<\body> <\hide-preamble> >> >> >> >> ||<\author-address> Radboud University Nijmegen, the Netherlands |<\author-note> The author was partially supported by the Netherlands Organization for Scientific Research (NWO). The current paper is an updated version of including the corrections in. >> \; <\abstract> Let be a positive >->> contraction. We prove that the following statements are equivalent in constructive mathematics. <\enumerate> The projection in > on the space :={x-T*x:x\L}> exists; The sequence )>> Cesąro-converges in the > norm; The sequence )>> Cesąro-converges almost everywhere. Thus, we find necessary and sufficient conditions for the Mean Ergodic Theorem and the Dunford-Schwartz Pointwise Ergodic Theorem. Bishop (p.233) put forward the following problem connected with the question of finding a constructive interpretation of ergodic theorems. as the union of two equal tanks of fluid, and as a motion of the fluid, which is supposed to keep the fluid confined to the tank in which it has been placed. Imagine that there may be a small leak, which would in fact allow the fluid in the two tanks to mix, but that we are not able to decide whether a leak actually exists. Since the leak if it exists, is small, there will be little mixing between the tanks after unit time (that is, under the transformation ), but after a long time (that is, under the transformation > for some large ) the mixing may be substantial.>> 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 substitute for the ergodic theorem. Bishop considered finding an - substitute to be `an important open problem', see (p55). His student Nuber found such an 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 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. 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 (p.65,p.159) and Dunford and Schwartz. We use as a general reference for constructive mathematics. > The following definitions will be used throughout this chapter. Let be an operator on a Banach space . Define the sum :=T> and the average :=S>. Define the subspaces :={f\:T*f=f}> and :={x-T*x:x\}>. An operator is called a > if T*x\<\|\|\>\\<\|\|\>x\<\|\|\>> whenever >. When is a vector space and are subspaces such that for every in there exist unique in and in such that , we write Z>. <\theorem> >Let be a contraction on a Banach space . The sequence )>> converges if and only if =\,> in which case \>A=P>,> where >> denotes the projection on parallel to . <\proof> First suppose that =\>>. Let >+h>>, where >\\> and >\\>. We claim that h> converges to >>. First consider for some \>; then the sequence f=(g-Tg)> converges to 0. When >, then there exists > such that f-(g-T*g)\<\|\|\>\\>, so for all ,> A(f-(g-T*g))\<\|\|\>\\>. Hence for large , Af\<\|\|\>\2\>. Consequently, the sequence f> converges to 0 whenever > and, using the notation above, h> converges to >>. Let >; then there exist >> in and >> in such that >+f>>. Consequently, f=f>+Af>> which converges to >> when tends to >. Now suppose that the sequence )>> converges to an operator . The equalities follow easily from the definition of the sequence )>>. Consequently, on and <\equation*> P=lim\>APlim\>P=P. If \> and \0>, then there exists > such that z-(u-T*u)\<\|\|\>\\>. Hence for all ,> A(z-(u-T*u))\<\|\|\>\\>. Because (u-T*u)> converges to 0 and for all ,> z=z> we see that z\<\|\|\>\\>. Consequently, \={0}>. To see that > observe that <\equation*> (I-T)(I+T+\+T)=I-A for all in >. Define\ <\equation*> :=(I+T+\+T)x>; then \(I-P)x>. Consequently, =\>. Let be a Hilbert space and let be an operator on .> \ Let .> There exists a vector >> such that T*y,x=x,x>> if and only if the functional \T*y,x\> is normable. This follows from the Riesz representation theorem (p.419). If such a vector exists we will denote it by >x> even if the adjoint is not totally defined<\footnote> Classically, the adjoint of an operator is always totally defined. Constructively this is not the case. . <\theorem> Let be a contraction on a Hilbert space . Then the sequence )>> converges if and only if is located; in this case the sequence )>> converges to the orthogonal projection >> on .> <\proof> We first prove that and are orthogonal. Suppose that >, i.e. . We claim that the map \T*y,x\> is normable. Since T*y,x\|\\<\|\|\>x\<\|\|\>\<\|\|\>y\<\|\|\>> whenever \>, x\<\|\|\>> is an upper bound on the norm. On the other hand this upper bound is attained at . It follows that T>> and T>x\<\|\|\>=\<\|\|\>x\<\|\|\>>. Now, <\eqnarray*> T>x-x\<\|\|\>>||T>x-x,T>x-x\>>|||T>x\<\|\|\>+\<\|\|\>x\<\|\|\>-\x,T>x\-\T>x,x\>>|||T>x\<\|\|\>+\<\|\|\>x\<\|\|\>-\T*x,x\-\x,T*x\>>||>|T>x\<\|\|\>+\<\|\|\>x\<\|\|\>-2\x,x\=\<\|\|\>T>x\<\|\|\>-\<\|\|\>x\<\|\|\>=0.>>>> Consequently, >x=x> and so <\equation*> \x,(I-T)y\=\(I-T)>x,y\=0 for all in >. We see that and are orthogonal. Suppose that the sequence )>> converges. Theorem shows that =\>. Because and are also orthogonal, and are located. Conversely, suppose that is located. We know that \>>. We will prove that >\>. Let >>. Then (I-T)y,x\=0> whenever >. It follows that y,x\=\T*y,x\>, i.e. >x>. By a similar argument as above we see that . We conclude that =>>, so by Theorem the sequence )>> converges. Let )> be a measure space. A of is a partial function from a full set to a full set such that for all integrable sets , (A)> is integrable and (\(A))=\(A)>. If > is a measure-preserving transformation, then >f:=f\\> is a contraction on >. This shows that our result generalizes the possibly more familiar formulation of the Theorem. Bishop and Bridges (problem46, p.395) give the following version of the Mean Ergodic Theorem. Let be a unitary operator on a Hilbert space ; then for all H> the sequence x)>> converges if and only if the sequence Ax\<\|\|\>)>> converges. Let )> be a measure space. An operator on (\)> is an >->> contraction> if is a contraction on > that contracts the >>-norm on \L>> that is, f\<\|\|\>\\<\|\|\>T*f\<\|\|\>> and for all real numbers , m> whenever L> and m>. An operator on an ordered vector space is ><\footnote> When this order-theoretic definition differs from the definition of a positive operator on a Hilbert space. if \ 0> whenever 0>. When > is a measure-preserving transformation, then >> is a positive >->> contraction. Let be a positive >->>> contraction. Define the operator > by\ <\equation*> f:=supn>Af> for all in . Garcia's proof (p.8) of the following Theorem is constructive. Note however that we do not make any claims about >>. This operator is defined classically as >A>, but constructively >f> may not be a measurable function for all in >, i.e.we may not be able to find simple functions approximating >f>. In the constructive theory of measure spaces it is not always possible to compute the measure of the set \]\{x:f(x)\\}>. However, we can compute the measure for all but countably many >, such > are called admissible, see for details. <\theorem> Let be a positive contraction on (\)>. Let be a natural number. If \0> is admissible for f>, then <\equation*> f\\]>f\0. <\corollary> (p.51)]> alpha>Let be a positive >->> contraction. Let L>, > and \0> be admissible for f>. Then for all , <\equation*> \[Mf\\]\>f\\]>f. The following theorem is sometimes called the little Riesz theorem. The proof we give here is an adaptation of (Lemma 1.7.4). <\proposition> If is a positive >->> contraction, then can be uniquely extended to an > contraction for all 1>. <\proof> Because for all 1,> \L>> is a dense subset of ,> it is enough to prove that \ T*f\<\|\|\>\\<\|\|\>f\<\|\|\>> for all L\L>>. To achieve this goal we will prove that\ <\equation> T*f\T(f)>T(fp)> for all positive simple functions and 1>. Assume for a moment that we have done so and let be a positive simple function and 1>. Then \T(f)>, so (T*f)\<\|\|\>\\<\|\|\>T(f)\<\|\|\>\\<\|\|\>f\<\|\|\>> and thus T*f\<\|\|\>\\<\|\|\>f\<\|\|\>>. Observe that this inequality trivially holds for . Consequently, T*f\<\|\|\>\\<\|\|\>f\<\|\|\>> is impossible, i.e.T*f\<\|\|\>\\<\|\|\>f\<\|\|\>> holds for all 1>, even if we are unable to decide or 1>. It follows that for all 1,> is a positive >-contraction on the positive simple functions, and consequently also on the simple functions. The simple functions are dense and is a contraction, so the operator restricted to the simple functions can be uniquely extended to >. This extension agrees with on \L> for all 1.> We will now prove(T(fp)>) for a positive simple function and 1>. We will assume that 0]> is integrable. Since the simple functions with this property are also dense in >, we do not loose generality. We note that .> We define )> as usual. For all real numbers 0>: <\equation*> a*b\|p>+|q>. It follows that for all real numbers 0>: <\equation*> \|d>\|cp>+|dq>. Consequently, <\equation> T(f\\)\T(f)p>+T(\)q>.. Let be a full set on which() holds for rational and and thus by continuity for all 0>. Compute > such that f\m\\0>. Then mf.> Let \F> be a full set such that for all F> <\equation*> M>,m(T*f)(x)>, M(T\)(x)> and )>(x)\1>. Fix F>.If )(x)=0,> then mT(f)(x)=0=T(f)>(x)>.If )(x)=0,> then M*T(\)(x)=0\T(f)>(x)>.If )(x)\0> and )(x)\0>, then we define )>(x)> and )>(x)>. The right hand side of () equals +)=c*d>. Because > we obtain: <\equation> T(f)(x)\T(f)>T(\)>(x)\T(f)>(x).Tfp> We conclude that in any case T(f)>(x)> is impossible. It follows that (Tfp>) holds for all F>. Consequently, (T(fp)>) holds and we have thus completed the proof. From this point onwards we will assume that a positive >->> contraction is extended to > for all 1>. <\theorem> Let be a positive >->> contraction. Then for all >, 1> and L>: Mf\<\|\|\>\\<\|\|\>f\<\|\|\>>. <\proof> Fix ,> 1,> and L.> <\eqnarray*> Mfd\>||f(s)>p\d\d\(s)>>|||>\\f\\]>(s)d\d\(s)>>||>>|>\\[Mf\\]d\>>|||>>|>\f\\]>f*d\d\>>|||>\\f\\]>(s)f(s)d\(s)d\>>||>>|f(s)>\\f\\]>(s)d\d\(s)>>|||f(s)f(s)>\d\d\(s)>>|||fMfd\>>||>|\<\|\|\>f\<\|\|\>\<\|\|\>Mf\<\|\|\>.>>>> This last inequality follows from Hölder's inequality and the fact that f\Af>, so that f\L>. For all L,> f\M\|f\|> and thus Mf\<\|\|\>\\<\|\|\>M\|f\|\<\|\|\>\\<\|\|\>f\<\|\|\>.> A function \> is if\ <\equation*> f(\x+(1-\)y)\\f(x)+(1-\)f(y), whenever \[0,1]> and >. <\theorem> A (total) convex function from > to > 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 > to be the set of piecewise linear functions <\equation*> h(z)\ whenever y>. These functions are 0 when x> and 1 when y>. We define\ <\equation*> \(h)\. Then > is increasing<\footnote> This is most easily seen by a geometric argument considering the graph of a convex function. on the set > with the order inherited from the the functions from > to > that is, ,\)> is a profile. The profile theorem ensures that all but countably many points are smooth. In the present case this means that the function is differentiable at these points.\ <\lemma> > Let )> be a finite measure space and :\> be a convex function. If \f> are in >, then <\equation*> \(X)>f\(X)\\\f. <\proof> Let be in the domain of >. Then for all real numbers ,\ <\equation*> \(y)\\(x)(y-x)+\(x), and hence\ <\equation*> (f(t))\\(x)(f(t)-x)+\(x)> for all f>. By integrating we obtain <\equation*> \\f\\(x)f-x\(X)+\(x)\(X). If we take a sequence )>> in \> tending to (X)>f>, we obtain the inequality above. The > Ergodic Theorem can be proved classically for finite measure spaces using the pointwise ergodic theorem. Another proof (Thm.2.1.2) uses a non-constructive compactness argument. Our proof works not only for finite measure spaces, but also for >-finite measure spaces. We make some preparations. Let > be a finite measure. If p\q> and L>, then > is measurable and bounded by \1>, hence > is integrable and by taking (x):=x>> in Jensen's inequality we obtain <\equation> \<\|\|\>f\<\|\|\>\\<\|\|\>f\<\|\|\>\(X)->. We see that \L>. Let > be >-finite. If q\1> and M>, then <\equation*> f\<\|\|\>=\|f\|=\|f\|\|f\|\M\<\|\|\>f\<\|\|\>>. \ Consequently, <\equation> \<\|\|\>f\<\|\|\>\M>\<\|\|\>f\<\|\|\>>. <\theorem> > Ergodic Theorem]> Let 1> and )> be a finite measure space or let 1,q\1> and )> a >-finite measure space. Let be a positive >->> contraction. If the sequence )>> converges in >, then it converges in >. <\proof> Let L> and choose a simple such that f-g\<\|\|\>\|4>>. Let be a bound for . For all >: <\eqnarray*> Af-Af\<\|\|\>>|>|Af-Ag\<\|\|\>+\<\|\|\>Ag-Ag\<\|\|\>+\<\|\|\>Ag-Af\<\|\|\>>>||>||4>+\<\|\|\>Ag-Ag\<\|\|\>+|4>.>>>> If we show that Ag-Ag\<\|\|\>\0>, when \,> then f)>> is a Cauchy sequence in >. That Ag-Ag\<\|\|\>\0> follows from the fact that the sequence g)>> converges in > and the inequalities() and() for the case > is finite or > is >-finite and q>. The case > is >-finite and p\q> is more difficult. We will now proceed to consider this case. If 1,> L,> > and , then <\eqnarray*> Ah\<\|\|\>-\<\|\|\>Ah\<\|\|\>>|>|\<\|\|\>(I+T+\+T)Ah\<\|\|\>-\<\|\|\>Ah\<\|\|\>>>|||\<\|\|\>T(I+\+T)h\<\|\|\>>>||>|\<\|\|\>h\<\|\|\>\\<\|\|\>h\<\|\|\>.>>>> It follows that the sequence Ah\<\|\|\>)>> is , that is for each > and each \0>, there exists > such that Ah\<\|\|\>\\<\|\|\>Ah\<\|\|\>+\> for all N>. Let L\L.> Let > be the limit of g> in >. The function > is in >, because contracts the >-norm. We will prove that \>Ag=> in >. By looking at > we may assume that =0>. Because the sequence Ag\<\|\|\>)>> is essentially decreasing, it is enough to find for each \0> and >, an k> such that Ag\<\|\|\>\\>. We will now proceed to find such . Let =|4>>. Take an integrable set > such that g\>\<\|\|\>\\>. We recall that g\0> in .> We can thus compute > such thatA>g\<\|\|\>\\\(B)-1>>. By()\ <\equation*> (A>g)\>\<\|\|\>\\<\|\|\>(A>g)\>\<\|\|\>\(B)>\\>>. Now <\equation*> A>g\<\|\|\>\\>orA>g\<\|\|\>\\-\>. In the former case we are done, so we may assume that A>g\<\|\|\>\\-\> and thus (A>g)\>\<\|\|\>\\-2\>. We choose \B> such that (A>g)\-B>\<\|\|\>\\-3\>. Compute \n> such that A>g\<\|\|\>\\\(B)-1>>. Then A>g\-B>\<\|\|\>\\>. We continue in this way until we find > such that A>g\<\|\|\>\\>. That such an exists we see as follows. Choose > such that <\equation*> (()-1)\<\|\|\>g\<\|\|\>+\\K(\-3\). For all K,> A>g\<\|\|\>\\-\> or A>g\<\|\|\>\\.> Suppose that for all K,> A>g\<\|\|\>\\-\.> Define =\> and =0>. Remark that for all ,> <\eqnarray*> \|g\|>|>|M\|g\|\-B>>>||>|A>\|g\|\-B>>>||>|A>g\-B>.>>>> It follows from the Dominated Ergodic Theorem that for all K,> <\eqnarray*> \<\|\|\>g\<\|\|\>>|>|M\|g\|\<\|\|\>>>||>|(A>g)\-B>\<\|\|\>>>|||\<\|\|\>(A>g)\-B>\<\|\|\>>>||>|-3\)+\<\|\|\>g\<\|\|\>-\.>>>> It follows that there exists K> such that A>g\<\|\|\>\\>. Consequently, g\> in .> Finally, we observe that even if we do not know whether q> or q> we can show that the sequence > converges in >. To see this we observe that either q+\> or q>, the latter case has been treated before. In the former case, the sequence also converges in \>> as we proved before, and we can thus proceed as above. If > is >-finite, it is not true in general that if for some 1>, the sequence > converges to 0 in >, then the sequence > converges in >. To see this let (x):=x+1> on > with Lebesgue measure, >> and >. For all 1>, f> converges to 0 in >, but the sequence does not converge in >. > 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 >> is measurable. <\theorem> Let )> be a measure space. Let )>> be a sequence of linear operators from a Banach space to the space >-measurable real functions. Define for each ,> an operator > by x:=supn>\|Tx\|> for all >. Suppose that there exists a positive decreasing function from > to > such that \\>C(\)=0> and for all and : <\equation*> \[x\\\<\|\|\>x\<\|\|\>]\C(\), whenever \<\|\|\>x\<\|\|\>> is admissible for x>. Then the set of elements > for which the sequence x)>> converges a.e.is closed. <\proof> Suppose that a sequence > converges to in > in norm and that there exists a sequence )>> of measurable functions such that for all ,> z\f> a.e. We will prove that the sequence z)>> converges a.e. For natural numbers and \Y> and > put <\equation*> \(\,x):=supn,m\b>\|Tx(\)-Tx(\)\|. Then <\equation*> \|\(\,z)-\(\,z)\|\\|\(\,z-z)\|\2(z-z). Let \0>. Choose a natural numbers such that |2>\<\|\|\>z-z\<\|\|\>)\\> and choose a natural number and an integrable set Y> such that (D)\\> and for all k>: <\equation*> {\:\|Tz(\)-f(\)\|\|2>}\D. For all a\k,> <\eqnarray*> [\|\(\,z)\|\2\]>|>|[\|\(\,z)\|\\]>>|||+\[\|\(\,z)-\(\,z)\|\\]>>||>|(D)+\[2(z-z)\\]>>||>|+C(|2>\<\|\|\>z-z\<\|\|\>)\2\.>>>> Construct ascending sequences )>> and )>> of natural numbers such that <\equation*> C(\2\<\|\|\>z-z>\<\|\|\>)\\2 and <\equation*> \[\|Tz>-f>\|\\2]\\2 for all k>. Put <\equation*> \ {\:\|\,k>(\,z)\|\\2}>. The set is integrable and (E)\>2\2=2\>. For \-E> and k>: <\equation*> \|Tz(\)-Tz(\)\|\\. Consequently, the sequence z)>> converges a.e. Bishop (p.230) gave a constructive proof of Lebesgue's Differentiation Theorem. Using Banach's principle one can give another constructive proof, see (p.101).\ When 1> and is a positive >->> contraction, we define ={f\L:T*f=f}> and ={T*f-f:f\L}.> <\theorem> > Let > be a >-finite measure. Let be a positive >->> contraction. If =\>, for some 1>, then for all 1>, the sequence f)>> converges a.e.for all L>. <\proof> Suppose that =\>, for some 1>. First let 1>. Theorem and Theorem show that =\>. Suppose that , where L>, and L>\L.> The set of these is dense in >. Because contracts the >>-norm, \>Af=g> a.e. For all admissible \0,> <\eqnarray*> \[Mf\\]>||[]>>|f\\]>f>>|||[Jensen]>>|f\\]>f>\[Mf\\]>>>||>|f\<\|\|\>\[Mf\\]>.>>>> Consequently, \[Mf\\]>\\<\|\|\>f\<\|\|\>.> Define f=supn>\|Af\|,> for all L.> Substituting =\\<\|\|\>f\<\|\|\>> and observing that f=Mf> when 0,> we obtain <\equation> \[f\\\<\|\|\>f\<\|\|\>]\\. For all L,> f\\|f\|,> so inequality() holds for all L.> Banach's principle shows that the sequence f)>> converges a.e.for all in >. Finally we remove the assumption that is strictly greater than 1. Let 1>. The set \L> is dense in > , so one can apply Banach's principle since inequality() above holds for all 1>. The two-step argument above is used because in general we do not have convergence in the >-norm. The space > has an awkward geometrical structure: it is not uniformly convex. <\theorem> Let 1>. If the sequence f)>> converges a.e.for all in >, then ={x-T*x:x\L}> is located in >. <\proof> Let be an element of >. Without loss of generality we may assume that f\0> a.e. We claim that the sequence f)>> converges weakly to 0. We may assume that is a simple function. Since the sequence f)>> is bounded, the sequence(Af)g> converges to 0 whenever is a simple function. The inner product is continuous on ,> so Af,g\\0> for all L> that is, f)>> converges weakly to 0. Define \(I+T+\+T)>. By an argument similar to the proof of Theorem we see that for each , x> converges weakly to an element in the space L}> and thus <\equation*> L=\wcl{(I-T)x:x\L}, where wcl denotes the weak closure. Since x=(I-A)x> is a bounded sequence and the weak closure of a bounded convex inhabited subset coincides with its strong closure, by Lemma5.2.4 in, we see that =wcl{y-T y:y\L}>. It follows that the display sum above is orthogonal and thus that > is located. <\lemma> >Let be a uniformly convex and uniformly smooth Banach space and a bounded convex subset of . Then is located if and only if C }> exists for each normable linear functional f on E . <\lemma> >>For 1>, the space > is uniformly convex and uniformly smooth and each normable functional may be represented by a functional f*g>, for some in >, where +=1>. <\lemma> If > is located in some >, then both > and > are located in all >, where 1>. <\proof> Locatedness is a property of closed sets, so we may restrict to a set which is dense in all the >-spaces, for instance the bounded > functions or the simple functions. Moreover, an inhabited set is located if we can compute the distance (x,A)> for each . Let be an element of . Then (x,A)=\(x,A\B(x,\(x,a)+1)))>, where denotes the ball around with radius . Consequently, when considering located sets we may restrict to its bounded subsets.\ We will now apply Lemma. Since the simple functions are dense in all the >-spaces, if C}> exists for all in >, then this supremum exists for all in set of the simple functions, and thus for all in any >>, where \1>. It follows that is located in >>. In particular, if > are located in >, then it is located in >. Since these sets are orthogonal, they are thus both located in >. Consequently, they are both located in any >>. <\theorem> Let > be a >-finite measure and a positive >->> contraction. Denote by > the average T.> The following statements are equivalent: <\enumerate> The set ={x-T*x:x\L}> is located in >; The sequence )>> converges in >; For all 1>, the sequence )>> converges in >; There is 1> such that the sequence )>> converges in >; For all 1> and L>, the sequence f)>> converges a.e.; There is 1> such that for all L,> the sequence f)>> converges a.e. For a finite measure we may replace 1> by 1>. <\proof> (1)>(2) is Theorem. (4)>(3) is Theorem. (3)>(2) is trivial. (2)>(4) is trivial. (4)>(5) follows from Theorem and Theorem. (5)>(6) is trivial. (6)>(1) follows from Theorem. Some of these results have circulated in preprints for some time. They then appeared in my thesis. The results have been used in 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|bib|plain|newdatabase.bib,jsl-ergodic.bib,ergodic-cor.bib> <\bib-list|10> Jeremy Avigad and Ksenija Simic. Fundamental notions of analysis in subsystems of second-order arithmetic. , to appear. Errett Bishop. Mathematics as a numerical language. In , pages 53--71. North-Holland, Amsterdam, 1970. Errett Bishop and Douglas Bridges. , volume 279 of . Springer-Verlag, 1985. ErrettA. Bishop. . McGraw-Hill Publishing Company, Ltd., 1967. N.Dunford and J.T. Schwartz. Interscience Publishers, 1958. Hajime Ishihara and Luminiµa Vīµ . Locating subsets of a normed space. , 131(10):3231--3239, 2003. Ulrich Krengel. . Studies in Mathematics. de Gruyter, 1985. J.A. Nuber. A constructive ergodic theorem. , 164:115--137, 1972. J.A. Nuber. Erratum to `A constructive ergodic theorem'. , 216:393, 1976. Karl Petersen. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1983. Bas Spitters. . PhD thesis, University of Nijmegen, 2002. Bas Spitters. A constructive view on ergodic theorems. , 71(2):611--623, 2006. Bas Spitters. Corrigendum to: ``A constructive view on ergodic theorems'' J. Symbolic Logic (2006), no. 2, 611--623. , 71(4):1431--1432, 2006. Peter Walters. , volume79 of . Springer-Verlag, 1982. <\initial> <\collection> <\references> <\collection> |1>> alpha|> > > > > > > > 7 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > T(fp)|> Tfp|> > > > > > > > > > > <\auxiliary> <\collection> <\associate|bib> ergodic ergodic-cor Bishop:1967 Bishop:1970 Nuber:1972 Nuber:1976 Bishop/Bridges:1985 Krengel:1985 Dunford/Schwartz:1958 Bishop/Bridges:1985 Bishop/Bridges:1985 Bishop/Bridges:1985 Krengel:1985 Bishop/Bridges:1985 Krengel:1985 Krengel:1985 Walters:1982 Krengel:1985 Bishop:1967 Petersen:1983 IshiharaVita Bishop/Bridges:1985 IshiharaVita Spitters:phd AvigadSimic <\associate|idx> |> |> |> |> |> <\associate|toc> |math-font-series||1.Introduction> |.>>>>|> |math-font-series||2.The mean Ergodic Theorem>|>>> |.>>>>|> |math-font-series||3.Maximal Ergodic Theorems> |.>>>>|> |math-font-series||4.The Pointwise Ergodic Theorem>|>>> |.>>>>|> |math-font-series||Bibliography> |.>>>>|>