Internat. J. Math.
Math. S.
801
Vol. 3 No. 4 (1980) 801-808
A LEBESGUE DECOMPOSITION FOR ELEMENTS IN
A TOPOLOGICAL GROUP
THOMAS P. DENCE
Department of Mathematics
Bowling Green State University
Firelands Campus
44839 U.S.A.
Huron, Ohio
(Received May 3, 1979 and in revised form February 29, 1980)
ABSTRACT.
Our aim is to establish the Lebesgue decomposition for strongly-bounded
elements in a topological group.
In 1963 Richard Darst established a result giving
the Lebesgue decomposition of strongly-bounded elements in a normed Abelian group
with respect to an algebra of projection operators.
Consequently, one can establish
the decomposition of strongly-bounded additive functions defined on an algebra of
sets.
Analagous results follow for lattices of sets.
techniques yield decomposiitons for
elements
Generalizing some of the
in a topological group.
KEY WORDS AND PHRASES. Lebgue decomposition, projection operor, stronglybounded, topological group.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
Primy 22A10, 28A10, 28A45.
INTRODUCTION.
In 1963 R. B. Darst
2] established a result giving the Lebesgue decomposition
802
T.P. DENCE
of s-bounded elements in a normed Abelian group with respect to an algebra
projection operators.
Consequently, one can establish the decomposition of s-
bounded additive functions defined on an algebra of sets
ILl-
The set of cor-
responding restrictions of additive set functions defined on a lattice of sets
corresponds to a lattice of projection operators
15.
The analagous result on
3.
lattices is established by using the same techniques
More recently, Traynor
has obtained decompositions of set functions with values in a topological group
theorem for
elements in a topological group by the use of projection operators.
It is be-
lieved that this result would aid in obtaining decompositions of eperators on
non-locally convex lattices.
2. PRELIMINARIES
Let G be an Abelia. topological group uder addition, and let T be
I 1 on G.
algebra of projection operators
mean t
It2
tI ad define
partial ordering on
T,
t2
and t
be the set of all
/
{O c G,
t2
t2,
whence
n, defie nU
U
U.
It
t + t2 t 2
so T is a Boolean algebra of operators.
But, we
have t v t 2
G. For
tric neighborhoods about O
each positive integer
OU
lt2’.
which in tur has a lattice structure if we set
providing the sup and inf exist.
(’t2’)’
t 2 to
For t1, t 2 T define t
This relation induces a
to mean t
x
+
Then a subset
there exists an integer n such that H c
each
(n-1)U and y
y: x
Let
where
H C G is bounded if iven U
nU. It
would make sense to eve say
H c(m/n)U for this would mean nH c mU. We define an element f e G to be s-
-
bourvd (strongly bounded) if, for every sequence
elements, ti(f)
O. For
ti
each positive real number
epty subset of T with the properties
T of paiuise disjoint
x, Tx shall denote
a non-
LEBESGUE DECOMPOSITION IN A TOPOLOGICAL GROUP
803
I) tx e Tx and t T implies ttx
2)
e Ty implies t
Tx and
.
Several lemmas
L@%
trary g
I. Let tl,
G and U
2. If
ti
U
t2 s
//
T.
it(g)
Cauch
iS
U for arbi-
t2"
is a monotone sequence of elements of
in
T,
and if
f c G
G.
2nUn
since addition is continuous in
DEFINITION. T
uch
a_
t
s U implies
w write UO U and for each n > O we write Un to represant
c U. This is possible
where Un + Un C Un_ whence
some element of
a V
t2(g)
Suppose
Then
{ti(f)
iS s-botmded, then
Given
Tx, and
xv tye Tx+y.
G.
has Property A if given g
that if
b s T and
a,
(a’b)(g)
6
G and U s
U then
a(g)
//
than there axists
s V and (a +
a’b)(g) F V.
Note that Property A is a condition yielding information about the growth
of elements from
G; a
condition on the manner in which projections affect the
relative location of elements in symmetric neighborhoods.
.
We also look at a
smaller class of neighborhoods by selecting an arbitrary bounded set
and forming the sets
S
....
U, 2U,
U2,
I, 2,.
nU with n
and set
UI , U2,
Choosing
U from
we then form
equal to the set
n
It follows that
U
then there exists
LD@tA 3.
tI
possesses the following property inherited from
t2,
then
U
such that U
+ U
If T has Property A with respect to
t2(g)
U implies
t1(g)
,
h.
LSt f
and if
U for arbitrary g
-
G be s-bounded,
& positive integer n such that if
J
..
c U. This yields the result
t
{tk C
T and U
i > n
then
z,.
t2
T with
G and U
From now on we shall assume T has Property A with respect to
L4MA
if
Then there exists
804
T.P. DENCE
v
v
kn
ikj
For g
Lemma 3
Kuarantees that
%. If t(g)
F.
()n
[Akl
+ U
n for
"
()i"
er.
Ts
1/n}.
T
a W s
S(n,g)
follows. Set
I ()I + ()2 +
2n-1)/2U, d hen A2
bitr Ui, d set
C
e (sen
t
e
there
for I choices of W e
i
c
a
d coct
suce
n
pre=e el se
.
e U for all t
S(n,,)
+ W
,o bi,
T1/n, en
O for s t
t U e S(n,g)
U1 + U2 +
t(g)
e.
S(n,g) Is
no
ch at WI + W2 +
{U :
S(n,g)
G and n s N let
ven
Ak ch
ossible
+
C
0
ere es
> 0
con {%(l): t s T in},
sce t() # 0 for so t.
sc
d ch ctue
Let us denote this set W by W(n,g). This lemma implies that out of all the
neighborhoods
Since
tt(g}i
W(n,g) to
contalnin
t
t(g):
Tl/(n+l)
be nested,
W(n,g)
t e
T1/n , W(njg) is one of the "smallest."
iS contained in
D
t t(g):
T
t
11n
we can choose our
W(n+l ,g). Assuming this sequence of neighborhoods
cnverEes, we are led to defining the following function.
DEFINITION. Let
Y:G-
by
Y(g)
lira
W(n,g).
This function is the counterpart to the function y in
[2].
Our last le,ma is
the following.
L 6. Let G be complete and f e G be s-bounded.
associated sequence of neighborhoods as above that contain
Then given M > O there exists a decreasing sequence
1) if x > O then there
[ai
Let W(n,f) be an
It(f):
t e
T1/n.
in T such that
exists an integer i such that ai e
Tx,
and
LEBESGUE DECOMPOSITION IN A TOPOLOGICAL GROUP
2)
lira
ai(f) W;
PROOF. To
such that
ti(f) E W1
J
where W
just sketch %he essentials
W2 +
+
possible by the choice of
such that
+
+ W +
2
i > n
+
%+2
W(n,f). By Lemma
imp lies
V
Iemma 4 again to the sequence
n suc that
W(n,f),
v
kg
q
If
uj
-
V
-p
k
/
uj
nj+
uj)(f) +j+3.
k &
nj
is
W. rot
{nj
tk)(Z) .j.e
and k
T1/2nj+
/
Setting
jk
.3.
Applying
produces a positive
t)(Z)
V
nj. <
ti then
k/
nj < i
up
/
j<pk
tk
%i e T/2i+]
tk)(f) e
ikj
nl< kn2
Continuing this process we get an increasing sequence
such that
allWi.
there exists a positive integer n
tk
t
or
1,2,’",M+2. This
i
Wi,
tnl+l, tn1 +2’ "’" tn2’
/
and
the 1emma, we let
for all
ikJ
integer
o
lira
805
>
uj
i
j
>
ne.
of positive integers
wenever
p
q > nj.
implies
j
produces the desired
decreasing sequence.
We now can state and
THEOREM. Let
. .
our main decomposition result.
prove
G be an Abelian topological group, and let T be an algebra of
projection operators on
G. Assume
Tx,
and
are as before with G being
complete, and with T possessing Property A with respect to
bounded then there exists unique elements
If f e G is s-
h, s e G such that
1)f=h+s,
2) given U e
.I
there exists a positive real number x such that if t e TX
then t(h) e U,
3) given U
e
and
>
0 there exists t
PROOF. First, as counterparts to
T such that
t’(s)
e
U.
the classical Lebesgue decomposition
theorem, the element h is to represent the continuous portion of f, while s
represents the singular portion.
Again, to just sketch some of the essentials
of the proof, we bypass the uniqueness and, turning our attention to existence
806
T.P. DENCE
f satisfies condition (2) then there is nothing to prove.
note that if h
noting Y(f) by
we assume W(f) contains points other than O e G.
W(f),
i
Lemma 6, there exists a sequence
in T such that lim
i
Then, from
+
f
W1
f
De-
W2 ( f
allWi(f). Let s lira ali(f) G and fl f Sl itm ’Cf}. f
Y(fl ) 0, then fl satisfies (2) and the proof is completed because i(f)- s
implies
i ’(s 1)- O and thus fl is also s-bounded. So given U s and s O
for
there exists t e T
e such that
t’(s)
e
U, namely t
for large i.
i
If
fl
does
fl produces another sequence a2i T
such that llm a2i(f1)
Wl(f1) + W2(fl). Let s 2 lim a2i(f1) and f2 fl s2
+
lim
a2i ’(fl ) Then f2 is s-bounded and f f2 + (Sl + s2)" To show s s 2
ot satisfy
-
(2),
applying Lemma 6 to
(3) we let U
satisfies condition
’(s 2)
a2i ’( s 2)
(all
t
A
U for all i greater than
a2i’)(s I) (i
+
A
for large i.
and the proof is completed.
integer k,
O},
Y(fk
We have
ali’(s I) -, O
N.
a2i’)(s 2)
So if Y(f2
If
not,
Then
U.
(ally
Condition
O
a21)’(s
(3)
then let h
continue the process.
we are through.
and
’(s
exists a positive integer N such that
O. So there
i v a2i
> O.
and
+
U
and
s2
is satisfied by letting
f2
and s
+ s
2
s
If for some positive
Otherwise we obtain a sequence
(Sk, fk)
ea ences
s
i=1
of elements of T such that for each positive integer k we have
1) there exists a sequence
and
ald.
of positive reals where
T
2) s k
with
3)
fk fk-1
4)
sk
Sk
Wl(fk.ll
k
5) f
Xki i=1
fk
*
si
i=1
il aki’ (fk-l)’
* W2(fk.11 for all Wi(fk.1 )’
i-* O
LEBESGUE DECOMPOSITION IN A TOPOLOGICAL GROUP
In the end we will have our decomposition f
807
s
h + s with s
and
i
h
f
Toward this goal, although the steps shall be omitted, the next step is to show
lira sk
0 by showing that sk eventually
And then it must be established that
n
lt s
s
n
i’ (sk) Uk+
s
Assuming this, we then
shall, sl tlt s satisfies
then there exists a positive integer N such that
i
k
s i exists.
lira
n
s. I
f
to an arbitrarily selected U
i
for all i greater than some positive integer
e
lira
’d h
i
belongs
N
(s)
VN aki]
V
aki
.
si
c’I.t&on
Since
e
UI,
s
and fen
i=N+1
+
sj
V
aki
sj
-j>N
N
k:N(sJ)
+
[k
j=1
So, let t
V
k-N
fled. Now
h
aki
f
s
e
U2 +
s
U.
where
lira
+
i
UN+
+
U
N
ba
/ s.S
J>N
for large i
maxim1,---,MN
max_M1,-.-, M... and condition
and
fn
Y(fn
"O’"
Then
Y(h)
O
(3)is satisand the
decomposition is finished.
These results are Dart of the author’s dissertation from Colorado State
University.
REFERENCES
I. R.B.Darst.
A Decomposition for complete normed abelian groups with applications to spaces of additive set functions,
.A..er, ’lath._ Soc., 103
()
5-558.
2. R.B.Darst. The Lebesgue decomposition, Duke
Tran..s
Math..J.ournal
30 (1963) 553-556.
s.
T.P. DENCE
808
3. R.B.Darst. The Lebesgue decomposition for lattices of urojection operators,
Advances in Math I 19 75) 30- 33.
4. T. Dence.
A Lebesgue decomposition for vector valued additive set functions,
Pacific Journal of Eath.,
57 (1975) 91-98.
A Lebesgue decomposition with respect to a lattice of projection
operators, C...a.ad, Journal. of l.lath., 29(1977) 295-298.
5. T. Dence.
6. T. Traynor. Decomoosition of group-valued additive set functions, Ann. Inset.
Fourier (grenoble-) 22 (1972) fasc. 3, 31-1 40.
7. T. Traynor. The Lebesgue decomoosition for group-valued set functions, Tran___s
.Amer...Match.
Soc., 220 (1976) 307-319.