Pacific Journal of
Mathematics
CONVERGENT SEQUENCES OF FINITELY ADDITIVE
MEASURES
T SUYOSHI A ND Ô
Vol. 11, No. 2
December 1961
CONVERGENT SEQUENCES OF FINITELY
ADDITIVE MEASURES
TSUYOSHI ANDO
1. Introduction and Main theorem. Though sequences of countably
additive measures have been investigated by many authors, comparatively little attention has been paid to finitely additive measures in general.
The main purpose of this paper is to give a generalization of a classical
convergence theorem to the case of finitely additive measures and its
improvement.
Let <5$ be a tf-complete (infinite) Boolean algebra with the unit J.
9ft is the class of finitely additive measures on & with bounded variations, that is, real valued functions / i o n ^ with the following properties:
sup I pt(E) | < OD ,
fjt(O) = 0
and
fi(E U F ) + fJt(E f l f ) - ft(E)
for every E, F e £$. We shall call elements of 9ft simply measures.
Under the ordinary addition and scalar multiplication, 9ft is a linear
space. Moreover it is a universally continuous semi-ordered linear space
[6] ( = a conditionally complete vector lattice [2]) under the order relation: v ^ ft means v{E) ^ fJt(E) for every E e &m The symbols V
and A will denote supremum and infimum in 9ft respectively. We shall
write ft+ = M
j V 0, ft~ = ( — ft) V 0 and | ft | = ft V (—ft), then ft = fi+ — pr
and \ fi\ = ft+ + ft". For each subset © of 2ft its orthogonal complement,
i.e., {j"| |jw| A I v| = 0 for every v e @} will be denoted by &1. We
quote some results from the theory of vector lattices (see [6] chap. I).
@ is called normal, if @ = (&1-)1. Any orthogonal complement is normal.
Every normal subset is a direct summand, that is, 9ft = @ 0 @x (in
linear order sense). Thus each normal subset @ determines a linear
lattice homomorphism of 9ft onto @ which makes @ invariant. Following
[6] §5 this homomorphism will be denoted by [@], that is,
+ /9i;) = a[@]/i + /3[@]v
(a, y8 real) ,
V v) = [@]JH V [@]v ,
and [@]jM = // is equivalent to ft e @. When @ = ({v}-1)1 where {v} consists of a single measure v, the linear operator [@] will be denoted
simply by [v], ft is said to be absolutely continuous with respect to vy
Received June 2, 1960.
395
396
TSUYOSHI ANDO
if I v I (E) —> 0 implies /^(i?) —* 0. It is known that this is equivalent to
the relation \y\fi = ft. A measure ft is called countably additive, if
I p> I (\Jk=iEte) < S?=i I A* I (Ek) for every sequence {Ek}. The set of countably additive measures is normal. It will be denoted by S. Following
[8] we shall call measures in S 1 purely finitely additive.
A classical convergence theorem of countably additive measures can
be formulated as follows (see [7]):
Let {fjtk} be a sequence of measures such that limk^o/ik(E) exists and
is finite for every E e &.
If every [ik is absolutely continuous with
respect to a fixed countably additive measure v, then the function
fi(E) = \im.k^oo [jtk(E) is also a measure absolutely continuous with respect
to v, and the sequence has the following uniform absolute continuity:
I v I (JE7) —> 0 implies
sup
\fit\(E)-*0.
We shall prove the theorem without assumption of countable additivity. Since, as stated before, absolute continuity can be expressed in
terms of [@] the following theorem will give a more complete answer.
If lim^^fik(E) exists and is finite for every Ee ^ ,
pt{E) == imi^^^TciE) is a measure, and for each nor-
MAIN THEOREM.
then the function
mal ©
lim [&]fJLk(E) = [&][Ji(E)
Moreover the sequence {[@]/*fc} has the following
nuity:
v(E) —* 0 (for every v e S ) implies
for every E e & .
uniform
absolute conti-
sup | [@]/^ | (E) —> 0 .
2. Proof of Main theorem. In connection with uniform absolute
continuity we begin with some lemmas.
LEMMA
(*)
1. Let {fik} be a sequence of measures with the property:
lim p(Ek - # , ) = <)
for every monotone sequence {Ek} where
p(E) - sup I fik I (E) .
Then for each sequence {Fk} and e > 0 there exist two sequences {Gk}
and {Hk} such that
(1)
Gk = Fk U Fk+1 U • • • U Fj{k)
for some j{k) ,
CONVERGENT SEQUENCES OF FINITELY ADDITIVE MEASURES
397
(2)
Proof. Since the sequence FkJ == Ui=»^ (i = &) *s monotone (for
each fixed &), by property (*) there exists a sequence {j(k)} of positive
integers such that j(k) ^ i(& + 1) and
p(Fki - FfcJ(fc)) ^ s/2* for
i
We define the desired sequences by
Gfc = Ffc>jU)
and fl* = fl Gt
Then (1) and (2) are trivially satisfied.
As to (3)
u — FitJ{i))
=
= e.
This completes the proof.
REMARK.
< 1')
Likewise we can choose
j and {Hk} as follows:
Gk = FR n FK+1 n • • • n ^ (
( 3')
for some j(k) ,
/>(#* - G.) ^ e
LEMMA
A; = 1, 2,
2. For cray non negative measures v and \JL
where the infimum
is taken over all the sequences {Eh} such that
r
UJCEJC C E and lim^ o o v(£ - Ek) = 0.
Proof. First remark that limfc^co[^]/^(£rA) — [^]i«(£r) for every such
sequence, because [v\[x is absolutely continuous with respect to v. Since
[v]fjt(E) = inf jsup [v]fjt(Ek)\ g inf jsup
I A;
J
Ik
(because 0 g [v][i ^ //), the function ^(E) defined by the right side of
(4), is a measure with the property: [v\[x ^ [ix^ [i. If it is proved that
/*! itself is absolutely continuous with respect to v (i.e. [v][*i = /^), by
398
TSUYOSHI ANDO
the order-preserving property of [v] we have
Ml* = M(MAO ^ M A = A ^ Mi">
that is, [v]// = /^. Now suppose that fa is not absolutely continuous
with respect to v, then there exist a sequence {FK} and e > 0 such that
v(Fk) ^ l/2fc
(5 )
and
^(F fc ) > Se
k = 1, 2, . . . .
Since condition (*) is evidently satisfied for single fa, Lemma 1 guarantees
the existence of {Gk} and {Hk} with the properties (1), (2) and (3) (with
fa instead of p). From (1), (2), (3) and (5) it. follows
fa(Hk) = fa(Gk) - fa(Gk - Hk) ^ ft(F») - ftiG* - Hk) ^ 2s i.e.
By the definition of fa there can be chosen a double sequence {EkJ}
such, that
U Ekj dHk^
(7)
Hk+1 ,
Km v(Hk - Hk+1 - Ekj) = 0 ,
i
i—
sup M ^ i ) ^ A(fl* ~ fl*+i) + £/2fc
Writing D3 = \JUiEkj,
it follows
consequently we have
(8 )
lira viH, -D,)
= 0,
because by (5) and (1)
v{H%) £ v(Gt) £ ± v{F})
j =t
j-i
On the other hand, on account of (2), (6) and (7)
£ ± fa(Hk - Hk+1)
k=i
k
=
that is,
sup p.{D}) ^ /^(H,) - e .
3
Taking (8) into consideration, by the definition of fa.
fa(Hx) g sup fi(Ds) S fa(Hx) - s .
fc
= 1, 2, • • • .
CONVERGENT SEQUENCES OF FINITELY ADDITITE MEASURES
399
This contradiction establishes the assertion.
We shall reduce a proof of Main theorem to the case of a concrete
Boolean algebra. The simplest tf-complete (infinite) Boolean algebra is
the class ^Y* of all subsets of natural numbers. Phillips proved a special
case of our Main theorem when & — ^f \ The following Lemma due
to him is essential.
LEMMA 3 (Phillips). Let {<pk} be a sequence of measures on ^K.
If \imk^<pk(A) exists and is finite for every A e <yf^, then
lim <pk{{k}) = 0
fc->oo
where {k} is the set consisting of single k.
This is a slight modification of [1] p. 32 Lemma.
Up to this point the ^-completeness of & has not been used, however in the following Lemma it plays a decisive role.
4. If lim^^ fik(E) exists and is finite for every E e &',
then sup*; | ytk \ (I) < oo and (*) is satisfied.
LEMMA
Proof. As proofs of two assertions are similar, we confine ourselves
to the proof of (*). Supposing that (*) is not satisfied, we can choose
a sequence {Ek} and e > 0 such that E1dE2d • • • and
\fik\(Ek+1-Ek)>2s
k - 1 , 2 , ...
(taking a subsequence of {[ik} if necessary). Since in general (see [8])
there exists a sequence {Fk} such that
Fk c Ek+1 - Ek and | ftk(Fk) \ > s
k = 1, 2,
Writing <pk(A) = /^(U;e J^) for A e ^
(here the (/-completeness of &
is necessary), we obtain a sequence of measures on ^V~ with the property: \\mk^<pk{A) exists and is finite for every A e ^".
Then Lemma 3
shows that f*k(Fk) = <pk({k}) - ^ 0. This contradiction establishes the
assertion.
With these preparations we are now in position to prove Main
theorem.
Proof
of Main
theorem.
Since sup fc \f*k\(I)<
°° b y L e m m a 4, t h e
function [x{E) = Wm^^fjtJ^E) is a measure. Considering the sequence
{[ik —ta} instead, we may assume [x = 0. Define
400
TSUYOSHI ANDO
It is not difficult to see that v is a measure and [&]fik = M/^ fc = 1, 2, • • •
(see [6] § 5), hence we may also assume [©] = [v], and we shall prove
lim [v]fik(E) = 0
for every E e ^
.
By Lemma 2 for each J57 there exist sequences {Ckj} and {DkJ} such that
U Cw c E ,
U ^ c £ ,
lim v(.E - Cw) = lim v(E - Dk)) = 0 ,
j-*OO
J-»OO
lim ^+(CW) = [i»K(B) ,
and
lim f£i(Dtj) = [v]ft(E)
fc
= 1, 2, • • • .
Since
y(£7 - Cs, n Dkl) £ v{E - Cw) + v(^ - i?w) ^
0
by Lemma 2 we obtain
mt(Etj)
£ lim fit(Ct}) = [v\
where Ek3 = Cfcj n I>fcJ, similarly
lim //fc(Sfcj) = [v]fiu(E)
k = 1, 2,
Writing Ffcj — H L i ^ ^ the similar arguments show
lim v(E - FkJ) = 0
and
lim /i?(F4i) - [y]i"?(JS7)
i = 1, 2, • • •,fc„
and similarly for \pt). By a diagonal method we can find a subsequence
^} of {Fkj\ such that
v ( S - F«) < 1/2*
i = 1,2,
and
lim ^(F,) - M^ + (F)
fc
= 1, 2,
and similarly for {/^^}. Since condition (*) is satisfied by Lemma 4,
there exist sequences {Gk} and {Hk} with the properies (I'), (2') and (3').
CONVERGENT SEQUENCES OF FINITELY ADDITIVE MEASURES
401
On account of (1'), by the similar way as above, it is not difficult to
see that
lim nt(G5) = [v]pt(E)
fc
= 1, 2,
and similarly for {fa}, hence by subtraction
(9)
lim fik(Gj) = [v]ME)
fc
= 1, 2, •
On the other hand, for each a > 0 (*) and (2') guarantee the existence
of n = n(e) such that
/ ( ^ -Hn)^e
for j ^ rc ,
consequently by (3')
(10)
I MHn) - MGJ)
I^ ^
- Hn) + p{H, - 6,) ^ 2s
for j 2> n .
Then from (9) and (10) it follows
I fik(Hn)
- [v]ftk(E)
I = l i m I fik(Hn)
-
fr(Gj)
\ ^ 2 e fc = 1 , 2 , • • • .
Since lim^0Oi«A.(iif7l) — 0 by hypothesis, combining this with the above, we
obtain Hm^[v]fJtk(E) ^ 2s. The arbitrariness of e implies limfc_oo[v]^fc(£r) =
0.
Next we shall turn to a proof of the second assertion. Here we
may again assume [@] = [v]. First remark that the sequence {M/4}
satisfies (*). Given a sequence {Fk} with the property: v{F1c) < l/2fc k —
1,2, •••, applying Lemma 1 to this sequence and flV]/^}, for any e > 0
we can find sequences {Gk} and {H^} with properties (1), (2) and (3) (with
p(E) = supfc I [v]^ I (E)). Since (*) and the absolute continuity of every
Mi"* (with respect to v) imply p{Hk) - ^ » 0, we obtain ^mi^^piF^ ^ 2s,
because
p(Fk) £ p(Gk) g ^(ff.) + p(Gk - Hk) £ p(H>) + 2e
k = 1, 2, . . . .
The arbitrariness of e establishes the assertion.
REMARK. When &f is moreover complete, our Main theorem can
be deduced from a result of Grothendieck [3] by the following way. By
means of the theory of Boolean algebra, & can be represented by the
class of open-closed subsets of a compact Stonian space Q. Then by
the natural way 9Ji may be considered as the dual of the Banach space
C(Q) (the space of continuous functions on Q with the supremum norm).
Grothendieck proved that if [ik is (7(5111, C(£?))-convergent, then it is also
a(%R, 9ft')-convergent, where 2J1' is the dual of 9JI and o( • • •) denotes
the weak topology. Since every operator [@] is a(3Ji, 9Ji')-continuous,
402
TSUYOSHI ANDO
the first part of our Main theorem follows immediately.
part is also stated there.
The second
3. Corollaries. An immediate corollary (which is a direct generalization of the classical theorem in question) is as follows.
COROLLARY 1. Every normal set of 9ft is sequentially complete
under the weak topology defined by &'.
Let {Ex} be the set of all atoms in & and
<$ = {ft e 8 11 pt I (EK) = 0 for every Ek} .
Then ^3 is normal. A modification of a recent result of Kaplan [5] § 9
shows that the closure of 21 with respect to the topology in question
coincides with ^3 © 8 1 . In this regards, the following special case of
Corollary 1 is of some interest.
COROLLARY 2. The set of purely finitely additive measures is
sequentially complete under the weak topology defined by &.
4* Sequences of purely finitely additive measures. In connection
with Main theorem a question arises whether ftk - ^ > 0 implies | fik | - ^ > 0.
When sup^efiSS/^i?) > 0, for every E e ^ , Halperin and Nakano [4]
proved that the property "ftk e 8 , f t - ^ > 0 implies | fik |^^> 0 " is equivalent to the atomicity of ^ . We shall treat this problem in 8 1 , namely,
does [J.k—^0 (all ftk being purely finitely additive) imply | / ^ | - ^ > 0 ?
The answer is negative.
THEOREM. There exists a sequence {fjtk} of purely finitely additive
measures such that
lim fjtk(E) = 0
for every E e &
but
Proof. As in § 2 we shall reduce the proof to the case & =
Banach (see [1] p. 83) proved that there exists a positive measure <pQ
on ^
invariant under translation, that is, g>0(I0) — 1 and
0 ^ <pJ(A) = <P*(TA)
for every A e ^V
where IQ is the unit of ^
and TA = {j + 11 j e A}.
sequence {q>k} recurrently by the formula
<pk+1(A) = <pJA n U Bk+1J) -
<PJA
n U
We define a
CONVERGENT SEQUENCES OF FINITELY ADDITIVE MEASURES
403
where
Bkj = {i I i = i
mod 2*} .
From the arguments in [4] it results
(11)
lim <pk(A) = 0
for every A e <yV~
A;->oo
but
\cpk\ = cpQ
ft
Let {JFJ be a sequence in & with the property
(12)
UF* = I
and
= l, 2, •••
FfcnFj = 0
(fc =£ i) .
A;
On account of the representation theorem of Boolean algebra (see [6]
§ 8; [2] Chap. X) there exists a sequence {vk} of two-valued (say 1 and
0) measures on & such that
8kJ
k,j = 1,2, ••• .
We construct t h e desired sequence {fik} from {<pk} and {vk} by the formula:
fik(E) = cpk{A)
where A = {j\ v5(E) = 1}. From (11) and (12) it results
lim (ik(E) = 0
for every E e & .
but I //* I = fi0 k = 1, 2, • • •. There remains to prove pure finite additivity
of fik. For this purpose it is enough to prove it for fiQ only. Invariance
of <p0 under translation shows
3)
- <PQ({J}) - <Po({i\) =
ftW)
i, i = 1, 2, • • •
hence
^ o ( ^ ) - g ^ o ( ^ ) ^ i"o(/) - 1
i, fc = 1,2, . . .
finally
th{Fs) = 0
i - 1, 2, . . .
Since ^0(-^) = 1> this implies pure finite additivity (cf. [8] § 4).
REFERENCES
1. M. M. Day, Normed linear spaces, Berlin, 1958.
2. G. Birkoff, Lattice theory, New York, 1948.
3. A. Grothendieck, Sur les applications lineaires faiblement compactes d'espace du type
C(K), Canad. J. Math., 5 (1953), 129-173.
404
TSUYOSHI ANDO
4. I. Halperin, and H. Nakano, Discrete semi-ordered linear spaces, Canad. J. Math., 3
(1951), 293-298.
5. S. Kaplan, The second dual of the space of continuous functions, II, Trans. Amer.
Math. Soc, 9 3 (1959), 329-350.
6. H. Nakano, Modulared semi-ordered linear spaces, Tokyo, 1950.
7. O. Nikodym, Sur les suites convergentes de fonctions parfaitement additives d'ensemble
abstrait, Monat. fur Math. Phy., 4 0 (1933), 427-432.
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A.
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Pacific Journal of Mathematics
Vol. 11, No. 2
December, 1961
Tsuyoshi Andô, Convergent sequences of finitely additive measures . . . . . . . .
Richard Arens, The analytic-functional calculus in commutative topological
algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Michel L. Balinski, On the graph structure of convex polyhedra in
n-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
R. H. Bing, Tame Cantor sets in E 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cecil Edmund Burgess, Collections and sequences of continua in the plane.
II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
J. H. Case, Another 1-dimensional homogeneous continuum which contains
an arc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lester Eli Dubins, On plane curves with curvature . . . . . . . . . . . . . . . . . . . . . . .
A. M. Duguid, Feasible flows and possible connections . . . . . . . . . . . . . . . . . . .
Lincoln Kearney Durst, Exceptional real Lucas sequences . . . . . . . . . . . . . . . .
Gertrude I. Heller, On certain non-linear opeartors and partial differential
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calvin Virgil Holmes, Automorphisms of monomial groups . . . . . . . . . . . . . . .
Wu-Chung Hsiang and Wu-Yi Hsiang, Those abelian groups characterized
by their completely decomposable subgroups of finite rank . . . . . . . . . . .
Bert Hubbard, Bounds for eigenvalues of the free and fixed membrane by
finite difference methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D. H. Hyers, Transformations with bounded mth differences . . . . . . . . . . . . . . .
Richard Eugene Isaac, Some generalizations of Doeblin’s
decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
John Rolfe Isbell, Uniform neighborhood retracts . . . . . . . . . . . . . . . . . . . . . . . .
Jack Carl Kiefer, On large deviations of the empiric D. F. of vector chance
variables and a law of the iterated logarithm . . . . . . . . . . . . . . . . . . . . . . . .
Marvin Isadore Knopp, Construction of a class of modular functions and
forms. II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gunter Lumer and R. S. Phillips, Dissipative operators in a Banach
space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nathaniel F. G. Martin, Lebesgue density as a set function . . . . . . . . . . . . . . . .
Shu-Teh Chen Moy, Generalizations of Shannon-McMillan theorem . . . . . . .
Lucien W. Neustadt, The moment problem and weak convergence in L 2 . . . .
Kenneth Allen Ross, The structure of certain measure algebras . . . . . . . . . . . .
James F. Smith and P. P. Saworotnow, On some classes of scalar-product
algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dale E. Varberg, On equivalence of Gaussian measures . . . . . . . . . . . . . . . . . . .
Avrum Israel Weinzweig, The fundamental group of a union of spaces . . . . .
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