CONVERGENCE OF THE AVERAGES AND FINITENESS OF ERGODIC POWER FUNCTIONS IN WEIGHTED Ll SPACES

PEDRO ORTEGA SALVADOR Let (X, .F, u) be a finite meásure space . Let T : X -. X be a measure preserving transformation and let A f denote the average of Tk f, k = 0, . . . . n. Given a real positive function v on X, we prove that {A f} converges in the a .e . sense for every f in L1 (v dli) if and only if inf ;>_p v(Tx) > 0 a.e ., and that the same condition is equivalent to the finiteness of a related ergodic power function Pf for every f in L 1 (v dp.) . We apply this result to characterize, being T null-preserving, the finite measures v for which the sequence {A n f} converges a .e . for every f E L1 (dv) and to prove that uniform boundedness of the averages in Ll is sufficient for finiteness a .e . of Pr .


. Introduction
Let (X, .F, p) be a finite measure space and let T : X -" X be a measure preserving transformation .For every measurable function f on X we consider the averages where T3 .f(x)= f(T-1 . x), the maximal operator and the power function In [7], Martín Reyes and A. de la Torre characterized the positive measurable functions v such that {Aj} converges a.e. for all f in L 1 (v dp.) as those functions that verify (1 .1)inf v(Tix) > 0 a.e.>o (see also [15] for a ratio theorem) .
Section 3 of this note is devoted to give a simpler proof for this result and to prove a similar theorem for P,.It is seen that condition (1 .1) is also valid for P,.The main tools we use are Nikishin's theorem and conditional expectations which solve the problems derived from the non-invertibility of the transformation .These technics have been recently used to solve the problem of the convergente of the averages for p > 1 (see [8]) .
As a previous result, we have to state the weak type (1,1) for P, .This question was solved in [17] by Yoshimoto .Our approach is different, but suitable for our purposes .It was also treated in [5] and [11], but under more restrictive conditions .
Finally, in section 4 we work with a null-preserving transformation T and characterize the finite measures v for which the sequence.of the ergodic averages {Anf} converges a.e. for every f E L 1 (dv) as those measures with the property: there exists a measure 7 equivalent to v such that for every f E L 1 (dv) .

7({x
In [13], Ryll-Nardzewski characterized the finite measures v for which the ergodic averages {A,,,f} converge a.e. to a L 1-function for every f E L1 (dv) as those measures that verify Hartman's condition : there exists a constant K such that for every set E.
Our result is different from the Ryll-Nardzevsky's one, because we allow the limit function not to be in L 1 (dv) .This situation is possible as Dowker's example shows (see [1] and, for a two-dimensional version, see [12]) and, therefore, our condition is strict1y weaker than Hartman's condition .
As a corollary, we prove that uniform boundedness of the averages is a sufficient condition for finiteness of P, for every f E L1 .This result is a L1 version of theorem 3.1 in [10].Other referentes about P, are [14] and [16] .
We will need two lemmas and several results about the operators P,., q,. and Q,., where qr is defined on functions on N, the set of the natural numbers, by gra (2 ) = and Qr on functions on X by 00 1/r Qrf(x) -~~If (Tkx) Ir(k + 1 ) _ -rl k-0 Lemma 1.Let k be a natural number .Then, there exists a countable family üi) For every i, there exists a natural number s(i) with 0 _< s(i) <_ k such that the sets {T -jBi : 0 < j < s(i)} are pairwise disjoint and such that if s(i) < k then T-I-s(i)A = A for every subset A of Bi .Consequently, for every subset A of Bi k=o 2 .Previous results where C(i) is the least integer bigger than or equal to (k+1)(1+s(i))-1 .
For the proof see lemma (2 .10) in [9] changing Th by T-h.Lemma 2. Let (X, .F, M) be a finite measure space and let {.Fn } be a de- creasing sequence of sub-o-algebras .Let .F~. = n, .Fn and denote by En che condicional expectation with respect to .F .If {fn} is an a. e. convergent se quence of functions such that Ifnl <_ C a.e .and f is che a.e .limit of fn then E,,,,f is the a. e. limit of Enfn .
This lemma follows from theorem 7.6 in [6] and the decreasing martingale theorem .
Theorem 1. qr is of weak type (1, 1) with respect to che counting measure on N .
Proof.The proof is the same as the one of theorem (3.8) in [10] with obvious changes derived from the facts that we are working in N and that lemma (3.2) (in [10]) is not necessary.Theorem 2. Q,. is of weak type (1,1) .
Proo£ It follows inmediately by theorem 2, the ergodic theorem and the well-known inequality PTf 5 CMf + QTf.
Remark .Note that theorems 2 and 3 do not peed finiteness of the measure space.

. Main result
Theorem 4. Let (X, .P, p) be a finite measure space.Let T : X ---> X be a measure preserving transformation.Let v be a positive measurable function on X .The following are equivalent : a) The sequence {Anf} converges a. e. for all f in L' (v dp) .b) Eko IAk+lf -Ak f l' < oo in the a. e. sense for all f in L' (v dp) .c) Eko(k+ 1) -TITkflT < oo in the a.e.sense for all f in Ll (vdp) .d) Mf < oo a. e. for all f in L 1 (vdp) .e) There exists a positive measurable function u such that f {x :mf(x)>al u dh <-.\ -l fx lf¡v dp for all A > 0 and all f in Ll (v dp) .
f) There exists a positive measurable function u such that supk> _o f{x :Akflx)>al udp <'\ -1 fx lf1vdp for all A > 0 and all f in L l (vdp) .
g) There exists a positive measurable function u such that f{ .:p f(x)>, \1 u dp <_ \l fx lf¡v dp for all A > 0 and all f in Ll (v dp) .h) There exists a positive measurable function u such that f{x :Qr f(x)>a} u di, < A-l fx lf1v dp for all A > 0 and all f in L' (v dp) .i) infi>ov(Tix) > 0 a.e .
Proof-Implications a) => d) and e) => f) are clear.d) implies e), b) implies g) and c) implies h) by Nikishin's theorem (see [2] pages 536-537 and [3]).Nikishin's theorem needs the continuity in measure of the operators M, PT and QT from Ll (v dM) to LI (dp) .This condition follows by theorem 1.1.1 in [4], page 10. f) =* i) We may assume u < 1 .Let k be a nonnegative integer.Let {Bi} be the sequence of sets associated to k by lemma 1 .Fix i and let A be a measurable subset of Bi .Let R = Uo<j<kT -'A = Uo<j<s(i)T-'A .It is clear that R is contained in {x : Ak(XA)(x) >-C(i)(k+ 1)Then f), lemma 1 and the fact that T is m.p.t .give k k The above inequality has been proved for a measurable subset A of Bi.Since X = U¡Bi, it is clear that the inequality is true for every measurable subset A of X and therefore if Ek is the conditional expectation with respect to the sub-Q-algebra T-k.F we have Ek (k + 1) -1 1: Tjul (x) <-Tkv(x) a.e.x E X .j=o Taking lim inf when k tends to infinity, Birkhoff's theorem and lemma 2 give Eu(x) < lim k inf Tkv(x) a.e.x E X, ---+oo where Eu is the conditional expectation of u with respect to the sub-u-algebra of the invariant sets.
Since Eu is positive a.e., we obtain infk>o v(Tkx) > 0 a.e. .g) => i) We may assume u <_ 1 .Let k be a natural number and {Bi } be the sequence of sets given by lemma 1. Fix i with s(i) > 0 and let A be contained in Bi .Let R = Uo<j<kT -'A = Uo<j<,,(i)T-' A. Let's see that R-A is contained in {x : P,(XA)(x) ?(1 + s(i))-i} .

R A A Since u < v we have
Recall that we have been working with s(i) > 0. But if s(i) = 0 the last inequality is trivial.
Now, the same argument used in the above implication gives i).h) => i) Let k, {Bi}, A and R as in f) =~> i) .It is easy to see that R is contained in {x : Q,(XA)(x) > (1+s(á))-1}.Then, the argument follows as in f) => i).
i) => a) The proof of this implication can be seen in [7] .We include it for this section to be selfcontained .
Finally, i) => b) and i) => c) by the same argument that the above but using theorems 3 and 2 respectively in place of Birkhoff's theorem.

Convergence of the averages and finiteness
of P,. in the general case Theorem 5 .Let (X, .P, v) be a finite measure space and let T : X -r X be a null-preserving transformation .The following statements are equivalent : a) There exists a measure y equivalent to v such that for every f E L1 (dv) .Let L be a Banach's limit (for instante see [6]) and define M, is well defined by (4.1) .h is an invariant measure and it is absolutely continuous with respect to v. Let v be the Radon-Nikodym derivative dw/dv, D = {x : v(x) :,A 0} and Y = nn>OT-nD.It is clear that p(X -Y) = 0 and TI Y applies Y in Y. Therefore we have that vly is equivalent to the invariant measure hl y .Then it follows by theorem 4 that the averages {Akf} converge and Mf, Q, f and Pf are finite a.e.(v) in Y for every f E L1(dv) .
To prove the a.e.(v) convergente of {Akf} and the finiteness of Mf, Q,f and Pf on X -Y we shall first state that for almost all x (v) in X there exists n such that Tnx E Y.If this property is not true, then thére exists B with v (B) > 0 such that for every i, B is contained in T-i (X -Y) .Then for every k -y (B) :5 (k+ 1) -1 1: -y(T-'(X -Y)) = f AkXX-Y d y i=o x and the properties of Banach's limits give y(B) :5 L ({Ix AnXX-Y dy}) .=p(X -Y) = 0, which goes against v(B) > 0 since ,y and v are equivalent .Let x be in X -Y and let n be an integer verifying Tnx E Y. Let k _> n.Then Since T'x E Y and T applies Y in Y, when k tends to infinity we obtain finite limits .Therefore, we have proved that {Akf (x)} converges a.e. and that Mf (x) and QT .f(x) are finite a.e. for every f in LI (dv).Then, since P,f < CMf + QTf we obtain the finiteness of P,.. Corollary.Let (X, .F, v) be a finite measure space and let T : X --> X be a null-preserving transformation.If supk>o ~JAk il I < oo then {Akf} converges a. e. and Mf, Q,. f and P,.f are inite a. e_ for every f E Ll (dv) .