Internat. J. Math. & Math. Sci.
VOL. 12 NO. (1989) 809-820
8O9
SUBLINEAR FUNCTIONALS ERGODICITY AND FINITE
INVARIANT MEASURES
G. DAS
Department of Mathematics
Utkal University
Vani Bihar
Bhubaneswar
Orissa, India
and
B.K. PATEL
Department of Mathematics
Khallikote College
Berhampur
Orissa
India, Pin Code
760001
(Received January 21, 1986)
By introducing a sublinear functional involving infinite matrices, we esta-
ABSTRACT.
Further,
blish its connection with ergodicity and measure preserving transformation.
we characterize the existence of a finite invariant measure by means of a condition in-
volving the above sublinear functional.
KEY WORDS AND PHRASES. Sublinear functional, Banach limit, conservative, regular,
trans-
lative, null invariant, measure preserving, ergodic.
1980 AMS SUBJECT CLASSIFICATION CODES.
58FII
I. INTRODUCTION AND DEFINITIONS.
Let
be the set of all real bounded sequence
Linear functional
i)
(xn) _> O,
ii)
(Xn+l) (xn)
lim___ Xn --< (xn)
n+
iii)
if
on
=
Xn_
O, n
_< lim
n/
n
lxl
normed by
suplx n
n>O
0,1,2
Xn
almost convergent and we write; F
{x
n
are called Banach limit [I] satisfying the conditions,
If there is a number for all Banach limits
quence
{x
lim x
is almost convergent with
n
s
F-limit
#, the sequence
x
{x
n
is called
It is shown by Lorentz [2] that a ses, if and only if
i+n-
lim
n+
uniformly in
i.
-n ki
Xk
s
(I.I)
810
G. DAS AND B. K. PATEL
Let,
A
such that
a
A
called
(a
k(i)
n,
0, if any
n,
summable to
AIn the case
(F).
The sequence
{x
is
n
(i)
(1.2)
s
Xk
an,k
and in this case we write:
i
uniformly in
is a negative integer.
n,k,i,
0,1,2...
i
if
s
k=O
lim
n/
method
be a sequence of real or complex matrices for each
lim x
a
(i)
n,k
A
If
s,
n
a
n,k’
s(A).
n
and
(i < k < i+n)
1/n+I
A
x
or
0
otherwise,
(A)
reduces to the
then we obtain the usual summability method (A).
It is
significant to note that there does not exist any regular method (A) equivalent to
Theorem 11 and 12)
method (F) (See Lorentz [2]
(A)
The method
s
,i
n,k
u+l
r=
a
r,k
is called
x-> s => x-> s
conservative if
(A), r_egular,
if
The following characterization of regular matrices is due to Stieglitz [4J.
s
method
a
reduces to the almost summability method introduced bv King [3].
(A)
then
In the case
(A)
is called regular if and only if the following conditions hold:
kZ__0 lan,(i)Ik
for all
and there exists an integer
(i)
lim
a
(i)
n,k
i
(1.3)
-> 0,
an,(i)k
kZ__0 an,k
lim
and
such that
m
kE__0
sup
i_>0, n_>m
n
(i 4)
uniformly in
0
for fixed
k
(I 5)
i
uniformly in
i
(I 6)
We write
I111
A
The matrix
m
n
,
d
(i)
translative if
o
(1.7)
(1.8)
called positive, if
2
0
(1.9)
nki
wrte,
X+
The matrix
la
()
(an,k-1 an,k())
(i)
n,k
n,k
we
k0
(i)
k=O
()
a
X
n->O
here
s
The matrix
For real
+/-->0
is called
n--
unifoIy
sup
max
(X,O),
X
max(-X,O).
positive, f
([)- O, unfoly in
an,k
is called almost
lim
kO
(1.10)
The
SUBLINEAR FUNCTIONALS ERGODICITY AND FINITE INVARIANT MEASURES
(x, F, m)
Let
if
0<->T-IA)
m(A)
for all
(A)
A
X
be a measurable
is called null invarian
m
T-nA
A
m(A) =0,
=>
equivalent to measure m, if
T is called measure preserving
is called
The transformation
m(T-IA),
The set
X
The measure
It is conservative, if
F
A
m(A)
0.
F
A
A measure
0, for
or invariant, if
m(X/A)
O,
F.
A
and
n
m(A) <-->
or
be a finite measure space and let, T;
(This is assumed throughout).
transformation.
811
F
A
It is called ergodic if,
T-IA, T-2A
F
T-IA.
A
is called invariant, if
T-IA
A =>m(A)=O
It is called
It is called weakly
1,2,3 ...}
wandering, if there is an increasing sequence of positive integers {r k, k
A
wandering, if
T-rlA
A,
such that
T-r2A
0
exists a
there exists a
{qn
sequence of measures
are mutually disjoint.
are mutually disjoint.
> 0
for
continuous if,
A
is called
uniformly
m(A) <
such that
0
A measure
q(A)
if for each
m continuous
for all
is called
m(A) <
such that
qn(A)
q
m
< e.
A
0, there
g
n
Write:
i+n-I
t(x)
{I
Let
,t}
lim
n
ki
sup
i
x k,
(l.ll)
Xke
,
denote the set of linear functionals
known (see Sucheston [5] Das and Misra [6]) that
{l,t}
and
limits on
{I
such that
,t}
is unique if and only if
#(x) ! t(x).
It is
is the set of all Banach
(x)
-#(-x)
and this hap-
pens when
i+n-I
as
ki
,
n
x
uniformly in
A
Let
sequences.
R
lim
n
sup
i
i)
k
k 0 an
IIAII
x
<
(x) ! R(x),
x
kO
(i)
Xk
in
i.
We now state a lemma.
Let
(a)
x
lim
n
I
I.
By
such that
(1.13)
satisfying (1.13).
It is easily
(*.4)
R(x)
an,k
, uniformly
LEMMA I.
by
is unique if and only if
#
-R(-x)
n
on
(R)
be the set of all linear functional
and this happens if and only if
as
I
I
( 12)
It is easy to see that it is a sublinear functional on
-R(-x) !
{I,R}
R:
k
Hahn-Banach theorem there exists a linear functional
seen that
Then we define,
I
x
is finite valued.
Let
.
Lorentz [2] calls all such sequences as almost convergent
i
be real and such that
R(x)
Since, for all
+a limit
k
x
<
I
then
R(x) ! lim x n
a limit
812
G. DAS AND B. K. PATEL
A
if and only if
is real, regular and almost positive.
-t(-x) -<-R(-x) <-R(x)
(b)
A
if and only if
(c)
If
t(x)
is almost convergent to
x
lim
2
_<
is regular, almost positive and translative.
k0
a
x
s, then
uniformly in
s
k
ERGODI CITY.
In this section, we establish that the ergodicity and invariance can be established
in terms of summabi.lity of a particular sequence and thus generalizes a result of
(Sucheston !-5], Theorem 3) involving almost convergence.
We now examine the foIlowing conditions:
(I)
For soe
(II)
lira
()
rgON 1.
(al
,
C)
(X,
get
()
(c)
c)]
re(B) Ore(c),
n
re(B) re(C)
)
be a fnite measure space and let
>
e
need the following lena for the proof of the theorem
A
Let
s(x
Then
Then
is regular, almost positive, and translat[ve, then
2.
la
()
If
Lg
0,1,2
ergodic and measure preserving.
()> ()
() (I) > T s ergodc
is translative.
() If
(b)
13
F
[ V B, C e
s
T
[m (T-nB
m(T-nB n
k0 an,k
u,iformly in
(III)
{I,R}
{I,R}
e
0
s (x
Xn+l
n
s:
n)
1
1
be the shft operator
Xn+2"
Then
I,(SX)
(a)
<-
*(x)
lxll li
sup
i
where
Let
(b)
d
A
(i)
n,k
is defined by
R(x-SX)
(i) R(SX- x)
(c)
(I 8)
be translative, then for
(ii) (Sx)
k 0 [dn(i),k
x
0
(x)
R(x)
R(Sx)
Let, further
lim
a
(i)
n,k
0
fixed
k
uniformly in
i
Then
(d)
P
R(j 0 sr3x)
ere
PROOF:
r
0
p.R (X)
1,rl,r 2
r
P
is a sequence of fixed positive integers.
Since
R(Sx-x)
lim
s?p
(i)
k 0 an, k
(SXk_Xk)
(2 I)
SUBLINEAR FUNCTIONALS ERGODICITY AND FINITE INVARIANT MEASURES
li-m
i)
k
k=Zo dn
sup
n
i
813
Xk
It follows that
IR(Sx-)I
_<
[[x[[ li--m
k_0_ [dn()[
sup
n
Now as
is linear, we obtain
sx and
(b) (i), (ii)
follows from (2.2), and changing the role of
b (i).
Changing the role of
r
Lastly
R(S
R
Since,
Sx
R(S
x
x
A
sx
and
is trans-
x
in
R(x)
So (c) follows.
R(Sx).
R(x)
x, we obtain
r
+ S 2X
r
rlx + S 2 x)
and
When
is sublinear,
R(Sx-x+x) < R(Sx-x) + R(x)
R(Sx)
by
in (2.2) and (2.3) we obtain (a).
x
follows from (a).
(2.2) (b) (ii)
(2.3)
(sx-x) _< R(sx-x)
(x)
(sx)
Changing the role of
lative
(2.2)
r
r
+ 2x) _< R(S x-x) + R(S 2 x-x) + 2R(x).
x
i.e.
r
R(S
Ix +
r
S
2
r
2R(x)
x)
R(S
r
Ix
x) + R(S
2x
(2.3)
x)
But,
R(S
rl x
x)
1Tm
sup
n+
i
(i)
an,k (Xk+r
k-E0-
(i)
Xk
(an,k-rl n,k(i))
(i)
(i))x k n-I
lim sup
kZO I’) Xk]
n,k
n
krl (an,k-rl
(i)
lim sup
by (2.1)
-an,k(i) x
(an,k_rl
-wrl
i)
l
=_ Xk rl-l_
sup kZ_r
j-EO (an (i)k_j_l an k-j
kO
lim sup
n
i
a
x_
a
a
i
n
n
i
im
i
n/
<
[[x[[
r
R(S
Hence,
2xr
R(s
x)
n
"." A
0
Similarly
j
i
is translative)
0
r
Ix + S 2x)
_< 2R(x),
x
i
Again, since
r
2R(x)
R(2x- S
R(x
S
rlx
r
x- S
2
r
x
+ S
r
+ R(S
S
x
2
r
+ S 2 x)
r
x) + R(S
r
x+S
2
x)
Proceeding as above, we have
r
R(S
2R(x)
Hence,
r
R(S
x
r
x
+ S
2
x)
r
+ S 2 x)
2 R(x)
I
x e-
x
(d) follows by repeated application of (2.4).
1
(2.4)
814
G. DAS AND B. K. PATEL
PROOF OF THEOREM I.
Let (II) hold.
(a)
Then
[-m(T-nB
-R
C)]
0
[m(T-nB
R
n C)]
Since
-R(x) _<
Q(x) _< R(x)
x e
I
It follows that
m(T-nB
C)]
m(B)" re(C)
n= 0,1,2
This proves (II)=> (I).
(b)
Take,
T-IB
B,
B-lin
x/B=
C
(I).
Hence it follows that
m(B)
either
i.e.
Writing, C
m(B)
(0)
0
0
m(B
or
m(B
I)
I)
O.
T is ergodic.
X
(I), we obtain
in
[m(T-nm)]
B
Replacing
T
by
-I B
in (2.6), we obtain
[m(T-n-IB)]
A
If further,
m(T-IB)
[m(T-n-IB)]
-I
m(T
(2.7)
m(X)
is translative, by Lemma 2 (b)
0 < m(X) <
Again, since
(2.6)
re(X)
re(B)
,
[m(T-IB)]
it follows from (2.6) and (2.7) that
re(B)
B)
Hence, (I) => (III)
(c)
B
(III) => (II).
In veiw of (a) and (b), it is enough to show that
F
m(B)
such that
0
m(T-nBoC)
qn(C)
k
E
Take any fixed
F
(2.8)
is an invariant measure and
q
m
q.
is almost positive
lim
for x
C
(qn(C))
q(c)
A
and
0,1,2
n
re(B)
We now show that
Since,
{I,R}
Define, for
kE__O
an-I k
i)
x
0
k
uniformly in
i
(2.9)
I.
Write
(x)
lm
sup
n
i
k=Zo
a
+ (i)
n,k
Xk
So, by (2.9)
R(x)
Since,
x
_> 0 =>
x _> 0
Again, since
R(qn(C))
_> 0
m
R+(x)
R+(x)
> 0
R(x) _> 0
is a measure,
(2.10)
qn(C)
Z 0.
So it follows from (2.10) that
SUBLINEAR FUNCTIONALS ERGODICITY AND FINITE INVARIANT MEASURES
R
Since
815
is sublinear, we have
[-qn(C)]
-R
0.
F.
q(c) Z O, C
Now, it follows from (2.5) that
Let
F
B.i
be a countable sequence
Then
of disjoint sets.
q(il B i)
[qn(il Bi)]
[i qn(Bi)] (’.
[qn(Bi)]
=i
m is a measure)
is continuous linear functional)
(0
is countably additive and hence it is a measure.
So, q
Next,
[m(T-nBa-IC)
re(B)
[m(T-n+IBC)
m(B)
q(T-Ic)
(’."
T
is a measure
preserving)
[qn-I (C)
is shift invariant by Lemma 2 (b),
Since
[qn(C)]
c
q(C)
This proves that
are of measure
is an invariant measure.
q
0
.
I.
or
Since
and
m
q
T
Site
is ergodic, the invariant sets
are invariant measures, an invariant mea-
sure is determined by the value it takes on invariant sets (See Sucheston [8], Theroem,
it follows that
q
m.
Now, we have
q(C)
Hence,
m(r-nBoC-]
m(C)
{T-nB n C},
is unique on
n
[m(T-nBnC)]
i.e.
m(B)
m(C)
{I,R}
But,
0,1,2
m(B)
has unique value
if and only if
R(x) =-R(-x)
%
Hence, it follows that
i.e.
3.
R[m(T-nBo C)] -R[-m(T-nBo C)]
m(B)
(II) holds and hence proves (c) completely.
re(C)
EQUIVALENT MEASURES.
Many necessary and sufficient conditions have been determined for the existence of
equivalent invariant measures (see Sucheston [7], [8], Mrs. Dowker [9], Calderon [10],
and Hajian and Kakutani [11]).
In the pointwise ergodic theorem of Birkhoff [12], it
was necessary to take invariant measure, but Halmos
[13] has shown that even if a mea-
sure is null invariant and conservative, an equivalent measure need not exist.
Sucheston [7], [8] has used Banach limit technique to prove the existence of invariant
measures
We now generalize some of the theorems of Sucheston [5] involving almost con-
vergence and some results of Mrs. Dowker on
(C,I) convergence and establish the exise
tence of invariant measure by using linear functional
{I=,R}
We now prove
THEOREM 2.
Let
A
be a real matrix such that
tive and translative.
Let
(x,
F, m)
IIAII
<
and let
be a finite measure space and
A
be almost posi-
T
be a measurable
816
G. DAS AND B. K. PATEL
Then, the following condition are equivalent.
transformation.
(I)
There exists an equivalent finite invariant measure.
(II)
For some
e
{I=,R}
re(B) > 0
(III)
re(B) > 0
(I)
PROOF.
m.
=>
R
(II).
B e F
and all
[m(T-nB)]
0
[m(T-nB)]
0
Suppose that
p
is an invariant measure which is equivalent to
Suppose that (II) fails to hold.
B e
Then there exists a
such that
m(B) > 0
and
[m(T-nB)
But, since
0
-R(-x) ! (x) ! R(x)
lim x
n->oo
<
n
B
m(T-nB)
x
n
F
[m(T-nB)
0
lira
E
and by Lemma
i=
R(x) < lim
n-co
-R(-x)
it follows that for all
But, since
x
m(T-nB)
lim
> 0, it follows that
n/
m(T-nB)
lim
O.
{x
Hence, there exists a sub sequence
m(T-nkB)
lim
k+
is equivalent to
p
Since
p(B) > 0
Since
p
Hence
p(B)
such that
0
m
and
we obtain
lim
k+
p(T-nkB)
0.
is invariant, we have
p(r-nkB)
O.
p(B)
(I) =>
This is a contradiction and this proves the fact that
(II)=> (III)
Let II hold
[m(T-nB)]
(III)
k}
(I)
> 0
R
Since,
[m(T-nB)]
[m(T-nB)]
> 0
Suppose (III) holds and (I) fails.
< R
[m(T-nB)]
(II).
it follows that
Since Condition (I) is equiva-
lent to non-existence of weakly wandering set (See Sucheston [7], Theorem 6) it follows
that there exists positive integers
such that
_r
B, T
B,
are mutually disjoint.
lim
n->o
I, r I, r
2
0
-r
T-r2 B,
T
k
and a set
B
(i)
-k
X)]
uniformly in i
R[m(X)]
m(X) R(1)
it follows that
m(X).
Again
m(X)
B
Since,
kE=0 an,k
R[m(T
r
R
[m(T-nX)]
> R
s
[m(jO T-rJB)]
E
F
with
m(B) >0
SUBLINEAR FUNCTIONALS ERGODICITY AND FINITE INVARIANT MEASURES
m(X)
(i)
suPi k=ZO an’k
+lim
n
lira sup
X
Where
m(T
k
-k
B),
m(X) _>
S
kO
a
n ,k
s
T _rj_kB
[3tJ0o=
m
817
sr3xk
j=0
is a shift operator.
Then by Lemma 2(d),
R(X)
S
(3.1)
Then it follows from (3.1) that
re(X) _>
Since
m(T
-k
S)
m(T-kB)
R
s
(3.2)
0 by hypothesis and
contradicts (3.2).
s
is an arbitrary positive integer.
This
This proves (III)=> (I).
In the next theorem we give yet another characterization of existence of invariant
measure in terms of the sublinear functional R(x).
THEOREM 3. Let A satisfy the condition of Theorem 2. Let (x, F, m) be the finite
measure space.
Then there exists an invariant measrure equivalent to measure
m
X,
on
if and only if,
(i)
m is null-preserving.
(ii)
T
(iii)
n__ZO
is conservative.
an,k(i)
m(T-kB)
converges uniformly in
for every
i
B e F
Again, whenever it has equivalent invariant measure, then the map
q(B)
defined by
with
[m(T-nB)]
is an invariant measure equivalent to
and agrees
m
on invariant sets.
m
PROOF:
R
q
NECESSITY
Let us assume that
B
Then
is
m
{I,R}
e
Write for
admits an invariant equivalent measure
m
[9] p. 125).
continuous (See Halmos
[m(T
q(B)
-n
B)]
We want to show
(a)
q
(b)
q
is a
(c)
q
is invariant.
is a measure
m
continuous
As in the proof of Theorem 1, we can show that
q(B) _> O, for all
B e
F
It is easy to show that
B,C e
Since
F
m(T-nB)
q
is
q(B) < q(C)
is linear, it also follows that
e > O,
m-continuous, for given
So
C =>
B
< e
m-continuous.
when
5 > 0
(B)
q
is finitely additive.
(T-nB)
< 5
The countably additivity of
m(T-nB)
and
m
and
< e =>
Next,
q(B)
[m(r-n-IB)]
[m(T-nB)]
[m(T-n-IB) m(T-nB)] (
q(B)
m-continuity of
Halmos [9] p. 39).
q(T-IB)
Since
is
Such that
is linear).
<
.
q (See
818
G. DAS AND B. K. PATEL
[m(T-n-lB)
_< R
lim sup
n
i
an ()--O, V
Since
li
sup
n
i
<
[a n (i)
k=O
(i)
m(T-kB)
B)
m(T-kB)
an,k
,k-1
k__Z0 [d n l)l
k
re(X)
0 as
(i)
an,k in(T-k-
i
lira sup
n
i
is translative,
A
Since
and
n
k--ZO
m(T-nB)]
n
uniformly in
i.
Hence,
q(T
-I
B) < q(B).
T
Changing the role of
q(B)
-I B
q(T
B, we obtain
and
-I B)
Hence,
q(T
q
i.e.
-!
-1
B
B.
q
m
[m(T-IB)]
[m(B)]
m(B).l)
mB)
Hence (Sucheston [8], Theorem 2)
on invariant sets.
[m(T-nB)]
q(B)
T
Then,
qB)
q(B)
q
is invariant under
Now if T
So
B)
{I,R}
But
is unique.
q
m
is unique if and only if
on
F.
Thus
R(x) =-R(-x)--
and this happens if and only if
(i)
k=Z0 an, k m(T-km)
lim
n-
q(m)
uniformly in i.
q(T-IB)
Now since,
B
E
F
i.e.
so
re(B)
0
=>
B F
q(B),
m(T-IB)
and
q
m
on
F,
we have
m(T-IB)
re(B),
=0
is null-preserving
m
(See Sucheston [7], Theorem 6) existence of invariant measure is equivalent to
non-existence of weakly wandering sets and non-existence of weakly wandering sets is the
Again
same as conservativeness of
T.
SUFFICIENCY:
Let (i), (ii) and (iii) hold.
q(B)
lim
k=EO
Define
(i)
an,k
Then it can be proved as before that
prove
q
is equivalent to
re(B)
0
q(B)
lim
m.
=>
Since
m(T
-I
B)
m(T-kB)
q
ia an invariant measure.
T
is null preserving,
O.
Then
uniformly in
i.
k__ZO
an,(i)k m(T-kB)’-
0
So only we have to
819
SUBLINEAR FUNCTIONALS ERGODICITY AND FINITE INVARIANT MEASURES
q(B)
Conversely, let
0.
Write;
a
n=l
A
,
q(A )=
i=n
T
q
n
q( i
I T-iB
i
q (T
in
q(B)
B.
T-iB)
i=n
iB)
Then
(q
is a measure)
(q is invariant)
=0
,
agrees on invariant sets, we have m(A
0.
0 => m(B)
tive by recurrence theorem m(B/A
and
Since, q
Hence
q
m
is equivalent to
O.
Since,
T
is conserva-
m.
REFERENCES
I.
BANACH, S., Theioris des Opirations Lineaires, New York, 1955.
2.
LORENTZ, G. G., A Contribution to the Theory of Divergent Sequence, Acta. Math., 80
(1948), 187-190.
3.
KING, J. P., Almost Summable Sequences, Proc. Amer. Math. Soc., (6)I_/7, 1966, 12191225.
4.
STIEGLITZ, M.,
5.
SUCHESTON, An Ergodic Application of Almost Convergence Sequence, Duke Math. Jour.,
30(1963) 417-422.
DAS and MISRA, Sublinear Functional and a Class of Conservative Matrices (Under Com-
6.
7.
8.
Eine Verallgemeineinerungdes Begriffder Fastkonvergenz, Mathematica
Japonicae, 18(1973), 53-70.
munication).
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SUCHESTON, On the Ergodic Theorem for Positive Operators, Z. Wahrscheinlichkeits
Theorie, Und. Verw. Gebiete, B(1966), 215-218.
DOWKER, Y. N., On Measurable Transformations in Infinite Measure Space, Annals. of
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I0. CALDERON, A., Surles Measures Invarionts, C. R. Acad. Sci. Paris., 240(1955), 19601962.
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Amer. Math. Soc., 110(1964), 136-151.
9.
12. BIRKHOFF, G. D., Proof of the Ergodic Theorem, Proc. of National Aca. of Science,
U.S.A., 17(1931), 656-660.
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