SUBLINEAR FUNCTIONALS ERGODICITY AND FINITE INVARIANT MEASURES

By introducing a sublinear functional involving infinite matrices, we esta- blish its connection with ergodicity and measure preserving transformation. Further, we characterize the existence of a finite invariant measure by means of a condition in- volving the above sublinear functional.

I. INTRODUCTION AND DEFINITIONS.

Let
be the set of all real bounded sequence {x normed by lxl suplx n n>O n Linear functional on = are called Banach limit [I] satisfying the conditions, i) If there is a number for all Banach limits #, the sequence x {x is called n almost convergent and we write; F lim x s It is shown by Lorentz

k(i)
Let, A (a be a sequence of real or complex matrices for each i 0,1,2... n, {x is such that a 0, if any n,k,i, is a negative integer.The sequence n, n called A summable to s if (i) Xk s (1.2) lim k=O an,k n/ uniformly in i and in this case we write: Alim x s, or x s(A).
n n (i) In the case a 1/n+I (i < k < i+n) and 0 otherwise, (A) reduces to the n,k method (F).If A A a then we obtain the usual summability method (A).It is n,k' significant to note that there does not exist any regular method (A) equivalent to method (F) (See Lorentz [2] Theorem 11 and 12) In the case a a n,k u+l r= r,k then (A) reduces to the almost summability method introduced bv King [3].
The following characterization of regular matrices is due to Stieglitz [4J.The method (A) is called regular if and only if the following conditions hold: kZ__0 lan,(i) Ik for all n and i -> 0, ( and there exists an integer m such that (i) (i 4) sup kE__0 an,k The matrix For real X we wrte, X+ max (X,O), X max(-X,O).
The matrix is called almost positive, f ([)-O, unfoly in lim kO an,k (1.7) (1.8) (1.9) (1.10) Let (x, F, m) be a finite measure space and let, T; X X be a measurable transformation.
(This is assumed throughout).Let {I ,t} denote the set of linear functionals , such that #(x) !t(x).It is known (see Sucheston [5] Das and Misra [6]) that {I ,t} is the set of all Banach limits on and {l,t} is unique if and only if (x) -#(-x) and this hap- pens when i+n-I ki x k +a limit as n , uniformly in i Lorentz [2] calls all such sequences as almost convergent sequences.Let A be real and such that IIAII < .Then we define, R: Since, for all x I R is finite valued.It is easy to see that it is a sublinear functional on I.By Hahn-Banach theorem there exists a linear functional Let {I,R} be the set of all linear functional satisfying (1.13).It is easily and this happens if and only if (i) Xk a limit kO an,k as n , uniformly in i.
We now state a lemma.(b) -t(-x) -<-R(-x) <-R(x) _< t(x) if and only if A is regular, almost positive and translative.
(c) If x is almost convergent to s, then lim k0 a x k s uniformly in 2 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) ) T s ergodic and measure preserving.rgON 1. get (X, , ) be a fnite measure space and let la Then Changing the role of sx and x in (2.2) and (2.3) we obtain (a).When A is trans- lative (b) (i), (ii) follows from (2.2), and changing the role of sx and x in by b (i).Changing the role of Sx and x, we obtain R(x) R(Sx).So (c) follows.
i.e.We now show that q is an invariant measure and m q.
Since, A is almost positive for x k E I.
F be a countable sequence i of disjoint sets.Then q(il B i) [qn(il Bi)] [i qn(Bi)] ('.m is a measure) =i [qn(Bi)] (0 is continuous linear functional) So, q is countably additive and hence it is a measure. Next, This proves that q is an invariant measure.Site T is ergodic, the invariant sets are of measure 0 or I. Since m and q 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 (II) holds and hence proves (c) completely.
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 exis- tence of invariant measure by using linear functional e {I=,R} We now prove THEOREM 2. Let A be a real matrix such that IIAII < and let A be almost posi- tive and translative.Let (x, F, m) be a finite measure space and T be a measurable transformation.Then, the following condition are equivalent. (I) There exists an equivalent finite invariant measure. ( m(T-kB) 0 as n uniformly in i.Hence, -I q(T B) < q(B).
-I Changing the role of T B and B, we obtain -I q(B) q(T B) Hence, -! q(T B) q i.e. q is invariant under T [m(T-nB)] is unique.But {I,R} is unique if and only if R(x) =-R(-x)-- q(B) and this happens if and only if (i) lim k=Z0 an,k m(T-km) q(m) nuniformly in i.Now since, q(T-IB) q(B), B F and q m on F, we have m(T-IB) re(B), B E F so re(B) 0 => m(T-IB) =0 i.e. m is null-preserving Again (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 same as conservativeness of T.

SUFFICIENCY:
Let (i), (ii) and (iii) hold.Define (i) m(T-kB) q(B) lim k=EO an,k Then it can be proved as before that q ia an invariant measure.So only we have to prove q is equivalent to m.Since T is null preserving, -I re(B) 0 => m(T B) O.
[2] that a se- n quence {x is almost convergent with F-limit s, if and only if n i+nlim -n ki Xk s (I.I) n+ uniformly in i.
LEMMA I. Let x I then (a) lim x < R(x) !lim x n n if and only if A is real, regular and almost positive.

I
. Suppose that p is an invariant measure which is equivalent to m. Suppose that (II) fails to hold.Then  there exists a B e such that m(B) it follows that for all B E F 0 [m(T-nB) lim m(T-nB) But, since lira m(T-nB) > 0, it follows that n/ lim m(T-nB) O. Hence, there exists a sub sequence {x k} such that lim m(T-nkB) 0 k+ Since p is equivalent to m we obtain p(B) > 0 and lim p(T-nkB) 0. k+ Since p is invariant, we have p(r-nkB) p(B) Hence p(B)O.This is a contradiction and this proves the fact that (I) => (II).(II)=>(III)Let II hold Since, [m(T-nB)] < R [m(T-nB)] it follows that [m(T-nB)] > 0 R [m(T-nB)] > 0(III) (I) Suppose (III) holds and (I) fails.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 r 0 The measure m is called null invarian