2, ... increasing sequences к

of sets in

. We may assume th a t A k cz A k+l for n, h =

,

, ... Let TJ*

be an arbitrary neighbourhood in Ц*. In view of 1° there is an increasing sequence of indices (kn) such th a t Gn\ A k e TJ* for all n = 1,2, ... Since {JGn = VJAk we have by 1° th a t //-lim A k = Hence by arbitrari-

П

ness of the neighbourhood TJ*, p-limGn = \J G n.

71

3° If M € 22, then for every TJ* t U* there exists a set G e 22a such th a t M cz G and G \ M e U*.

Since 22 is the closure of 22, there exists a set A t 22 such th at M J-Ae TJ*.

Thus by the Definition 1.1 there is a set E e

a such th a t M j - A cz E

and Ее TJ*. Taking G = A vj E we obtain a set in

a such th a t M cz G

and G J M cz E, what implies G \ M e TJ*.

4° If {Mn) is an increasing sequence of sets in ^ , then //-lim Mn = [J M n..

П

Let Ü* be an arbitrary neighbourhood in U*. Let ns choose a neigh­

bourhood V* e H* such that F * + F * a JJ. In view of the continuity of the operation of union of sets there exists a sequence ( V*) of neighbour­

hoods in U* having the following properties : T, ZeV* implies Tv Ze V*

and T, Ze V*+1 implies Tv Ze V* for n = 1 , 2 , . . .

By 3° there exists a sequence (Gn) of sets in 9Ha such th a t Mn c Gn П

and Gn\ M ne V * for n — 1,2, ... Let for n = 1, 2, ... G'n = {JG{. For

n n i =1

all n = 1, 2, ... we have th a t (G'n\ M n) = U ( ^ \ Ж га) c U ((тг- \Ж г-)e V*

i = 1 г = 1

By 2° p-limG'n = LJGn. Thus for sufficiently large n, {{JGn) J-G^eV*

and hencen

( \ J G n) \ M n = ( \ J G n) + G ’n + ( G^\ Mn)eV* + V* c U*.

But { v M n) + M n c= ( KjGn) \ M n for all n = 1, 2, ... and so ( u d f j - f + Mn e JJ* for sufficiently large n, what in view of arbitrariness of JJ* e U*, finishes the proof.

Now we are able to prove our extension theorem.

1.7. T

. Let M be a ring of sets and p an exhaustive measure defined on 91 with values in complete, T

topological abelian group. Then p.

can be extended to a measure p on the а-ring of sets ffl.

P ro o f. The measure p is uniformly continous function on M in //-topology of I I (9). Indeed, for any sets A, Be 92 we have the formula:

p(A) — p(B) — p ( A \ B ) — p ( B \ A ) . Thus if for an arbitrary neighbour­

hood U e U we choose a neighbourhood Ye U such th a t F + F c JJ, then for any two sets A, Be 9t with A - j B e V*, p(A) — p(B)e U.

Since the group X is complete and T

, the function p can be extended to a continous function p on 9t ([1], Chapter 2, § 3, n.2). By Corollary 1.5 and Theorem 1.6, 9t is .a cr-ring of sets.

For any two sets M , N e 9 t with M r \ N = 0 , we obtain the formula

p ( M kj X) — p(M )J-p(N) from the identity p ( A kj B) = p(A) + p(B) —

— p ( A n B ) by passing to the /г-Hmit successively with A->3I and B-+N and exploiting continuity of p and continuity of operations of union and intersection of sets.

The fact th a t // is actually a measure on ffl follows from Theorem 1.6- in view of continuity of p in //-topology of 11(9).

Closing this section we shall prove some approximation proj)erty of the measure // constructed in Theorem 1.6, which will be used the sequel.

1.8. L

. Let Me 9$ For an arbitrary neighbourhood Ve U there exist sets F e 9 d and Ge 9 a such that F с M c G and p(N)e U for all set#

N e 9 , N c G \ F .

Extension of measure 77

P ro o f. Let Fc H be such a neighbourhood th a t 7* -j- 7* c U*.

Since M belongs to the closure of 0t in /г-topology, there are the sets A e 0t and Ee3$a such th a t Ж + А < = ■ E and E*V*. Taking F = A \ E and G = A kj E we obtain sets Fe3#d and Ge3$a such th a t F <= M c G and G \ F c= E. I t remains to prove th a t for any set N e Зё if N cz E, then у (Ж) e Ü.

But у(Ж) is the /г-limit of /л(A), where АеЗё and A tends to Ж in /г-top­

ology. So we can confine ourselves to the sets A e

and A e A -f F* c= E + + 7* с 7* + V* c U*.

2. In this section we give some remarks on the nature of the exten­

sion obtained in Theorem 1.7, connected with the notion of measure- theoretic completeness.

2.1. D

. A set A is called у -null iff A e ^ and every set B e В a A satisfies y(B) = 0. A family of all /г-null sets will be denoted by AT (/л).

A measure /г is called complete iff every subset of /г-null set belongs to 3# (and hence is /г-null).

Exactly as in the classical case (i.e. when A is a group of real numbers (see [3]; § 13, Theorem B)), one can prove:

2.2. T

. I f 3% is a о-ring of sets, then the family 0t of all sets of the form M = A + T, where A e 3% and T is a subset of у -null set, is a а-ring of sets. And the set function defined by /г(M) — /г (A) is a complete measure on 3%.

2.3. D

. Measure /г described in Theorem 2.2 will be called a completion of measure у and cr-ring 0t a completion of a-ring 3%.

Let /г be an exhaustive measure defined on a ring of sets 01 with values in complete, T

topological abelian group and let у be an extension of у obtained in Theorem 1.7. By Sf we denote the c-ring of sets generated by

and by v the restriction of у to Sf.

2.4. T

. Ж(/г) = {0}, where {0} is the closure of {0} in y-top- ology.

P ro o f. Suppose Ж € "(0} and let U be an arbitrary neighbourhood in If. Let us choose Ve U such th a t V* + 7* c U*. Let Me3% and M <= Ж.

In the limit y(M) — y- lim у (A) we may confine ourselves to the sets

A zSt,A ->M _____

A t J and A € M + 7*. B ut since Ж e {0} we have th a t Ж e V*, arid hence M e 7*. Thus A e and Ac 7 * + 7* c U* and consequently y{A)e U and in the limit y(M)e U. But th a t means, in view of arbitrariness of Ue

U, th a t y ( 31) = 0.

Suppose now Ж

and let Z7*e H*. By Lemma 1.8 there are

sets F e 0tô and G e

a such th a t F <= Ж <= G and for every M e 3& with

M a G \ F , p { M ) e U. For any set A e M and A cz G we have /и(A) — p [ A n n (G \F )) -\ -p (A n F ). B ut since A n F cz Жe Ar (p) we obtain th a t p(A)

= p ( A n ( G \ F ) )e U. But this means, in view of arbitrariness of AeS#, A cz G, th a t G e U* and consequently Же Ü*. In view of arbitrariness of

U*e U* we obtain th a t Же {

}.

2.5. C

. Measure p is complete.

P ro o f. Let T а ЖеАг (р). By Theorem 2.4, Же U* for any U*e H.

Hence Te U* for any U*e H* what means th a t Те {0} cz

t.

2.6. C

. ST cz 9t ( where ST is the completion of а-ring ST).

P ro o f. In view of Corollary 2.6 it suffices to show th a t JT[v) c JT(p).

Suppose ЖеМ(г) and let Tie H. By Lemma 1.8 there are sets F e

t&

and Ge L M e such th at F cz Ж cz G and for every set Me ST with M cz G \ F , v(M)e U. For every set A e

t and A cz G we have th a t p{A) = v ( A n n (G \F f)- \- v{A nF ). And since A n F e ST and A n F cz Жe JT{v) we obtain th a t ( a {A) = r(A n (6 J\F ) ) e U. This means, by arbitrariness of a set A e m, A cz G, th a t Ge U* and consequently Же U*. Now, by arbitrariness of the neighbourhood U e U, we obtain th at Ж e {0} = A'(p).

2.7. T

. Let zero of the group X be the intersection of a countable family of neighbourhoods of zero. Then p =v, where v is a completion of v.

P ro o f. In view of Corollary 2.6 it remains to show th a t 01 cz ST and p(M) = v ( M ) for all Me m .

Let (Un) be a sequence of neighbourhoods in U such th a t p Un = {0}.

Let M be an arbitrary set in m. By Lemma 1.8 there are the sets F ne

t

and Gne ma (n = 1, 2, ...) such th a t F n с: M cz Gn and for every set Же m with Ж a Gn\ F n, р{Ж)е Un. Take F = { J F n. Now M = F -\-(M\F), F e ST and M \ F cz f} (G n\ F n). But p (Gn\ F n)e ST and for every I c f with Ж cz 0 ( G n\ F n), р(Ж)е p TJn = {0}. Hence the set (~}(Gn\ F n) is /7-null and consequently r-null. Thus MeSS and p(M) = p(F) = v(F)

= v(M).

In general or-ring m is strictly greater than ST, what can be illustrated by the following example:

2.8. E

. Let the set Ü be uncountable and let m be the smallest algebra of subsets of Ü containing all finite sets. Let A be a group of all real functions defined on Q with the topology of pointwise con­

vergence. Let for any set Ae0t, p{A) denotes the characteristic function of A. Then p is an exhaustive measure with values in complete, T 0, topo­

logical abelian group. О'-Bing 0i is simply the family of all subsets of Q

and p the characteristic function subsets of Q. But since the set Ü is

uncountable, ST Ф m and consequently ST Ф m {ST = ST for M(v) — {0}).

Extension of measure

3. M. Sion’s extension theorem may be stated as follows:

3.1. T

. Let 3% be a ring of sets and у an exhaustive measure defined on the ring of sets 3$ with values in complete, T

topological abelian group. Then:

(1) For every set TeH(3$) there exists a limit rj{T) = lim y {A) in

Aeât,A <= T

the sense of ordering of the set {A \ A e âiï, A cz T } by the relation В < A iff В cz A.

(2) For every set TeH(3ê) there exists a limit £(T) = lim y(T) in

Ge3êaG=>T

the sense of ordering of the set {G\Ge 3$a, G => T} by the relation Q < 6r iff G c Q.

(3) A class of all sets in H(3%) such that for every set ТеН(Зё), £(T)

= Ç(Tr\M) -\-£(Т\М) is a а-ring of sets containing 0t.

(4) The restriction y

of | to 3#Q is a measure extending у and y.

The aim of this section is to prove the following

3.2. T

. 3

= 3$ and y

= y, where and y

has the same meaning as in Theorem 3.1 and 3% and у as in Theorem 1.7.

P ro o f of the theorem will be produced in four steps.

1° Let Те H(3$) and Ue H. There exists a set Ge 3ia such th at T cz G and for all Me3

with M cz G \ T , y

(M)e U.

By the definition of £ there exists a set Ge 3$a such th at T cz G and for every Qe3?a with T cz Q cz G, Ç(T) — y(Q)e U. If Me3

and M a G \ T we may confine ourselves in the limit lim y{Q) to the sets Qe Зёа such th a t

Thus we obtain th a t

—

But from the definition of 3%Q, £ ( T cj M) = £(T) + ! ( M) and hence y

(M) = £(M)eU.

2 ° i 0 c m.

Let M e 3/t%. Since 3&a cz 3%, it suffices to show th at for any U e U there are sets Ge3#a and Ee3#a such th a t M cz G, G \ M cz F and for all sets Ae3% with A cz E, y(A)e U.

Let Ue U be such th at У + 7 c U. By 1° there exist sets Ge3êa and E e 3&a such th a t M c: G, G \ M с E and for any set N e 3

with N cz G \ M or N cz E \ ( G \ M ) , y

(N)eV. Now for any set A e M with A cz E we have, by the definition of J?0, th a t

y (A) = y

( A n ( G \ M ) ) + y

( A \ { G \ M ) ) e V + V cz U.

((G\M)e 3

since 3

is a tr-ring containing 3ê).

3° 3# cz 3t0.

Let Me&. By Lemma 1.8, for every Ue VL there exist sets F Q€&d and G0e 0ta such th a t F

с M c G

and every set AeM, A c G

\ F

satisfies ju(A)e U. Thus, by the definition of rj, for any set Qe&e and Q c Go^F0, r){Q)e Ü.

Since rj restricted to is the restriction of a measure, we have th at for any sets G, HeAêa and F e & d such th at F с H, the following identity holds :

r](G) = T]((GnH)Kj(G\F)) = r](GnH)Fr](G\F)~7](Gn(H\F)).

Thus, if F

c F с M с H c G0, then

rj(G) — rj{Gr\H) — r]{G\F)e Ü.

Let T be an arbitrary set in H(£%). If we shall pass to the limit in the last formula (in the sense of suitable direction) simultaneously with G tending to T, G n H tending to T n M and G \ F tending to T \ M we obtain th a t i(T) - К Т п Ж ) — £ ( Т \ М ) е U, what in view of arbitrariness of a set T € H (Щ and U e U implies th a t Me

4° = [л.

It suffices to show th a t /л

is a continous function on & in /^-topology.

And it follows from the formula

p

{ M ) - p

(N) = ju

( M \ N ) - f i

( N \ M ) (for all M, N e ^ 0) th a t /и

is uniformly continuous function on . Indeed, if M , N e and iTf + Ae U* for some U*e U*, then, from the definition of /^-topology, there is a set Ее Ma such th a t M - \ - N c E and for every A e M with A с E, ju(A)e U. Thus for any Get%a if G c E , then rj(G)e Ü. But in the limits lim r}{G) and lim ?](G) we may confine ourselves to the sets

Geâêa, G ^ M \ N Ge&a,G = > N \M

GeM„ with G c E and thus obtain th at

ju

( 3 I \ N ) — ju

( N \ 3 I ) e U + U .

R eferences

[1] N. B ou rb ak i, Topologie générale, Chapters 1 and 2, Paris 1961; Chapter 3, Paris 1960; Chapter 9, Paris 1958.

[2] L. D rew n ow sk i, Topological ring o f sets, continuous set functions, integration, I, II, III, Bull. Acad. Polon. Sci. 20 (1972), p. 269-276 and p. 277-286, 20 (1972), p. 439-445.

[3] P. R. H alm os, Measure theory, New York 1950.

[4] M. Sion, Outer measures with values in a topological group, Proc. Lond. Math.

Soc. 19 (1969), p. 89-106.