1.Introduction

Bishop [4] (p.233) put forward the following problem connected with the question of finding a constructive interpretation of ergodic theorems.

He concluded that Birkhoff's Ergodic Theorem is non-constructive.

Bishop proved, using so-called upcrossings, a version of the Chacon-Ornstein Theorem. This theorem is a generalization of Dunford and Schwartz's version of the Pointwise Ergodic Theorem, a result which extends Birkhoff's Ergodic Theorem. In Bishop's ergodic theorem a limit is proved to exists in a constructively very weak sense. Bishop's result is a so-called equal-hypothesis substitute for the ergodic theorem. Bishop considered finding an equal-conclusion substitute to be ‘an important open problem', see [2] (p55). His student Nuber [8][9] found such an equal-conclusion substitute for Birkhoff's Ergodic Theorem. His proof uses measure theoretic techniques and seems to work only for measure-preserving transformations. We use functional analytic techniques to give necessary and sufficient conditions for von Neumann's Mean Ergodic Theorem and the Dunford and Schwartz version of the Pointwise Ergodic Theorem to hold.

In the context of Bishop's constructive mathematics [3] we prove that for the Mean Ergodic Theorem to hold it is sufficient that the projection on the space of invariant functions exists. Conversely, from the convergence of the sequence in the conclusion of that theorem we obtain the projection. We also show that the Mean Ergodic Theorem is sufficient to prove the Dunford and Schwartz version of the Pointwise Ergodic Theorem, and again a converse is also true. The aim of this paper is to make these claims rigorous, see Theorem 16.

The paper is organized as follows. We first prove a Mean Ergodic Theorem. Then the Maximal Ergodic Theorem and Banach's Principle are proved and used to prove the Pointwise Ergodic Theorem. Our presentation loosely follows that of Krengel [7] (p.65,p.159) and Dunford and Schwartz [5]. We use [3] as a general reference for constructive mathematics.

2.The mean Ergodic Theorem

The following definitions will be used throughout this chapter.

Let T be an operator on a Banach space X . Define the sum S _n : = ∑

When X is a vector space and Y,Z are subspaces such that for every x in X there exist unique y in Y and z in Z such that x = y + z, we write X = Y⊕Z.

Theorem 1. Let T be a contraction on a Banach space X. The sequence (A_n)_n∈N converges if and only if X = M⊕N, in which case lim_n→∞A_n = P_M, where P_M denotes the projection on M parallel to N.

Proof. First suppose that X = M⊕N . Let h = h _M + h _N , where h _M ∈ M and h _N ∈ N . We claim that A _n h converges to h _M . First consider f = g - T g for some g ∈ X ; then the sequence A _n f =

Let f∈X; then there exist f_M in M and f_N in N such that f = f_M + f_N. Consequently, A_nf = f_M + A_nf_N which converges to f_M when n tends to ∞.

Now suppose that the sequence (A_n)_n∈N converges to an operator P. The equalities T P = P = P T follow easily from the definition of the sequence (A_n)_n∈N. Consequently, P = 0 on N and

If z∈M∩N and ε>0, then there exists u∈X such that ‖z - (u - T u)‖<ε. Hence for all n∈N, ‖A_n(z - (u - T u))‖<ε. Because A_n(u - T u) converges to 0 and for all n∈N, A_nz = z we see that ‖z‖≤ε. Consequently, M∩N = {0}.

To see that (I - P)x∈N observe that

for all n in N. Define

Let H be a Hilbert space and let T be an operator on H . Let x ∈ H . There exists a vector x ^∗ such that langle T y , x rangle = langle x , x ^∗ rangle if and only if the functional y ↦⟨ T y , x ⟩ is normable. This follows from the Riesz representation theorem [ 3 ] (p.419). If such a vector exists we will denote it by T ^∗ x even if the adjoint is not totally defined

Theorem 2. [Mean Ergodic Theorem]Let T be a contraction on a Hilbert space H. Then the sequence (A_n)_n∈N converges if and only if N is located; in this case the sequence (A_n)_n∈N converges to the orthogonal projection P_M on M.

Proof. We first prove that M and N are orthogonal. Suppose that x∈M, i.e. T x = x. We claim that the map y↦⟨T y,x⟩ is normable. Since |langleT y,xrangle|≤‖x‖‖y‖ whenever y∈H, ‖x‖ is an upper bound on the norm. On the other hand this upper bound is attained at x. It follows that x∈DomT^∗ and ‖T^∗x‖ = ‖x‖. Now,

Consequently, T^∗x = x and so

for all y in H. We see that M and N are orthogonal.

Suppose that the sequence (A_n)_n∈N converges. Theorem 1 shows that H = M⊕N. Because M and N are also orthogonal, M and N are located.

Let (X,μ) be a measure space. A measure-preserving transformation of X is a partial function from a full set to a full set such that for all integrable sets A, τ(A) is integrable and μ(τ(A)) = μ(A). If τ is a measure-preserving transformation, then T_τf: = f∘τ is a contraction on L₂. This shows that our result generalizes the possibly more familiar formulation of the Theorem.

Bishop and Bridges [3] (problem 46, p.395) give the following version of the Mean Ergodic Theorem. Let T be a unitary operator on a Hilbert space H; then for all x∈H the sequence (A_nx)_n∈N converges if and only if the sequence (‖A_nx‖)_n∈N converges.

3.Maximal Ergodic Theorems

Let ( X ,μ) be a measure space. An operator T on L ₁ (μ) is an L₁-L_∞ contraction if T is a contraction on L ₁ that contracts the L _∞ -norm on L ₁ ∩ L _∞ — that is, ‖ f ‖ ₁ ≤‖ T f ‖ ₁ and for all real numbers m , | T f |≤ m whenever f ∈ L ₁ and | f |≤ m . An operator T on an ordered vector space is positive

Let T be a positive L₁-L_∞ contraction. Define the operator M_n by

for all n in N. Garcia's proof [7] (p.8) of the following Theorem 3 is constructive. Note however that we do not make any claims about M_∞. This operator is defined classically as sup_k∈NA_k, but constructively M_∞f may not be a measurable function for all f in L₁, i.e. we may not be able to find simple functions approximating M_∞f.

In the constructive theory of measure spaces it is not always possible to compute the measure of the set [f<α]≔{x:f(x)<α}. However, we can compute the measure for all but countably many α, such α are called admissible, see [3] for details.

Theorem 3. [Hopf's Maximal Ergodic Theorem] Let T be a positive contraction on L₁(μ). Let n be a natural number. If α≥0 is admissible for M_nf, then

Corollary 4. [Wiener [7] (p.51)]Let T be a positive L₁-L_∞ contraction. Let f∈L₁, n∈N and α>0 be admissible for M_nf. Then for all n,

The following theorem is sometimes called the little Riesz theorem. The proof we give here is an adaptation of [7] (Lemma 1.7.4).

Proposition 5. If T is a positive L₁-L_∞ contraction, then T can be uniquely extended to an L_p contraction for all p≥1.

Proof. Because for all p≥1, L₁∩L_∞ is a dense subset of L_p, it is enough to prove that ‖T f‖_p≤‖f‖_p for all f∈L₁∩L_∞. To achieve this goal we will prove that

for all positive simple functions f and p>1.

Assume for a moment that we have done so and let f be a positive simple function and p>1. Then (T f)^p≤T(f^p), so ‖(T f)^p‖₁≤‖T(f^p)‖₁≤‖f^p‖₁ and thus ‖T f‖_p≤‖f‖_p. Observe that this inequality trivially holds for p = 1. Consequently, ‖T f‖_p>‖f‖_p is impossible, i.e. ‖T f‖_p≤‖f‖_p holds for all p≥1, even if we are unable to decide p = 1 or p>1. It follows that for all p≥1, T is a positive L_p-contraction on the positive simple functions, and consequently also on the simple functions. The simple functions are dense and T is a contraction, so the operator T restricted to the simple functions can be uniquely extended to L_p. This extension agrees with T on L₁∩L_p for all p≥1.

We will now prove ( 1 ) for a positive simple function f and p >1. We will assume that Y : = [ f >0] is integrable. Since the simple functions f with this property are also dense in L

It follows that for all real numbers c,d>0:

Consequently,

Let F be a full set on which (2) holds for rational c and d and thus by continuity for all c,d>0. Compute M, m∈R ⁺ such that M≥f≥mχ_Y>0. Then f≤m^{1 - p}f^p. Let F'⊂F be a full set such that for all x∈F'

Fix x ∈ F '.

If T ( f ^p )( x ) = 0, then T ( f )( x )≤ m ^{1 - p} T ( f ^p )( x ) = 0 = T ( f ^p )

From this point onwards we will assume that a positive L₁-L_∞ contraction is extended to L_p for all p≥1.

Theorem 6. [Dominated Ergodic Theorem] Let T be a positive L₁-L_∞ contraction. Then for all n∈N, p>1 and f∈L_p: ‖M_nf‖_p≤

Proof. Fix n∈N, p>1, and f∈L

A function f:R→R is convex if

whenever λ∈[0,1] and x,y∈R.

Theorem 7. A (total) convex function from R to R has an non-decreasing derivative which is defined in all but countably many points.

Proof. To prove this we will use Bishop's profile theorem, a constructive substitute for the classical lemma stating that every non-decreasing real function is continuous in all but countably many points. Like Bishop we define the set E to be the set of piecewise linear functions

whenever x<y. These functions are 0 when z⩽x and 1 when x⩾y. We define

Lemma 8. [Jensen's inequality] Let (X,μ) be a finite measure space and ϕ:R→R be a convex function. If f,ϕ∘f are in L₁, then

Proof. Let x be in the domain of ϕ'. Then for all real numbers y,

and hence

for all t∈Domf. By integrating we obtain

The L_p Ergodic Theorem can be proved classically for finite measure spaces using the pointwise ergodic theorem [14]. Another proof [7] (Thm. 2.1.2) uses a non-constructive compactness argument. Our proof works not only for finite measure spaces, but also for σ-finite measure spaces. We make some preparations.

Let μ be a finite measure. If 1≤p<q and f∈L_q, then |f|^p is measurable and bounded by |f|^q∨1, hence |f|^p is integrable and by taking ϕ(x): = x

We see that L_q⊂L_p.

Let μ be σ-finite. If p≥q≥1 and f≤M, then

Consequently,

Theorem 9. [L^p Ergodic Theorem] Let p,q≥1 and (X,μ) be a finite measure space or let p>1,q≥1 and (X,μ) a σ-finite measure space. Let T be a positive L₁-L_∞ contraction. If the sequence (A_n)_n∈N converges in L_q, then it converges in L_p.

Proof. Let f ∈ L _p and choose a simple g such that ‖ f - g ‖ _p <

If we show that ‖A_ng - A_mg‖_p→0, when m,n→∞, then (A_nf)_n∈N is a Cauchy sequence in L_p. That ‖A_ng - A_mg‖_p→0 follows from the fact that the sequence (A_ng)_n∈N converges in L_q and the inequalities (4) and (5) for the case μ is finite or μ is σ-finite and p≥q. The case μ is σ-finite and 1<p≤q is more difficult. We will now proceed to consider this case. If p≥1, h∈L_p, n,m,l,r∈N and m = l n + r, then

It follows that the sequence (‖A_nh‖_p)_n∈N is essentially decreasing, that is for each n∈N and each ε>0, there exists N∈N such that ‖A_mh‖_p≤‖A_nh‖_p + ε for all m≥N.

Let g∈L_q∩L_p. Let g^~ be the limit of A_ng in L_q. The function g^~ is in L_p, because T contracts the L_p-norm. We will prove that lim_n→∞A_ng = g^~ in L_p. By looking at g - g^~ we may assume that g^~ = 0.

Because the sequence (‖ A _n g ‖ _p ) _n∈N is essentially decreasing, it is enough to find for each η>0 and k ∈ N , an n > k such that ‖ A _n g ‖

Now

In the former case we are done, so we may assume that ‖A_n1g‖

We continue in this way until we find N∈N such that ‖A_nNg‖

For all N≤K, ‖A_nNg‖

It follows from the Dominated Ergodic Theorem that for all N≤K,

If μ is σ-finite, it is not true in general that if for some p>1, the sequence A_n converges to 0 in L_p, then the sequence A_n converges in L₁. To see this let τ(x): = x + 1 on R with Lebesgue measure, T = T_τ and f = χ_[0,1]. For all p>1, A_nf converges to 0 in L_p, but the sequence does not converge in L₁.

4.The Pointwise Ergodic Theorem

The following principle, called Banach's principle, will be used as follows: we prove that a sequence of operators converges almost everywhere (a.e.) on a dense set and then conclude that the sequence converges a.e. on the whole space.

The proof of the following theorem would be easier if we could prove constructively that M_∞ is measurable.

Theorem 10. [Banach's Principle] Let (Y,μ) be a measure space. Let (T_n)_n∈N be a sequence of linear operators from a Banach space X to the space μ-measurable real functions. Define for each n∈N, an operator M^~_n by M^~_nx: = sup_k≤n|T_kx| for all x∈X. Suppose that there exists a positive decreasing function C from R to R such that lim_α→∞C(α) = 0 and for all x and n:

whenever α‖x‖ is admissible for M^~_nx. Then the set of elements x∈X for which the sequence (T_nx)_n∈N converges a.e. is closed.

Proof. Suppose that a sequence z_n converges to z in X in norm and that there exists a sequence (f_n)_n∈N of measurable functions such that for all m∈N, T_mz_n→f_n a.e. We will prove that the sequence (T_mz)_m∈N converges a.e.

For natural numbers a and b, ω∈Y and x∈X put

Then

Let ε>0. Choose a natural numbers n such that C (

For all b≥a≥k,

Construct ascending sequences (n_m)_m∈N and (k_m)_m∈N of natural numbers such that

and

for all n≥k_m. Put

The set E is integrable and μ(E)≤∑

Bishop [4] (p.230) gave a constructive proof of Lebesgue's Differentiation Theorem. Using Banach's principle one can give another constructive proof, see [10] (p.101).

When q≥1 and T is a positive L₁-L_∞ contraction, we define M_q = {f∈L_q:T f = f} and N_q = cl{T f - f:f∈L_q}.

Theorem 11. [Pointwise Ergodic Theorem] Let μ be a σ-finite measure. Let T be a positive L₁-L_∞ contraction. If L_q = M_q⊕N_q, for some q≥1, then for all p≥1, the sequence (A_nf)_n∈N converges a.e. for all f∈L_p.

Proof. Suppose that L_q = M_q⊕N_q, for some q≥1. First let p>1. Theorem 9 and Theorem 1 show that L_p = M_p⊕N_p. Suppose that f = g + T h - h, where g∈L_p, T g = g and h∈L_∞∩L_p. The set of these f is dense in L_p. Because T contracts the L^∞-norm, lim_n→∞A_nf = g a.e. For all admissible α>0,

Consequently, αμ[M_nf≥α]

For all f∈L_p, M^~_nf≤M^~_n|f|, so inequality (6) holds for all f∈L_p. Banach's principle shows that the sequence (A_nf)_n∈N converges a.e. for all f in L_p.

The two-step argument above is used because in general we do not have convergence in the L₁-norm. The space L₁ has an awkward geometrical structure: it is not uniformly convex.

Theorem 12. Let p≥1. If the sequence (A_nf)_n∈N converges a.e. for all f in L_p, then N = cl{x - T x:x∈L₂} is located in L₂.

Proof. Let f be an element of L₂. Without loss of generality we may assume that A_nf→0 a.e. We claim that the sequence (A_nf)_n∈N converges weakly to 0. We may assume that f is a simple function. Since the sequence (A_nf)_n∈N is bounded, the sequence∫(A_nf)g converges to 0 whenever g is a simple function. The inner product is continuous on L₂, so ⟨A_nf,g⟩→0 for all f,g∈L₂ — that is, (A_nf)_n∈N converges weakly to 0.

Define B _n ≔(

Lemma 13. [6]Let E be a uniformly convex and uniformly smooth Banach space and C a bounded convex subset of E. Then C is located if and only if sup{f(z) : z ∈ C } exists for each normable linear functional f on E .

Lemma 14. [3][6]For p>1, the space L_p is uniformly convex and uniformly smooth and each normable functional may be represented by a functional f↦∫f g, for some g in L_q, where

Lemma 15. If N is located in some L_p, then both M and N are located in all L_p, where p>1.

Proof. Locatedness is a property of closed sets, so we may restrict to a set which is dense in all the L_p-spaces, for instance the bounded L₁ functions or the simple functions. Moreover, an inhabited set A is located if we can compute the distance ρ(x,A) for each x. Let a be an element of A. Then ρ(x,A) = ρ(x,A∩B(x,ρ(x,a) + 1))), where B(x,r) denotes the ball around x with radius r. Consequently, when considering located sets we may restrict to its bounded subsets.

Theorem 16. Let μ be a σ-finite measure and T a positive L₁-L_∞ contraction. Denote by A_n the average

1. The set N = cl{x - T x:x∈L₂} is located in L₂;
2. The sequence (A_n)_n∈N converges in L₂;
3. For all p>1, the sequence (A_n)_n∈N converges in L_p;
4. There is p>1 such that the sequence (A_n)_n∈N converges in L_p;
5. For all p≥1 and f∈L_p, the sequence (A_nf)_n∈N converges a.e.;
6. There is p≥1 such that for all f∈L_p, the sequence (A_nf)_n∈N converges a.e.

For a finite measure we may replace p>1 by p≥1.

Some of these results have circulated in preprints for some time. They then appeared in my thesis [11]. The results have been used in [1] in the context of reverse mathematics.

I would like to thank Wim Veldman for his advice during the PhD-project. Finally, I would like to thank the referee for comments that helped to improve the presentation of the paper.

Bibliography