Ergodic theorem for a Loeb space and hyperfinite approximations of dynamical systems
aa r X i v : . [ m a t h . C A ] A p r Ergodic theorem for a Loeb space and hyperfinite approximationsof dynamical systems
L.Yu. Glebsky, E.I. Gordon, C.W. Henson
Abstract
Although the G.Birkhoff Ergodic Theorem (BET) is trivial for finite spaces, this does not help inproving it for hyperfinite Loeb spaces. The proof of the BET for this case, suggested in [14], works,actually, for arbitrary probability spaces, as it was shown in [16]. In this paper we discuss the reasonwhy the usual approach, based on transfer of some simple facts about arbitrary large finite spaces oninfinite spaces using nonstandard analysis technique, does not work for the BET. We show that the theBET for hyperfinite spaces may be interpreted as some qualitative result for very big finite spaces. Weintroduce the notion of a hyperfinite approximation of a dynamical system and prove the existence ofsuch an approximation. The standard versions of the results obtained in terms of sequences of finitedynamical systems are formulated.
Introduction
Many applications of nonstandard analysis (NSA) to the investigation of infinite structures arebased on embedding these structures in appropriate hyperfinite structures . The latter inheritmany properties of finite structures due to the transfer principle of NSA. The effectiveness of thedescribed applications of NSA is based on the fact that many properties can be proved in a morestraightforward way for finite structures, than for the corresponding infinite ones.In the framework of standard mathematics the same idea is implemented by an approximation ofan infinite structure by finite ones, e.g. by an embedding of this infinite structure into an appropriateinductive (projective) limit of finite structures. However, in some problems a construction ofinductive and projective limits is either impossible or inappropriate, while the hyperfinite structuresand their nonstandard hulls can be effective. One can find many such examples in stochasticanalysis, functional analysis, harmonic analysis and mathematical physics (see e.g. the books[1, 7, 18, 9] and the bibliography therein).The book [19] contains the exposition of Probability Theory based not on the Kolmogorov’saxiomatics but on the theory of hyperfinite measure spaces considered by the author as a formal-ization of the notion of a very big finite probability space. In the author’s opinion this approachmakes the foundation of probability theory more transparent and more natural when comparedwith the classical approach.In this paper we discuss the NSA approaches to the Birkhoff Ergodic Theorem (BET). Weremind the formulation of this theorem. See [1, 18] for the basic notions of NSA. As a rule a hyperfinite structure used in this paper can be considered asan ultraproduct of finite structures. heorem 1. Let ( X, Σ , ν ) be a probability space and let T : X → X be a measure preservingtransformation and f ∈ L ( X ) . Set A k ( f, T, x ) = 1 k k − X i =0 f ( T i x ) . Then1) there exists a function ˆ f ( x ) ∈ L ( X ) such that A k ( f, T, x ) → ˆ f ( x ) as k → ∞ ν -almosteverywhere (a.e.);2) the function ˆ f is T -invariant, i.e. ˆ f ( T x ) = ˆ f ( x ) , for almost all x ∈ X ;3) R X f dν = R X ˆ f dν . We denote a dynamical system in X by a triple ( X, ν, T ) whose terms satisfy the conditions ofTheorem 1. In this paper two types of probability spaces are considered:1) a
Loeb space , [1, 18] that is a standard probability space constructed from a hyperfinite setendowed with the counting probability measure;2) a
Lebesgue space that is a probability space, which is isomorphic modulo measure 0 to thesegment [0 ,
1] with the standard Lebesgue measure, or to a countable space with an atomic measure,or to the direct sum of the previous two.It is known that every compact metric space with a Borel probability measure is a Lebesguespace (see e.g. [3]).We call a dynamical system in a Lebesgue (Loeb) space a
Lebesgue (Loeb) dynamical system .The known proofs of Theorem 1 are not very difficult, but rather tricky (see e.g. [2]). Since thespecialists in dynamical systems are interested mainly (if not to say, only) in dynamical systems inLebesgue spaces, it would be worthwhile to simplify (to make more straightforward) the proof ofthe BET at least for Lebesgue dynamical systems.Two attempts to apply NSA to the BET in Lebesgue spaces are known [14, 12].The NSA approach to a proof of the BET for Lebesgue dynamical systems due to Kamae [14]consists of two parts.1. The proof of the BET for a Loeb dynamical system.2. The derivation of the BET for a Lebesgue dynamical system from the BET for a Loebdynamical system.It is natural to expect that the proof of the BET for a Loeb space is simpler, than its proofin the general case, since a Loeb space is constructed from a hyperfinite probability space. TheBET for hyperfinite probability spaces follows from the BET for finite probability by the transferprinciple, and for the last class the BET is trivial.However, the triviality of the BET for finite spaces is not conductive to a simple proof of theBET for Loeb spaces. The proof of the BET for a Loeb space, due to Kamae [14], is not easier,than the known proofs of the general case. In fact, it was shown in [16] that Kamae’s proof isapplicable for the general case after a slight modification.In Section 1 we give some explanation why the BET for finite spaces does not help to prove theBET for Loeb space. We suggest an interpretation of the BET for a Loeb space as some statement2n Internal Set Theory [20] about the stabilization of ergodic means in a hyperfinite space (Theorem3). Regarding the notion of a hyperfinite set as a formalization of the notion of a very big finiteset, we may consider Theorem 3 as a qualitative statement about the behavior of ergodic means ofsome specific functions on a very big finite probability space.In the framework of classical mathematics, the notion of a very big probability space can beformalized using sequences of finite probability spaces, whose cardinalities increase to infinity. Asa rule it is rather difficult to obtain, or even to formulate, results in probability theory in terms ofsuch sequences. So, dealing with very big finite probability spaces, one prefers to use a transitionto infinite probability spaces according to Kolmogorov’s axiomatic. However, it is often difficult tointerpret statements about standard infinite probability spaces in terms of very big finite probabilityspaces.NSA suggests some methods to reformulate results about hyperfinite structures in terms ofsequences of finite structures. The standard ”sequence” versions of the results about the behaviorof ergodic means in a hyperfinite probability space obtained in Section 1, are discussed in theconcluding subsection of this section. It is interesting that Theorem 3 does not have any readableand intuitively clear standard ”sequence” version, though its interpretation in terms of very bigprobability spaces is clear and simple. We describe how the statement of Theorem 3 can be seenin computer experiments.Another attempt to prove the BET for Lebesgue spaces using NSA is contained in the paper [12]by A.G. Kachurovskii. This paper is written in the framework of E. Nelson’s Internal Set Theory(
IST ) [20] without any use of Loeb spaces. The author makes an attempt to deduce Theorem 3from his theorem about the fluctuations of ergodic means (see Theorem 5 below). This derivationcontains a serious gap that is filled here with the help of Loeb spaces. Actually A.G.Kachurovskiican deduce only Proposition 7 from Theorem 5. This proposition is equivalent to the BET for aLoeb space. However, it is not clear, whether it is sufficient for the proof of the BET for Lebesguespaces the approach of [12]. Theorem 5 is very interesting by itself. Its proof (that is much moredifficult, than those of the BET) was transformed in the proof of standard theorem about estimatesof ε -fluctuations of ergodic m! eans [13]. We show that, unlike Theorem 3, Theorem 5 has a simplestandard ”sequence” version (Theorem 7).In Section 2 we discuss the second part of NSA approach to the BET - the derivation of the BETfor Lebesgue dynamical systems from the BET for Loeb dynamical systems. Though the abovediscussion shows that the BET for Loeb dynamical systems is not simpler than for the generalcase, the both parts of NSA approach to the BET are interesting in their own right. The secondpart is interesting from the point of view of approximation of Lebesgue dynamical systems by finiteones. The approach to approximation suggested here differs from the most popular approach inergodic theory based on Rokhlin’s Theorem (see e.g. [3]). Rokhlin’s approximations have manyinteresting applications to ergodic theory, especially to problems connected with the entropy ofdynamical systems. The disadvantage of Rokhlin’s approximations is caused by the difficulty oftheir construction that makes them near on impossible to be used in computer simulations ofLebesgue dynamical systems. In this paper we suggest a definition of approximation of a Lebesguedynamical system by a hyperfinite dynamical system (Definition 4 below, see also Definition 2) andprove the existence of such approximation for an arbitrary Lebesgue dynamical system (Theorem8). This theorem allows us to prove the universality of Loeb dynamical systems, i.e. the existence ofmeasure preserving epimorphism from a Loeb dynamical system to a Lebesgue dynamical system.3he existence of such epimorphism obviously implies the BET for Lebesgue dynamical systemsassuming the BET for Loeb systems to have been already established. The universality of Loebsystems was proved in [14]. However, the proof presented here is simpler than one of Kamae andallows to construct hyperfinite approximations for concrete dynamical systems.The definition of a hyperfinite approximation of Lebesgue dynamical systems can be reformu-lated in terms of an approximation of Lebesgue dynamical systems by finite dynamical systems inmany particular cases, but not in the full generality. However, for the case of a Lebesgue dynamicalsystem, whose measure space is a compact metric space and the measure preserving transformationis continuous, the standard sequence versions of Definitions 2, 4 and Theorem 8 are quite simple(see Definition 3, Proposition 18 and Theorem 10).As examples, we construct in this paper finite approximations for shifts of the unit interval andfor the invertible Bernoulli shift and study the behavior of ergodic means for these approximations.It is interesting that the ergodic means of approximations of irrational shifts of the unit intervalstabilize at the average of a function for all infinite n . Theorem 9 states that the same is true forall hyperfinite approximations of any uniquely ergodic dynamical system.The NSA approach to Rokhlin’s finite approximations of Lebesgue dynamical systems will bediscussed in another paper. Some preliminary results were announced in [8].The authors are grateful to Boris Begun, Karel Hrbacek, Peter Loeb and Edgardo Ugalde forhelpful discussions of various parts of this paper. Recall that a number α ∈ ∗ R is said to be finite if | α | < a for some a ∈ R , otherwise α is said tobe infinite. For every finite α ∈ ∗ R there exists a unique standard real ◦ α such that α ≈ ◦ α . It iscalled the standard part or a shadow of α . We use also the following notations: α ∼ ∞ for infinite α , α ≫ α β ≫ α for αβ ≈ ∗ N ∞ for ∗ N \ N .First we modify a trivial proof of the BET for finite spaces for the case of hyperfinite spaces.Throughout this article we consider a hyperfinite set Y such that | Y | = M , where M is aninfinite number ( M ∈ ∗ N ∞ ) and an internal permutation T : Y → Y . We say that p ( x ) is the T -period of an element x ∈ Y if its orbit Orb ( x ) = { x, T x, ..., T k x.... } has exactly p ( x ) distinctelements. Let G : Y → ∗ R be an internal function. Denote A p ( x ) ( G, T, x ) (the average of G alongthe orbit of x ) by Av x ( G ).If there exist an element x ∈ Y that has a T -period M , then the T -period of any element in Y is equal to M . In this case we say that this permutations is transitive , since it generates thetransitive cyclic group of permutations of X .If T is transitive, then for every internal function G : Y → ∗ R one has obviously ∀ x , x ∈ Y Av x ( G ) = Av x ( G ) = Av ( G ). So, Av ( G ) = 1 M X y ∈ Y G ( y ) = 1 M M − X j =0 G ( T j x ) = A M ( G, T, x ) , (1)for every x ∈ Y .Denote the maximal value of G along the orbit of x by max x G . Proposition 1.
For an arbitrary internal function G : Y → ∗ R for all x ∈ Y and for all N ∈ ∗ N ∞ such that p ( x ) · max x | G | N ≈ (notation: N ≫ p ( x ) · max x | G | ) one has A N ( G, T, x ) ≈ Av x ( G ) . roof . It is enough to prove this statement for a transitive permutation T . In this case ∀ x ∈ Y p ( x ) = M, Av x ( G ) = Av ( M ) and N ≫ M · max | G | .Let N = qM + r , 0 ≤ r < M . Then A N ( G, T, x ) = (cid:18) qM + r − qM (cid:19) qM − X i =0 G ( T i x ) + 1 qM qM − X i =0 G ( T i x ) + 1 qM + r qM + r − X i = qM G ( T i x ) . Obviously qM qM − P i =0 G ( T i x ) = M M − P i =0 G ( T i x ) = Av ( G ) by (1). The first and the third terms areinfinitesimal since N ≫ M · max | G | ✷ We list now the necessary definitions and facts concerning Loeb spaces. We need here onlya particular case of a Loeb space, namely the Loeb space constructed from the hyperfinite set Y endowed with the uniform probability measure.Define the external finitely additive measure µ on the algebra P int ( Y ) of internal subsets of Y by the formula µ ( B ) = ◦ (cid:18) | B | M (cid:19) . The ℵ -saturation and the Caratheodory theorem imply the possibility to extend µ on the σ -algebra σ ( P int ( Y )) generated by P int ( Y ). The Loeb space with the underlying set Y is theprobability, space ( Y, P L ( Y ) , µ L ), where P L ( Y ) is the completion of σ ( P int ( Y )) with respect to µ and µ L is the extension of µ on P L ( Y ). The measure µ L is said to be the Loeb measure on Y . Ifnecessary we use the notation µ YL . We need the following property of the Loeb measure that followsimmediately from the ℵ -saturation. Proposition 2.
For every set A ∈ P L there exists an internal set B ⊆ Y such that µ L ( A ∆ B ) = 0 . Corollary 1. If A ∈ P L , then µ L ( A ) = 1 ⇐⇒ ∀ st ε > ∃ B ∈ P int ( Y ) ( B ⊆ A ∧ µ ( B ) > − ε ) Remark 1.
The right hand side of the equivalence in Corollary 1 can be formalized in the frame-work of
IST , while this is impossible for the left hand side. In works based on
IST (see e.g.[19, 12]) the right hand side of the above equivalence is used as a definition for a certain propertyto hold almost surely. In this article we say that a property holds µ L − a.e. to express the samestatement.For an arbitrary separable metric space R and an external function f : Y → R an internalfunction F : Y → ∗ R is said to be a lifting of f if µ L ( { y ∈ Y | F ( y ) ≈ f ( y ) } ) = 1 . Proposition 3.
A function f : Y → R is measurable iff it has a lifting. Recall that an internal function F : Y → ∗ R is said to be S -integrable if for all K ∈ ∗ N ∞ onehas 1 M X { y ∈ Y | | F ( y ) | >K } | F ( y ) | ≈ . (2)We need the following properties of S -integrable functions.5 roposition 4.
1) An S -integrable function is almost everywhere bounded.2) An internal function F : Y → ∗ R is S -integrable iff Av ( | F | ) is bounded and M P y ∈ A | F ( y ) | ≈ for every internal A ⊆ Y such that | A | M ≈ .3) An external function f : Y → R is integrable w.r.t. the Loeb measure µ L iff it has an S -integrable lifting F , in which case Z Y f dµ L = ◦ Av ( F ) . . We address readers to [1, 18] for the proofs of Propositions 2, 3 and 4.For the case of S -integrable functions one can get some information about behavior of ergodicmeans for infinite N M , using the same simple considerations. Lemma 1. If F : Y → ∗ R is an internal function, and Av ( | F | ) is finite, then for almost all x ∈ YAv x ( | F | ) is finite. Proof.
Let U = { x ∈ Y | Av x ( | F | ) ∼ ∞} . Let I = { x , . . . x k } be a selector , i.e. it intersectseach orbit of T by a single point. Consider an arbitrary internal set V ⊆ U and show that | V | M ≈
0. This will prove that µ L ( U ) = 0. Let B = { i ≤ k | Orb ( x i ) ∩ V = ∅} . Consider the set V ′ = S i ∈ B Orb ( x i ) ⊇ V . Then | V ′ | = P i ∈ B p ( x i ). To complete the proof of the lemma we prove that | V ′ | M ≈ λ = min { Av x ( | F | ) | x ∈ V } . Then λ ∼ ∞ . Suppose that | V ′ | M ≫
0. Then ∞ ∼ | V ′ | M · λ ≤ M X i ∈ B p ( x i ) · Av x i ( | F | ) = 1 M X x ∈ V ′ | F ( x ) | ≤ Av ( | F | ) , which contradicts the conditions of our Lemma. ✷ An internal function F : Y → ∗ R is said to be bounded if max {| F ( x ) | | x ∈ Y } is finite. The ℵ -saturation implies that F is bounded, if and only if all values of F are finite. Theorem 2.
Let T : Y → Y be an internal permutation. Then the following statements are true.1) For every S -integrable function F : Y → ∗ R , for every standard positive a ∈ R , and forevery infinite numbers K, L such that ◦ (cid:0) KM (cid:1) = ◦ (cid:0) LM (cid:1) = a one has A K ( F, T, x ) ≈ A L ( F, T, x ) for µ L -almost all x ∈ Y ;2) The statement 1) holds for all x ∈ Y such that p ( x ) M ≫ or max {| F ( y ) | | y ∈ Orb ( x ) } isfinite. In particular, the statement 1) holds for all x ∈ Y if T is a transitive permutation or F isa bounded function.3) If T is a transitive permutation, then for any internal function F such that Av ( | F | ) isfinite ( F may be not S -integrable here) and for any K ∈ ∗ N ∞ such that KM is infinite one has A K ( F, T, x ) ≈ Av ( F ) for all x ∈ Y . Proof . Assume
K > L and estimate | A K ( F, T, x ) − A L ( F, T, x ) | . It is easy to see that | A K ( F, T, x ) − A L ( F, T, x ) | ≤ (cid:18) L − K (cid:19) L − X k =0 | F ( T k x ) | + 1 K K − X k = L | F ( T k x ) | = U + V.
6e need to prove that U ≈ V ≈ B = { T k x | k = L, . . . , K − } . Then | B | M = K − LM ≈
0. Thus, V = M P y ∈ B | F ( y ) | ≈
0, due tothe S -integrability of F (item 2 of Proposition 4).One has U = (cid:18) ML − MK (cid:19) M L − X k =0 | F ( T k x ) | . Since ML ≈ MK ≈ a and M L − P k =0 | F ( T k x ) | ≤ M ([ a ]+1) M − P k =0 | F ( T k x ) | , to establish that U ≈
0, it isenough to prove that the right hand side of the last inequality is finite under conditions 1) and 2)of the theorem. This is obvious for any x such that F ↾ Orb ( x ) is a bounded function. It is enoughto discuss only the case of [ a ] = 0. In this case1 M M − X k =0 | F ( T k x ) | = d ( x ) · M p ( x ) − X k =0 | F ( T k x ) | + 1 M M − X k = d ( x ) · p ( x ) | F ( T k x ) | = S + S , where M = d ( x ) · p ( x ) + r ( x ) and r ( x ) < p ( x ).Since all terms S are distinct, one has S ≤ Av ( | F | ), which is finite due to the S -integrabilityof F .If d ( x ) is finite, then S is finite by the same reason. So, U ≈ | S − Av x ( | F | ) | = r ( x ) M Av x ( | F | )So S is bounded for µ L -almost all x ∈ Y by Lemma 1. This proves statement 1).To prove statement 3) set K = qM + r, r < M and notice that q is an infinite number. Then A K ( F, T, x ) = qM + r qM + r − P i =0 F ( T i x ) = (cid:16) qM + r − qM (cid:17) qM − P i =0 F ( T i x )++ qM qM − P i =0 F ( T i x ) + qM + r qM + r − P i = qM F ( T i x ) (3)Since T is a cycle of length M , one has qM qM − P i =0 F ( T i x ) = Av ( F ) and A K ( F, T, x ) = (cid:18) qMqM + r − (cid:19) Av ( F ) + Av ( F ) + 1 q + rM M qM + r − X i = qM F ( T i x ) ≈ Av ( F ) , (4)since Av ( | F | ) is bounded and | q + rM M qM + r − P i = qM F ( T i x ) | ≤ q + rM Av ( | F | ) ✷ Remark 2.
Due to (1) one has A N ( F, T, x ) ≈ Av ( F ) for NM ≈ T is a transitive permutation.7 roposition 5. For every S -integrable function F : Y → ∗ R and for every n ∈ ∗ N the function F n = A n ( F, T, · ) is S -integrable and for every n ∈ ∗ N ∞ the standard function ◦ F n is T -invarianta.e. Proof . For B ⊆ Y and for every n ∈ ∗ N one has1 M X x ∈ B | A n ( F, T, x ) | ≤ M X x ∈ B A n ( | F | , T, x ) = 1 n n − X i =0 M X x ∈ B | F ( T i x ) | ! (5)For B = Y each term in parentheses in (5) is equal to Av ( | F | ) since T i is a permutation for every i . Thus Av ( | A n ( F, T, · ) | ) ≤ Av ( | F | ), which is finite since F is S -integrable. For | B | M ≈ A n ( F, T, · ) is S -integrable for all n ∈ ∗ N by item 2) of Proposition 4.To prove T -invariance of ◦ F n for n ∈ ∗ N ∞ it is enough to show that G n ( x ) = F n ◦ T ( x ) − F n ( x ) ≈
0. Obviously G n ( x ) = n ( F ( T n x ) − F ( x )). Since F is S -integrable, it is bounded a.e. by (2). Thus G n ( x ) ≈ n is infinite. ✷ Remark 3.
Let B = { x | ∀ n ∈ ∗ N ∞ F n ( x ) ≈ F n ◦ T ( x ) } . It is not known if µ L ( B ) = 1 for every S -integrable function F ? It is obviously true if F is bounded everywhere. In this case B = Y . If ◦ ( n/M ) >
0, then ∀ x ∈ Y F n ( x ) ≈ F n ◦ T ( x ). This easily follows from the S -integrability of F . Proposition 6.
1) In conditions of the previous proposition let an internal function F be boundedon Y . For every x ∈ Y define the standard function f x : (0 , ∞ ) → R by the formula f x (cid:16) ◦ (cid:16) nM (cid:17)(cid:17) = ◦ F n ( x ) , (6) where ◦ (cid:0) nM (cid:1) > . Then f x is well defined and continuous on (0 , ∞ ) .2) The function ϕ ( x, a ) = f x ( a ) is a measurable function on Y × (0 , ∞ ) , where Y is the Loebspace and (0 , ∞ ) is the measurable space with Lebesgue measure dx .3) If T is a transitive permutation, then the function ϕ ( x, a ) satisfies the following equality ∀ a ∈ (0 , ∞ ) Z x ∈ Y ϕ ( x, a ) dx = ◦ Av ( F ) . Proof . Fix an arbitrary x ∈ Y and an arbitrary interval [ a, b ] ⊆ R such that a >
0. considerthe internal set Z [ a,b ] = { nM | n ∈ ∗ N } ∩ [ a, b ] and the internal function G [ a,b ] : Z [ a,b ] → R definedby the formula G [ a,b ] (cid:0) nM (cid:1) = F n ( x ). By Theorem 2 the function G [ a,b ] is S -continuous. This provesthe first statement of the proposition.To prove the second statement, one has to consider the function G [ a,b ] as an internal functionon Z [ a,b ] × Y .The third statement follows immediately from the obvious equality Av ( F n ) = Av ( F ) ✷ Remark 4.
The result of Proposition 6 can be observed in computer experiments with finitedynamical systems. The cardinality M , of a finite space Y in these experiments may not be verylarge, say M ∼ . To draw the graph of the function f x for a finite function F : Y → R onan interval (0 ,
1] one has to plot points (cid:0) nM , A n ( F, T, x ) (cid:1) for all n ≤ M . The fact that F is an S -integrable function, means that F should not be a δ -type function. For example, if max | F | ≪ M F is bounded), then we may assume that F satisfies the condition of S -integrability.In the Examples 3 and 5 below, the explicit formulas for functions f x are obtained for someconcrete hyperfinite dynamical systems.The following example shows that the condition of S -integrability of a function F in Theorem2 is essential and that the assumption of finiteness of Av ( | F | ) is not enough in the first part ofTheorem 2. Example 1 . Let Y = { , , . . . , M − } , where M is a infinite even number. Define thetransformation T : Y → Y by the formula T ( k ) = k + 1 ( mod M ) and the internal function F : Y → ∗ R by the formula F ( k ) = M δ k for all k < M . Then it is easy to see that A K ( F, T, k ) = (cid:26) MK , k > M − K , < k ≤ M − K We see that if
K < L < M , and k is such that M − L < k < M − K , then A K ( F, T, k ) A L ( F, T, k ) even if KM ≈ LM Here Av ( | F | ) = 1. However, F is not S -integrable by Proposition 4 (2), since M P k ∈{ } | F ( k ) | = 1.For the case of KM ≈ Proposition 7.
In conditions of Theorem 2 for any y ∈ Y the lim n →∞ A n ( ◦ F, T, y ) exists, if and onlyif one can find a number K ∈ ∗ N ∞ such that ∀ L ∈ ∗ N ∞ ( L < K = ⇒ A K ( F, T, y ) ≈ A L ( F, T, y ) ≈ lim n →∞ A n ( ◦ F, T, y )) . The proof of this proposition follows immediately from the following easy
Lemma 2.
Let { a n | n ∈ ∗ N ∞ } be an internal sequence. Then the standard sequence { ◦ a n | n ∈ N } converges if and only if there exist K ∈ ∗ N ∞ such that a K is a bounded number and ∀ L ∈ ∗ N ∞ ( L ≤ K = ⇒ ◦ a L = ◦ a K ) ✷ The following theorem is stronger then Proposition 7.
Theorem 3.
Let T : Y → Y be an internal permutation of the Loeb space ( Y, P L ( Y ) , µ L ) . Then forevery S -integrable function F : Y → ∗ R there exists an infinite hypernatural number N such that µ L -almost everywhere for all infinite hypernatural numbers L < N one has A L ( F, T, x ) ≈ A N ( F, T, x ) Remark 5.
According to Theorem 3 there exists an initial segment
A ⊆ ∗ N ∞ ∩ { K | KM ≈ } such that ergodic means of F stabilize on A almost surely. However, the example 2 below showsthat in general the ergodic means A K ( F, T, k ) may not stabilize almost surely on the whole set { K | KM ≈ } . In this case the function f x ( a ) corresponding to an S -integrable function F can notbe extended by continuity to the point a = 0. On the graph it can appear as a small cloud in thevicinity of zero. Theorem 3 implies that there can be found a large enough number L , however L ≪ M such that for every random number k ∈ (0 , M ) the set of points (cid:8)(cid:0) nM , A n ( F, T, k ) (cid:1) | n ≤ L (cid:9) looks on display like a horizontal line y = A L ( F, T, k ). This change of scale on the x-axis can beinterpreted as the observation of neighborhood of zero through a strong microscope. For some finitedynamical systems it is enough to take M ∼ and L ∼ to observe the described phenomenon. Here, as usual, δ i,j = 1(0), if i = j ( i = j ).
9o prove Theorem 3 first it is necessary to prove
Theorem 4.
Let f n : Y → R , n ∈ N be a sequence of measurable functions on Y , and F n : Y → ∗ R , n ∈ ∗ N be an internal sequence such that ∀ n ∈ N , F n is a lifting of f n . Then f n convergesto a measurable function f µ L -almost everywhere if and only if there exists K ∈ ∗ N ∞ such that µ L -almost everywhere ∀ N ∈ ∗ N ∞ , N < K = ⇒ F N ( x ) ≈ F ( x ) , where F is a lifting of f . The following lemma is an immediate consequence of the ℵ -saturation principle. Lemma 3. ∀ ϕ : N → ∗ N ∞ ∃ N ∈ ∗ N ∞ ∀ n ∈ N N < ϕ ( n ) . ✷ Proof of Theorem 4.(= ⇒ ) Let f n converges to f a.e. By Egoroff’s theorem ∀ k ∈ N ∃ B k ⊆ Y ( µ L ( B k ) ≥ − k ∧ f n ( x ) = ◦ F n ( x ) converges uniformly on B k ) . WLOG we may assume that B k is internal, | B k || Y | ≥ − k , and ∀ n, k ∈ N ∀ x ∈ B k F n ( x ) ≈ f n ( x )and F ( x ) ≈ f ( x ). Then ∃ st ϕ k : N → N ∀ st r ∀ st m > ϕ k ( r ) max x ∈ B k | F m ( x ) − F ( x ) | < r . Consider the internal set C kr = { N ∈ ∗ N | ∀ m ( N > m > ϕ k ( r ) = ⇒ ∀ x ∈ B k | F m ( x ) − F ( x ) | < r ) } The previous statement shows that C kr contains all standard numbers that are greater that ϕ k ( r ).Thus, there exists infinite N kr ∈ C kr . By Lemma 3 ∃ K ∈ ∗ N ∞ ∀ st k, r K < N kr . Obviously, this K satisfies Theorem 4( ⇐ =) Let B = { x ∈ Y | ∀ N ∈ ∗ N ∞ ( N ≤ K = ⇒ F N ( x ) ≈ F ( x ) } , A n = { x ∈ Y | f n ( x ) ≈ F n ( x ) } , n ∈ N , A = { x ∈ Y | F ( x ) ≈ f ( x ) } , C = B ∩ A ∩ T n ∈ N A n . By conditions of the theorem µ L ( C ) = 1. Fix an arbitrary x ∈ C , and an arbitrary r ∈ N . Theinternal set D = { n ∈ ∗ N | | F n ( x ) − F ( x ) | ≤ r } contains all infinite numbers that are less or equalto K . So ∃ n ∈ N ∀ n > n | F n ( x ) − F ( x ) | ≤ r } . Since F n ( x ) ≈ f n ( x ), the same holds for f n ( x )and ◦ F ( x ). Thus, f n converges to f = ◦ F a.e. ✷ . Proof of Theorem 3.
In conditions of Theorem 3 let f = ◦ F and f n ( x ) = A n ( f, T, x ) , n ∈ N and the internal sequence F n ( x ) = A n ( F, T, x ) , n ∈ ∗ N . Then f ∈ L ( µ L ) and F n is an S -integrablelifting of f n for all n ∈ N . By Theorem 1 f n converges to an integrable function ˆ f a.e. Let ˆ F bean S -integrable lifting of ˆ f . Then by Theorem 4 there exists K ∈ ∗ N ∞ such that µ L -almost surely ∀ N ∈ ∗ N ∞ N < K = ⇒ F N ( X ) ≈ ˆ F ( x ) ✷ . Remark 6.
Theorem 4 also makes it possible to deduce Theorem 1 for the Loeb space Y and itsinternal permutation T from the Theorem 3. We use the same notations as in the proof of Theorem3. If there exist K ∈ ∗ N ∞ such that a.e. ∀ N ∈ ∗ N ∞ N < K = ⇒ F N ( x ) ≈ F K , then by Theorem4 the sequence f n converges a.e. to f = ◦ F K . By Proposition 5 f is T -invariant a.e. and F K is S -integrable. So f is integrable ✷ { K ∈ ∗ N ∞ | KM ≈ } . Example 2
Consider the same Loeb space Y and the same transformation T as in Example 1.Fix an infinite K ∈ Y such that KM ≈
0. Let M = RK + S , where 0 ≤ S < K . Define the internalfunction G : Y → ∗ R by the formula G ( k ) = (cid:26) , mK ≤ k < ( m + 1) K, m < R, m is even0 , ( mK ≤ k < ( m + 1) K, m < R, m is odd) ∨ RK ≤ m < M The internal function G is bounded and, thus S -integrable.Fix an arbitrary N ∈ ∗ N ∞ such that N/K ≈ B = { k ∈ Y | ∀ n ≤ N G ( T n ( k )) = G ( k ) } . It is easy to see that µ L ( B ) = 1. Indeed, obviously B ⊇ A = S ≤ n ≤ R { ( n − K, . . . , nK − − N } and µ L ( A ) = ◦ (cid:0) RM ( K − N − (cid:1) = 1. Now it is easy to see that ∀ L ≤ N ∀ k ∈ B A L ( G, T, k ) = A N ( G, T, k ) = G ( k ). Thus, N satisfies Theorem 3.Let D = { k ∈ Y | mK ≤ k < mK + K } . Then, µ L ( D ) = . It is easy to see, that ∀ k ∈ D ∀ n ≤ K G ( T n ( k )) = G ( k ), thus, A [ K ]( G, T, k ) = G ( k ).To show that Theorem 2 fails for the galaxy { K ∈ ∗ N ∞ | KM ≈ } it is enough to provethat A K ( G, T, k ) A [ K ]( G, T, k ) almost everywhere on D . For every standard n consider the set D n = { k ∈ D | | A K ( G, T, k ) − G ( k ) | < n } . It is enough to prove that lim n →∞ µ L ( D n ) = 0. Since thecardinality of the set D n ∩ [ mK, ( m + 1) K ) is the same for all m < R , it is enough to calculate thecardinality of E n = D n ∩ (cid:2) , K (cid:3) . Recall that for k ∈ E n one has A [ K ]( G, T, k ) = G ( k ) = 1. On theother hand A K ( G, T, k ) = K − kK = 1 − kK . So, | E n | = (cid:2) Kn (cid:3) and µ L ( D n ) = ◦ (cid:0) R (cid:2) Kn (cid:3)(cid:1) / ( RK + S ) → ✷ The following example shows that one actually can obtain essentially distinct means for distinct a = ◦ (cid:0) KM (cid:1) . Example 3
Let
Y, T be the same as in the previous examples and F ( k ) = kM . For 0 ≤ a ≤ f : [0 , → R by the formula ψ a ( t ) = (cid:26) t + a , ≤ t ≤ − at + a − a (1 − t ) , − a < t ≤ ◦ A K ( F, T, k ) = (cid:26) ψ (cid:0) ◦ (cid:0) kM (cid:1)(cid:1) , KM ≈ , ∀ k < M − Kψ a (cid:0) ◦ (cid:0) kM (cid:1)(cid:1) KM ≈ a > , ∀ k ∈ X Notice, that in the first case stabilization holds almost surely in accordance with Theorem 3,while in the second one it holds for all k ∈ X in accordance with the Theorem 2.The function ψ ( x ) = for all x . This is the average of f ( x ) = x on [0 , ∗ f (cid:0) kM (cid:1) = F ( k ).For a > ψ a (cid:0) ◦ (cid:0) kM (cid:1)(cid:1) = ϕ ( k, a ), where ϕ ( x, a ) is the function defined in Proposition 6. Remarks on A.G. Kachurovskii’s paper [12]
In the paper [12] A. G. Kachurovskii proves the NSA version of his theorem about fluctuationsof ergodic means and undertakes an attempt to deduce the BET for Lebesgue spaces from thistheorem. However, his approach to the BET contains one gap, that we are going to discuss here. See [13] for the standard version of this theorem. In fact, the NSA version was the first one. N ∈ ∗ N ∞ . Following [19] we say that an internal sequence s : { , . . . , N } → ∗ R is convergentif there exists ξ ∈ ∗ R such that ∀ L ∈ ∗ N ∞ ( L ≤ N = ⇒ s ( L ) ≈ ξ ).For example, Theorem 3 states the existence of a number N ∈ ∗ N ∞ such that the sequence { A n ( F, T, x ) } Nn =1 is convergent for µ L almost all x ∈ Y .We say that a sequence s admits k ε -fluctuations if there exists a sequence of indices n < n < · · · < n k − , n k such that ∀ , i ≤ k | s i − − s i | ≥ ε . The following definition and proposition aredue to E. Nelson [19]. Definition 1.
We say that an internal sequence s is a sequence of limited fluctuation, if for everystandard ε > and for every K ∈ ∗ N ∞ the sequence s does not admit K ε − f luctuations . Proposition 8 ([19] Theorem 6.1) . If s is a sequence of limited fluctuation then there exists L ∈ ∗ N ∞ such that the sequence s ↾ { , . . . , L } is convergent. The following theorem is an obvious reformulation of Theorem 1A of [12].
Theorem 5. If ( Y, µ L , T ) is a transitive Loeb dynamical system, F : Y → ∗ R is an S -integrablefunction and N ∈ ∗ N ∞ , then the sequence { A n ( F, T, x ) } Nn =1 } is a sequence of limited fluctuation µ L -a.e. Using Proposition 8 one immediately deduces Proposition 7 from Theorem 5In [12] it is stated without proof that Proposition 7 implies Theorem 3. However, this implica-tion requires Theorem 4. Indeed, Proposition 7, implies the µ L -a.e. convergence of the standardsequence of functions { ◦ A n ( F, T, x ) | n ∈ N } , which by Theorem 4 implies Theorem 3. The exposi-tion in [12] is developed following book [19], i.e. it is based on Nelson’s Internal Set Theory (IST)[20]. On the other hand the proof of Theorem 4 is based on properties of external Loeb-measurablefunctions on Y . This theorem cannot be even formulated in IST. Is it possible to prove Theorem 3in IST with (or without) the help of Theorem 5, which is a theorem of IST? We suggest that theanswer is positive. We are obliged to K. Hrbacek for the following arguments in support of thisconjecture. In [15] the Hrbacek set theory (HST) for nonstandard analysis was introduced. Thistheory is a modification of Hrabcek’s theory of external sets [10]. It is proved in Chapter 6 of [15]that the σ -algebra Bor ( H ) generated by the algebra of internal subsets of a hyperfinite set H canbe defined in HST. Thus, Theorem 4 can be formulated and it is reasonable to suggest that it canbe proved HST as well as the deduction of Theorem 3 from Theorem 4. It is proved in [15] thatevery theorem about internal sets that can be proved in HST can be proved in IST. We do notcomplete these arguments here, since in any case, these facts about HST are very non-trivial, so itis hard to expect the existence of a proof of Theorem 3 as simple as the one based on applicationof Loeb measures.In the previous paragraph we discussed the proof of the implicationProposition 7 = ⇒ Theorem 3 . (7)Let us look on this implication from the point of view of Hyperfinite Descriptive Set Theory(HDST). The basic definitions, notations and fact concerning HDST can be found e.g. in Chapter9 of [15].Let Y and M be as above and H = { , , ..., M } . Consider the set U = { ( y, N ) ∈ Y × ( H ∩ ∗ N ∞ ) | ∀ L ∈ ∗ N ∞ ( L ≤ N = ⇒ A L ( F, T, y ) ≈ A N ( F, T, y )) } (8)12t is easy to see that U is a Π -set.Implication (7) is equivalent to the implication µ L ( { y ∈ Y | ∃ N ∈ ∗ N ∞ ( y, N ) ∈ U } ) = 1 = ⇒ ∃ N ∈ ∗ N ∞ µ L ( { y ∈ Y | ( y, N ) ∈ U } ) = 1 . (9)The following proposition easily follows from Theorem 1.1 of [11] Proposition 9.
If an arbitrary Σ -set U ⊆ Y × ( H ∩ ∗ N ∞ ) satisfies the following property ∀ ( y, N ) ∈ U ∀ L ∈ ∗ N ∞ ( L < N = ⇒ ( y, L ) ∈ U ) , (10) then the implication (9) holds for U . Proof . Let π Y : Y × H → Y and π H : Y × H → H be canonical projections. Suppose that U isa Σ -set that satisfies the antecedent of the implication (9). Then, by Theorem 1.1 [11], for every n ∈ N there exist an internal set U n ⊆ U such that µ L ( π Y ( U n )) > − n . Let N n = min π H ( U n ).Then, by Lemma 3, there exists a infinite number N that is less, than all N n . Since U satisfies theproperty (10), this N satisfies the consequent of the implication (9). ✷ Unfortunately, the set U defined by formula (8) is a Π -set, but not a Σ -set.So, it is natural to ask the following Question.
Does an arbitrary Π -set U that has the property (10) satisfy the implication (9)?More generally, under what conditions a set U that has a property (10) satisfies the implication(9)?A sufficient condition is given by the following theorem, that may be of interest also by itself. Theorem 6.
Let X be an internal set and λ a finitely additive internal probability measure on X .Let f : X → ∗ N ∞ be a countably determined (CD) function. Then there exist a set S ⊆ X suchthat λ L ( S ) = 1 , where λ L is the Loeb measure associated with λ , and a number K ∈ ∗ N ∞ such that f ( x ) ≥ K for all x ∈ S . Lemma 4.
Let Y ⊆ X be an internal set and g : Y → ∗ N be an internal function. Then thereexists a λ L -null set Y ′ ⊆ Y and a number K ∈ ∗ N ∞ such that ∀ y ∈ Y \ Y ′ ( g ( y ) ∈ ∗ N ∞ = ⇒ g ( y ) ≥ K ) . (11) Proof of Lemma 4 . Let e Y = g − ( ∗ N ∞ ). Since e Y is λ L -measurable, there exists a sequenceof internal sets { Y n ⊆ e Y | n ∈ N } such that λ L ( e Y \ Y n ) ≤ n . Take Y ′ = T n ∈ N e Y \ Y n , so λ L ( Y ′ ) = 0. Since Y n ⊆ e Y , we know that ∀ y ∈ Y n g ( y ) ∈ ∗ N ∞ . Since Y n and g are internalthere exists K n ∈ ∗ N ∞ such that K n = min g ( Y n ). By Lemma 3 there exists K ∈ ∗ N ∞ such that ∀ n ∈ N K ≤ K n . Thus, for every y ∈ ( Y \ Y ′ ) ∩ e Y one has y ≥ K , since y ≥ K n for some n . ✷ Proof of Theorem 6 . Since f is a CD-function, by Theorem 9.4.7 (i) of [15] there exists asequence of internal functions { g n | n ∈ N } such that f ⊆ S n ∈ N g n . Without loss of generality wemay assume that each g n ⊆ X × ∗ N . For every n ∈ N let Y n = dom( g n ). Applying Lemma 4 to( Y n , g n ) one gets a λ L -null set Y ′ n ⊆ Y n and a number K n ∈ ∗ N ∞ such that ∀ y ∈ Y n \ Y ′ n ( g n ( y ) ∈ ∗ N ∞ = ⇒ g n ( y ) ≥ K n ) . Again, by Lemma 3 one obtains K ∈ ∗ N ∞ such that K ≤ K n for all n ∈ N .13et S = X \ S n ∈ N Y ′ n , so that λ L ( S ) = 1. Since S ⊆ X = S n ∈ N Y n , for any y ∈ S there exists n such that f ( y ) = g n ( y ) ≥ K n ≥ K . ✷ Recall that, if a set V ⊆ A × B , then a set W ⊆ V is said to be a uniformization of V , if W isa function and dom( V ) = dom( W ). If ν is a σ -additive measure on A and dom( V )∆dom( W ) is a ν -null set, then W is a ν -a.e. uniformization of V . The following corollary is clear. Corollary 2.
Let ( X, λ ) be the same as in Theorem 6. Let U ⊆ X × ∗ N ∞ satisfy the followingconditions:1. λ L ( { y ∈ X | ∃ N ∈ ∗ N ∞ ( y, N ) ∈ U } ) = 1 ,2. ∀ ( y, N ) ∈ U ∀ L ∈ ∗ N ∞ ( L < N = ⇒ ( y, L ) ∈ U ) .Then U has a countably determined λ L -a.e. uniformization if and only if ∃ N ∈ ∗ N ∞ λ L ( { y ∈ X | ( y, N ) ∈ U } ) = 1 . ✷ Unfortunately, the existence of CD-uniformizations is proved only for sets with Σ -cross-sections,and there are known examples of sets with Π -cross-sections without CD-uniformizations. So, thiscorollary is not applicable to the set U defined by (8). Standard versions of the previous results
While in nonstandard analysis we use the notion of an infinite number (hyperfinite set) as aformalization of the notion of a very big number (finite set), in classical mathematics we use thesequences of numbers (finite sets) diverging to infinity to formalize these notions. For example, inprevious sections we considered a hyperfinite set Y and its internal permutation T : Y → Y . If wewant to treat the same problems in the framework of standard mathematics, we have to considera sequence ( Y n , T n ) of finite sets Y n whose cardinalities tend to infinity and their permutations T n . Similarly, internal functions F : Y → ∗ R correspond to sequences F n : Y n → R in standardmathematics.First, we discuss what property of such sequences correspond to the property of an internalfunction F to be S -integrable. The following proposition gives a reasonable answer to this question. Proposition 10.
Let Y n be a standard sequence of finite sets, such that | Y n | = M n → ∞ as n → ∞ .Then for an arbitrary sequence F n : X n → R the following statements are equivalent:1. For every K ∈ ∗ N ∞ the function ∗ F K is S -integrable.2. lim n,k →∞ M n X { x ∈ Y n | | F n ( x ) | >k } | F n ( x ) | = 0 (12)The proof can be obtained easily by application of the Nelson ’s algorithm [20] to the statement(2).A sequence F n that satisfies the statement (2) of Proposition 10 is said to be uniformly integrable .Proposition 10 leads to establishing the standard version of Theorem 2.14 roposition 11. In conditions of Proposition 10 let T n : Y n → Y n be a sequence of transitivepermutations and F n : Y n → R be a uniformly integrable sequence. Consider two sequences ofnatural numbers K n and L n such that K n M n is bounded, lim inf K n M n > and lim n →∞ K n L n = 1 . Then thefollowing two statements are true.1. For any ε > one has lim n →∞ M n · |{ y ∈ Y n | | A K n ( F n , T n , y ) − A L n ( F n , T n , y ) | ≥ ε }| = 0 (13)
2. . If T n is a sequence of transitive permutations or F n is a sequence of bounded functions, then lim n →∞ max y ∈ Y n | A K n ( F n , T n , y ) − A L n ( F n , T n , y ) | = 0 (14)Though the proof of Theorem 2 can be easily rewritten in (standard) terms of Proposition 11,we deduce here this proposition from Theorem 2, keeping in mind the following much more difficultanalysis of the standard version of Theorem 3. By the same reason we do not use the Nelson’salgorithm in our proof. Proof . We restrict ourselves to the case of transitive permutations T n . Other cases can betreated in a similar way.Let F be a non-principal ultrafilter over N . Consider the nonstandard universe ∗ U that isthe ultrapower of the standard universe U by F ( ∗ U = U N / F ). Let Y = Y n F , T = T n F , F = F n F , M = M n F , K = K n F , L = L n F be the classes of corresponding sequences in ∗ U . Then Y is a hyperfinite set of infinite cardinality, since lim n →∞ M n = ∞ , thus lim F M n = ∞ , T : Y → Y is an internal transitive permutation on Y , F : Y → ∗ R is S -integrable by Proposition 10, ◦ (cid:0) KM (cid:1) = lim F K n M n > ◦ (cid:0) KL (cid:1) = 1, so ◦ (cid:0) KM (cid:1) = ◦ (cid:0) LM (cid:1) >
0. Thus, the introduced nonstandard objectssatisfy all conditions of Theorem 2 (2). So, one has ∀ y ∈ Y | A K ( F, T, y ) − A L ( F, T, y ) | ≈ y ∈ Y | A K ( F, T, y ) − A L ( F, T, y ) | ≈
0. Thus, ◦ (cid:18) max y ∈ Y | A K ( F, T, y ) − A L ( F, T, y ) | (cid:19) = lim F max y ∈ Y n | A K n ( F n , T n , y ) − A L n ( F n , T n , y ) | = 0 . Since, this is true for an arbitrary non-principal ultrafilter F the equality (14) is proved. ✷ Let us discuss a standard sequence version of Theorem 3. Assuming that { F n : Y n → R } isuniformly integrable sequence, | Y n | → ∞ and T n is a permutation on Y n , we write in IST thestatement that for every infinite N the triple ( Y N , T N , F N ) satisfies Theorem 3: ∀ N ∈ ∗ N ∞ ∀ st ε > ∃ A ⊆ Y N ∃ L ∈ ∗ N ∞ (cid:16) | A || Y N | ≥ − ε ∧ ∀ K ∈ ∗ N ∞ ( K ≤ L = ⇒ = ⇒ ∀ y ∈ A A K ( F N , T N , y ) ≈ A L ( F N , T N , y ))) (15)We call the statement 15 the IST-sequence version of Theorem 3Since only the ℵ -saturation is needed in the proof of Theorem 3, to obtain the standardsequence version of this theorem one needs to write the statement about the validity of (15) in theultrapower U F for an arbitrary non-principle ultrafilter F ⊆ P ( N ). We skip this simple exercise. Itis clear, however that the existence of infinite L satisfying certain conditions, means the existence15or every non-principal ultrafilter F its own sequence L n such that lim F L n = ∞ . So, in this case,unlike the Proposition 11, it is impossible to get rid of limits along ultrafilters. This makes thestandard version of Theorem 3 very unclear intuitively.The formalization of Proposition 7 in the framework of IST is not simpler than the sentence(15). To obtain this formalization one should only place the quantifier ∀ y ∈ A in (15) between thequantifiers ∃ A ⊆ Y N and ∃ L ∈ ∗ N ∞ .The IST-sequence version of Theorem 5 is much simpler, than those of Theorem 3. In sakeof shortness let us restrict ourselves by sequences of ergodic means of length | Y N | . It means thatwe consider only sequences { A ( F, T, y ) | Y | n =1 in Theorem 5. Using Theorem 2 it is not difficultto deduce the general case of Theorem 5 from this particular case. Let Y n , T n and F n be asabove. Denote the number of ε -fluctuations in the sequence { A n ( F N , T N , y ) } | Y N | n =1 by F l ( ε, N, y ).Let U ( k, ε, N ) = { y ∈ Y N | F l ( ε, N, y ) ≤ k } . Notice, that U ( k, ε, N ) is an internal set.Then it is easy that Theorem 5 is equivalent to the following IST-sentence ∀ N ∈ ∗ N ∞ ∀ st ε > ∀ st δ > ∃ st k | Y N | | U ( k, ε, N ) | ≥ − δ Using Nelson’s algorithm it is easy to obtain the following standard sequence version of Theorem 5
Theorem 7.
Let Y n be a standard sequence of finite sets, such that | Y n | = M n → ∞ as n → ∞ ,let T n : Y n → Y n be a sequence of transitive permutations and let F n : Y n → R be a uniformlyintegrable sequence. Then ∀ ε, δ > ∃ N, k ∀ n > N M n | U ( k, ε, n ) | ≥ − δ. ✷ It would be interesting to obtain estimates concerning fluctuations of ergodic means for thistheorem similar to those obtained in [13].
Let X be a compact metric space with a metric ρ . It is known that for every ξ ∈ ∗ X there existsa unique standard element x ∈ X such that x ≈ ξ . Thus, the external map st : ∗ X → X such that ∀ ξ ∈ ∗ X st ( ξ ) ≈ ξ is defined. The map st (we also use the notation st X , if necessary) is calledthe standard part map. For a hyperreal number ξ ∈ ∗ R obviously one has ◦ ξ = st( ξ ). The map st is defined in this case not for all hyperreal numbers but only for bounded ones, since R is not acompact space. Definition 2.
Let X be a compact metric space, ν be a Borel measure on X and ϕ : Y → ∗ X bean internal map such that st ◦ ϕ : Y → X is a measure preserving map. We say that in this case ( Y, ϕ ) is a hyperfinite approximation (h.a.) of the measure space ( X, ν ) . In case of Y ⊆ ∗ X and the identical embedding Y we say that Y is a h.a. of ( X, ν ). Obviously,any h.a. (
Y, ϕ ) is equivalent to the h.a. ϕ ( Y ). 16f R is a complete separable metric space and f : X → R is a Borel measurable function, then f ◦ st ◦ ϕ : Y → R is a µ L -measurable function. A lifting F : ∗ X → ∗ R of the function f ◦ st ◦ ϕ issaid to be a lifting of the function f . By Proposition 3 every Borel measurable function f : X → R has a lifting.To formulate the standard version of this definition introduce the following notation. Let Z ⊆ X be a finite subset of X and δ Z = | Z | P z ∈ Z δ z , where δ Z is a Dirac measure at a point z ∈ Z , i.e. δ Z is a Borel probability measure such that for any Borel set A ⊆ X one has δ z ( A ) = 1 ⇐⇒ z ∈ A . Definition 3.
In conditions of Definition 2 let { Y n | n ∈ N } be a sequence of finite subsets of X .We say that the sequence Y n approximates the measure space ( X, ν ) if the sequence of measures δ Y n converges to the measure ν in the *-weak topology on the space M ( X ) of all Borel measures on X . Proposition 12.
In conditions of Definition 3 suppose that every open ball in X has the positivemeasure ν and every set of the positive measure ν is infinite. Then for every set A ⊆ X with ν ( A ) = 1 there exists a sequence Y n of finite subsets of X approximating the measure space ( X, ν ) such that ∀ n ∈ N Y n ⊆ A . Proof . Since X is a compact metric space, the space C ( X ) of all continuous functions on X isseparable. Then the space M ( X ) of all Borel measures on X is separable in the *-weak topology.By the Krein - Milman Theorem the convex set of probability measures in M ( X ) is the closure ofthe convex combinations of its extreme points, which are Dirac measures. So, the set of all convexcombinations with rational coefficients is dense in the set of probability measures. It is enoughto show that for any finite set E = { z , . . . , z k } ⊆ X , for any natural numbers { n , . . . , n k } , forany finite set of continuous functions { f , . . . , f m } ⊆ C ( X ) and for any ε >
0, there exists a set Y = { y , . . . , y n } ⊆ X , where n = n + ... + n k , such that ∀ i ≤ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X j =1 n j n f i ( z j ) − Z X f i dδ Y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε. (16)Since functions f i are uniformly continuous on X there exists σ > ∀ i ≤ m ∀ x , x ∈ X ρ ( x , x ) < σ = ⇒ | f i ( x ) − f i ( x ) | < ε and σ < min { ρ ( u, v ) | u, v ∈ E, u = v } . Then B σ ( z i ) ∩ B σ ( z j ) = ∅ for i = j . By the conditions of our proposition each set B σ ( z i ) ∩ A isinfinite. So, there exist a set Y = { y , . . . , y n , y n +1 , . . . , y n , . . . , y n k − +1 , . . . , y n k } ⊆ A such that y , . . . , y n ∈ B σ ( z ); y n +1 , . . . , y n ∈ B σ ( z ); . . . ; y n k − +1 , . . . , y n k ∈ B σ ( z k ). Obviouscalculation shows that Y satisfies (16) ✷ . Proposition 13.
A sequence Y n ⊆ X approximates a measure space ( X, ν ) in the sense of Defi-nition 3 if and only if for any N ∈ ∗ N ∞ the set Y N is a hyperfinite approximation of the measurespace ( X, ν ) . roof. = ⇒ Let Y n approximates ( X, ν ) and N ∈ ∗ N ∞ . Then for any f ∈ C ( X ) one has Z f (st( x )) dµ L = ◦ | Y N | X y ∈ Y N ∗ f ( y ) = Z X f dν (17)The first equality is due to ∗ f is a lifting of f ◦ st. The second follows from Definition 3 and fromthe nonstandard analysis definition of the limit of a sequence. Now st : Y N → X defines a measure ν ′ on X that is the image of the Loeb measure of Y . Due to (17) and the Riesz representationtheorem ν ′ = ν . ⇐ = Assume that st ↾ Y N : Y N → R is a measure preserving transformation for every N ∈ ∗ N ∞ .It is easy to see that for every function f ∈ C ( X ) the internal function ∗ f ↾ Y N is a lifting of f . So, ◦ | Y N | X y ∈ Y N ∗ f ( y ) = Z Y N ◦ ( ∗ f ) dµ L = Z Y N f ◦ st dµ L = Z X f dν. Thus, the equality (17) holds for every N ∈ ∗ N ∞ and by the nonstandard analysis definition of alimit one has lim n →∞ R X f dδ Y n = R X f dν ✷ Propositions 12 and 13 imply the following
Proposition 14.
Let ν be a non-atomic Borel measure on a compact metric space X such thatthe measure of every ball is positive. Then for every set A ⊆ X such that ν ( A ) = 1 there exists ahyperfinite set Y ⊆ ∗ A such that ( Y, st ) is a h.a. of ( X, ν ) . As it was mentioned in Introduction, in conditions of Proposition 14 the measure space (
X, ν ) isisomorphic modulo measure 0 to the measure space ([0 , , dx ), where dx is the standard Lebesguemeasure. This means that there exist a set B ⊆ X a set C ⊆ [0 ,
1] and a bijective map ψ : B → C such that dx ( C ) = ν ( B ) = 1 and the maps ψ, ψ − are measure preserving. Lemma 5.
In conditions of the previous paragraph let Y be a h.a. of ( X, ν )) . Then for every set D ⊆ [0 , with dx ( D ) = 1 there exists a bijective lifting G : Y → ∗ [0 , of the map ψ such that1. Z = G ( Y ) ⊆ ∗ D ;2. Z is a h.a. of ([0 , , dx ) .3. G − : Z → ∗ X is a lifting of ψ − . Proof . Let F : Y → ∗ [0 ,
1] be a lifting of ψ . Let σ = min { ρ ( u, v ) | u, v ∈ F ( Y ) , u = v } .Then 0 < σ ≈ ∀ u ∈ F ( Y ) B σ ( u ) ∩ F ( Y ) = { u } . Since ∗ ν ( B σ ( u )) > dx ( D ) = 1 the set B σ ( u ) ∩ ∗ D contains infinitely many points, and thus, there exists an internal set E u ⊆ B σ ( u ) ∩ ∗ D such that | E u | = | F − ( u ) | . Establishing bijection between F − ( u ) and E u for every u ∈ F ( Y ),we obtain the bijection G : Y → Z ⊆ ∗ D that is a lifting of ψ . Notice that since G and G − arebijections they are measure preserving maps between measure spaces ( Y, µ YL ) and ( Z, µ ZL ).To prove the second property of the set Z , one needs to show that st [01] ↾ Z : Z → [0 ,
1] is ameasure preserving map, i.e. that for every measurable set A ⊆ [0 ,
1] one has µ ZL ( st − , ( A ) ∩ Z ) = dx ( A ) (18)18ne has µ ZL ( st − , ( A ) ∩ Z ) = µ YL ( { y ∈ Y | G ( y ) ∈ st − , ( A ) } ) = µ YL ( { y ∈ Y | ◦ G ( y ) ∈ A } ) . (19)Since G is a lifting of ψ on has ◦ G ( y ) = ψ ( st X ( y )) for µ YL -almost all y . Thus, µ YL ( { y ∈ Y | ◦ G ( y ) ∈ A } ) = µ YL ( { y ∈ Y | ψ ( st X ( y )) ∈ A } ) = ν ( ψ − ( A )) = dx ( A ) , (20)since st X ↾ Y : Y → X and ψ : B → C ⊆ [0 ,
1] are measure preserving maps. The equality (18)follows from the equalities (19) and (20).To prove the third property of the set Z it is enough to show that st Y ( G − ( z )) = ψ − ( st [0 , ( z ))for µ ZL -almost all z ∈ Z . Since ψ is a bijection, the last equality is equivalent to the equality ψ ( st Y ( G − ( z ))) = st [0 , ( z ), which follows from the following sequence of equalities that hold for µ ZL -almost all z ∈ Z : ψ ( st Y ( G − ( z )) = st [0 , ( G ( G − ( z ))) = st [0 , ( z ) . ✷ Definition 4.
Let X be a compact metric space, ν be a Borel measure on X , τ : X → X bea measure preserving transformation of X and ( Y, ϕ ) be a h.a. of ( X, ν ) . Then we say that aninternal permutation T : X → X is a h.a. of the transformation τ if the following diagram Y T −−−−→ Y ϕ y ϕ y ∗ X ∗ τ −−−−→ ∗ X (21) is commutative µ L -a.e. We say also that the dynamical system ( Y, µ L , T ) is a h.a. of the dynamicalsystem ( X, ν, τ ) .In case of Y ⊆ ∗ X and and an identical imbedding ϕ the diagram (21) means that T is a liftingof τ i.e. ◦ T ( y ) ≈ τ ( ◦ y ) µ L -a.e. Theorem 8.
For every dynamical system ( X, ν, τ ) such that the measure space ( X, ν ) satisfiesthe conditions of Proposition 14, and for every h.a. Y of ( X, ν ) there exists a h.a. ( Y, µ L , T ) .Moreover, one can choose a h.a. T of τ to be a transitive permutation. Proof.
I. Here we prove the existence of a h.a. (
Y, µ L , T ) of the dynamical system ([0 , , dx, τ ).Let Y be an arbitrary h.a. of the measure space ([0 , , dx ). Let F : Y → ∗ [0 ,
1] be a lifting of τ .First we prove the following statement.(A) For every standard δ > there exists a permutation T δ : Y → Y such that |{ y ∈ Y | | F ( y ) − T δ ( y ) | < δ }| M ≈ . We deduce (A) from the Marriage Lemma. Fix a standard δ > y ∈ Y set S ( y ) = ∗ ( F ( y ) − δ, F ( y ) + δ ) ∩ Y . Let I be an arbitrary internal subset of Y . Set S ( I ) = S y ∈ I S ( y ) and B ( I ) = S y ∈ I ∗ ( F ( y ) − δ, F ( y ) + δ ). So, S ( I ) = B ( I ) ∩ Y . The internal set B ( I ) can be representedas a union of a hyperfinite family of disjoint intervals. Since the length of each of these intervals is19ot less than 2 δ , their number is actually finite. Let B ( I ) = n S i =1 ( ξ i , η i ), where intervals ( ξ i , η i ) arepairwise disjoint and n is standard.Consider the standard set C = n S i =1 ( ◦ ξ i , ◦ η i ). Then dx ( C ) µ L (st − ( ∗ C )). Obviously,st − ( ∗ C )∆ B ( I ) ⊆ n S i =1 (M( ◦ ξ i ) ∪ M( ◦ η i )) = M( ∂C ), where the monad of a number a ∈ [0 ,
1] is denotedby M( a ). Since the Loeb measure of the monad of any number is equal to 0 and so, M( ∂C ) = 0,one has dx ( C ) = ◦ (cid:16) | S ( I ) | M (cid:17) . Substituting [0 ,
1] for X , dx for ν and τ for ψ in (20) obtain dx ( C ) = dx ( τ − ( C )) = µ L ( F − (st − ( C ))). Since I \ F − (st − ( C )) ⊆ M( ∂C ), one has ◦ (cid:16) | I | M (cid:17) ≤ ◦ (cid:16) | S ( I ) | M (cid:17) .This means that if r I = max { , | I | − | S ( I ) |} , then r I M ≈
0. Let r = max { r I | I ⊆ Y } . Fix anarbitrary set Z ⊆ ∗ [0 , \ Y such that | Z | = r . For every y ∈ Y set S ′ ( y ) = S ( y ) ∪ Z and for anarbitrary I ⊆ Y set S ′ ( I ) = S y ∈ Y S ′ ( y ). Then S ′ ( I ) = S ( I ) ∪ Z , | S ′ ( I ) | = | S ( I ) | + r ≥ | I | , since | I | − | S ( I ) | = r I ≤ r . By the Marriage Lemma there exists an injective map θ : Y → S ′ ( Y ) = Y ∪ Z such that ∀ y θ ( y ) ∈ S ′ ( y ). Obviously | θ − ( Z ) | = | Y \ θ ( Y ) | ≤ r . So, there exists a bijective map λ : θ − ( Z ) → Y \ θ ( Y ). Define T δ : Y → Y by the formula T δ ( y ) = (cid:26) θ ( y ) , y ∈ Y \ θ − ( Z ) λ ( y ) , y ∈ θ − ( Z )Notice that | θ − ( Z ) | ) M ≤ rM . By construction of T δ one has ∀ y ∈ Y \ θ − ( Z ) | T δ ( y ) − τ ( y ) | < δ . Since µ L ( θ − ( Z )) ≤ rM ≈
0, the statement (A) is proved.Let S ( Y ) be the set of all internal permutations of Y . Consider the external function f : N →S ( Y ) such that f ( n ) = T n . By ℵ -saturation the function f can be extended to an internal function¯ f : { , . . . , K } → S ( Y ) for some K ∈ ∗ N ∞ . Internal function g ( n ) = |{ y ∈ Y | | F ( y ) − ¯ f ( y ) |≥ n }| M assumesonly infinitesimal values for all standard n . By Robinson’s Lemma there exists L ∈ ∗ N ∞ such that g ( L ) ≈
0. set T = ¯ f ( L ). Then µ L ( { y ∈ Y | T ( y ) ≈ F ( y ) } ) = 1. Since F is a lifting of τ , the sameis true also for T ( y ). This proves I.We have to prove now that a h.a. T of τ can be chosen as a cycle of maximal length.II. Fix a permutation T : Y → Y that is a h.a. of τ and represent it by a product of pairwisedisjoint cycles, including the cycles of length 1 (fix points): T = ( y ...y n )( y ...y n ) ... ( y b ...y bn b ) , (22)where y ij ∈ Y is the j -th element in the i -th cycle and b is the number of cycles. So, b X i =1 n i = M = | Y | . (23)We assume also that n ≥ n ≥ · · · ≥ n b . Consider the cycle C = ( y ...y n y ...y n ...y b ...y bn b ) (24)By (23) C is a cycle of length M , i.e. a transitive permutation.20et B = { y ∈ Y | C ( y ) = T ( y ) } . | B | = b = M X n =1 a n , (25)where a n is the number of cycles of length n .III. Recall that a point x ∈ [0 ,
1] is said to be an n -periodic point of the transformation τ ifits orbit under this transformation consists of n -points: x, τ x, . . . , τ n − x . A point x is said to be τ -periodic if it is n -periodic for some n . The transformation τ is said to be aperiodic if the set ofperiodic points has measure zero. It is well-known that every measure preserving automorphism τ of a Lebesgue space X defines the partition of this space by τ -invariant Lebesgue subspaces ofaperiodic and n -periodic points. So, it is enough to prove our statement for the case of aperiodictransformation τ and for the case of n -periodic transformation τ .Suppose that the transformation τ is aperiodic. Let us prove that under this assumption thecycle C defined in the part II is a h.a. of τ .Let P n ( T ) ⊆ Y be the set of all n -periodic points of T and let P n ( τ ) ⊆ X be the set of all n -periodic points of τ . Since T is a lifting of τ it is easy to that for every standard k the followingrelations T ( y ) ≈ τ ( ◦ y ) , . . . , T k ( y ) ≈ τ k ( ◦ y ) (26)hold µ L -a.e. on Y . So, for every standard n P n ( T ) ⊆ st − ( P n ( τ )) up to a set of the Loeb measurezero. Since dx ( P n ( τ )) = 0, one has M | P n ( T ) | ≈
0. Obviously, | P n ( T ) | = na n . Thus, for everystandard n one has M · a n ≈
0. By the Robinson’s Lemma there exists an infinite N such that M N P n =1 a n ≈ . . Obviously M ≥ M P n = N +1 a n · n ≥ ( N + 1) M P n = N +1 a n . So, M M P n = N +1 a n ≤ N +1 ≈ M · | B | = M M P n =1 a n ≈ . Thus, µ L ( B ) = 0, C ( y ) = T ( y ) µ L -a.e. and C approximates τ .IV. Suppose now that τ is n -periodic. We prove first that a h.a. T of τ also can be chosen to be n -periodic. The relations (26) imply that for almost every point y ∈ Y if y has a standard periodwith respect to T , then this period is a multiple of n . Indeed, if y satisfies (26), and its standardperiod is nq + r for 0 < r < n , then ◦ y = ◦ T nq + r ( y ) = τ nq + r ( ◦ y ) = τ r ( ◦ y ), which is impossiblesince τ is n -periodic. By ℵ -saturation, there exist an internal set I ⊆ Y such that µ L ( I ) = 1 anda number N ∈ ∗ N ∞ such that for every point y ∈ I , whose period is less, than N , this period is amultiple of n .Consider the representation (22) of T and set n i = nq i + r i , r i < n for each i ≤ b . Let Y ′ ⊆ Y be the set obtained by deleting from Y the last r i elements of the i-th cycle for each i ≤ b . Theset Y ′ has the Loeb measure equal to 1. Indeed, all the deleted elements either belong to the set Y \ I , whose measure is 0, or to a cycle whose length is greater, than N . The number of thesecycles does not exceed MN and the number of deleted points in each such cycle is less, than n . Sothe Loeb measure of the set of these points is also equal to 0. Since µ L ( Y ′ ) = 1 the pair ( Y ′ , st) isa h.a. of [0 , Y ′ defines also the permutation T ′ : Y ′ → Y ′ such that T ′ = ( y ...y n · q )( y ...y n · q ) ... ( y b ...y b n · q b ) . (27)Notice, that actually the number of cycles in T ′ may be less, than b , since in case of q i = 0 the i -th cycle is empty. However, the dynamical system ( Y ′ , µ L , T ′ ) is a h.a. of the dynamical system( X, ν, τ ). Indeed, let D = { y ∈ Y ′ | T ( y ) = T ′ ( y ) } . Then D ⊆ { y ∈ Y ′ | T ( y ) ∈ Y \ Y ′ } ⊆ − ( Y \ Y ′ ). Thus, µ L ( D ) ≤ µ L ( Y \ Y ′ ) = 0 To obtain an n -periodic h.a. of τ it is enough to spliteach cycle in the representation (27) in cycles of length n . Indeed, let the obtained cycle be T ′′ = ( z , ..., z n )( z n +1 , ..., z · n ) . . . ( z ( i − · n +1 , ..., z i · n ) . . . ( z ( K − · n +1 , ..., z K · n ) , where K = | Y ′ | /n . It is easy to see that T ′′ ( y ) = T ′ ( y ), only for the points z i · n . Notice, that µ L ( { z i · n | i ≤ K } ) = n > n -periodicity of τ , for almost all of thesepoints one has T ′ ( z i · n ) = z in +1 ≈ τ ( ◦ z i · n +1 ) = τ n ( ◦ z ( i − · n +1 ) = ◦ z ( i − · n +1 . At the same time T ′′ ( z i · n ) = z ( i − · n +1 by the definition. Thus, T ′′ ( y ) ≈ T ′ ( y ) for almost all y .V. To complete the proof of the theorem for X = [0 ,
1] we need to consider the case when allorbits of T have the same standard period n . In this case M = N · n .It is easy to see that there exists a selector I ⊂ Y (see the proof of Lemma 1) that is dense in ∗ [0 , M ( I ) = ∗ [0 , A of such intervals, thereexists a selector that intersects each interval from A . The existence of a dense selector follows fromthe ℵ -saturation.Let I = { y < y < · · · < y N } be a dense selector. Here < is the order in ∗ [0 , I in ∗ [0 ,
1] for every k < N one has y k ≈ y k +1 . Obviously, the transformation T can berepresented by a product of pairwise disjoint cycles as follows: T = ( y , ..., T n − y )( y , ..., T n − y ) . . . ( y N , ..., T n − y N )Consider the following cycle S of the length M : S = ( y , ..., T n − y y , ..., T n − y . . . y N , ..., T n − y N )Since for every k ≤ N holds T n ( y k ) = y k , one has ◦ S ( T n − ( y k )) = ◦ y k +1 = ◦ y k = ◦ T n ( y k ) = ◦ T ( T n − ( y k )) = τ ( ◦ T n − ( y k ))for almost all k . Thus, ◦ S ( y ) = τ ( ◦ y ) for almost all y and the cycle S is a h.a. of τ .We proved actually that for every h.a. Y of ([0 , , dx ) there exists an internal set Y ′ ⊆ Y with µ L ( Y ′ ) = 1 and a permutation T ′ : Y ′ → Y ′ such that the hyperfinite dynamical system( Y ′ , µ L , T ′ ) is a h.a. of the dynamical system ( X, ν, τ ) and T ′ is a transitive permutation of Y ′ (see Part IV of this proof). To obtain a transitive h.a. T : Y → Y of τ , set T ′ = ( z , ..., z | Y ′ | ) and Y \ Y ′ = { u , ..., u | Y \ Y ′ | } and consider the cycle of the length | Y | T : ( z , ..., z | Y ′ | , u , ..., u | Y \ Y ′ | )Since µ L ( { y ∈ Y | T ′ ( y ) = T ( y ) } ) = 0 the transformation T is h.a. of τ .VI. The statement of the theorem for the case of an arbitrary dynamical system ( X, ν, τ )satisfying the conditions of Proposition 14 follows immediately from Lemma 5. Indeed, let a set B ⊆ X , a set C ⊆ [0 , ψ : B → C and a bijective lifting G : Y → ∗ [0 ,
1] of ψ satisfy the conditions of Lemma 5. Then λ = ψτ ψ − : [0 , → [0 ,
1] is a measure preservingtransformation. Fix an arbitrary h.a. Y of the measure space ( X, ν ). Then by Lemma 5 thehyperfinite set Z = G ( Y ) is a h.a. of ([0 , , dx ). By the results proved in the parts I-V, there22xists a permutation S : Z → Z that is a h.a. of λ . Then it is easy to see that the permutation T = G − SG : Y → Y is a h.a. of τ . Obviously, if S is a transitive permutation, then T is atransitive permutation as well. ✷ The following corollary of Theorem 8 is obvious.
Corollary 3.
For every Lebesgue dynamical system ( X, T, ν ) , there exists a transitive Loeb dynam-ical system ( Y, T ) and a measure preserving map ψ : Y → X . In other words, every Lebesgue dynamical system is a homomorphic image of an appropriatetransitive Loeb dynamical system.
Proof . It is enough to take any transitive h.a. of (
X, T, ν ) for (
Y, T ) and st : Y → X for themap ψ ✷ Corollary 3 immediately allows to deduce the ergodic theorem for Lebesgue spaces from ergodictheorem for Loeb space. This was done in the paper [14]. Theorem 2 of [14] is stronger thanCorollary 3. It states that each Lebesgue dynamical is a homomorphic image of any transitiveLoeb dynamical system.On the other hand Theorem 2 of [14] does not imply (at least, immediately) the existence of ahyperfinite approximation of a Lebesgue dynamical system. The notion of a hyperfinite approxima-tion has a simple standard interpretation (see the end of this section) and might be of interest byitself, in particular, from the point of view of computer simulations of dynamical systems. Hyper-finite approximations of many important dynamical systems can be easily described (see examples5 – 7 below), while Loeb preimages of these systems obtained by Theorem 2 of [14] do not haveany simple description.In what follows we say that a number N ∈ ∗ N is M -bounded , if the number NM is bounded. Forany y ∈ Y and any k ∈ ∗ N the set { T k y, T k +1 y, ..., T k + N − y } is said to be an N -segment of theorbit of y . The N segment { y, T y, ..., T N − y } is said to be initial.It is easy to see that even a transitive dynamical system ( Y, T ) is never ergodic. Indeed, forany y ∈ Y and N ∈ ∗ N ∞ such that 0 < ◦ (cid:0) NM (cid:1) < N -segment A of the orbit of y is obviouslya T -invariant set, since µ L ( A △ T ( A )) = 0). Every transitive system is equivalent to a systemconsidered in the examples 1-4. Examples 3 and 4 show, that for every transitive system ( Y, µ L , T )there exist an S -integrable function F and numbers N, K ∈ ∗ N ∞ such that NM , KM are bounded and A N ( F, T, y ) A K ( F, T, y ) for all y in some set of a positive Loeb measure.In the few following propositions and examples 5 - 7 we discuss the behavior of ergodic means A N ( F, T, y ) for M -bounded N ∈ ∗ N ∞ in the case, when a Loeb dynamical system ( Y, µ L , T ) isa h.a. of a Lebesgue dynamical system ( X, ν, τ ) and F is an S -integrable lifting of a function f ∈ L ( ν ).The following proposition is an easy corollary of Theorem 3. Proposition 15.
In conditions of the previous paragraph let ˜ f = lim n →∞ A n ( f, τ, x ) and let e F be an S -integrable lifting of ˜ f , then there exists an N ∈ ∗ N ∞ such that µ L -a.e. ∀ K ∈ ∗ N ∞ ( K < N ⇒ A K ( F, T, x ) ≈ e F ( x )) . Corollary 4.
Let T be a transitive permutation and let τ be a non-ergodic transformation. Considera function f ∈ L ( ν ) such that the set B ⊆ X of all x ∈ X satisfying inequality lim n →∞ A n ( f, τ, x ) = Av ( f ) has a positive measure ν . Then there exist infinite M -bounded N, K such that for almost all y ∈ st − ( B ) one has A N ( F, T, y ) A K ( F, T, y ) . roof . Let e f = lim n →∞ A n ( f, τ, · ) and e F be the same as in Proposition 15. By this propositionthere exists N ∈ ∗ N ∞ such that NM ≈ A N ( F, T, y ) ≈ e F ( y ) µ L -a.e. Thus, A N ( F, T, y ) Av ( f )for µ L -almost all y ∈ st − ( B ).On the other hand, since T is a cycle of length M , by Theorem 2 one has A K ( F, T, y ) ≈ Av ( F ) ≈ Av ( f ) for all y ∈ Y and for all K such that KM ≈
1. Thus, A K ( F, T, y ) A M ( F, T, y ) for µ L -almostall y ∈ st − ( B ). ✷ Let X be a compact metric space. Consider a hyperfinite set Y ⊆ ∗ X . This set defines a Borelmeasure ν Y on X by the formula ν Y ( K ) = µ L ( st − ( K ∩ Y )). Obviously Y is a h.a. of the measurespace ( X, ν Y ). Let T : Y → Y be an internal permutation that is S -continuous on A for some (notnecessary internal) set A ⊆ Y with µ L ( A ) = 1, i.e. ∀ a , a ∈ A ( a ≈ a = ⇒ T ( a ) ≈ T ( a )) . (28)Notice that since st − ( st ( A )) ⊇ A and µ L ( A ) = 1, the set st( A ) ⊆ X is a measurable set w.r.t. thecompletion of the measure ν Y , which we denote by ν Y also, and ν Y (st( A )) = 1.Define a map τ T : X → X such that τ T (st( y )) = st( T ( y )) for y ∈ A and τ T ↾ X \ st( A ) is anarbitrary measurable permutation of the set X \ st( A ). Proposition 16.
The map τ T preserves the measure ν Y . Proof . Replacing, if necessary, A by T n ∈ N T n ( A ) we may assume that A is invariant for permu-tation T . Then, obviously, st( A ) is invariant for τ T .Consider a closed set B ⊆ X . We have to prove that ν Y ( τ − T ( B )) = ν Y ( B ). One has ν Y ( τ − T ( B )) = ν Y ( τ − T ( B ) ∩ st( A )) . It is easy to check that τ − T ( B ) ∩ st( A ) = st( T − (st − ( B )) ∩ A ) . Thus, ν Y ( τ − T ( B )) = µ L (cid:0) st − (st( T − (st − ( B )) ∩ A )) (cid:1) = µ L (cid:0)(cid:0) st − (st( T − (st − ( B )) ∩ A )) (cid:1) ∩ A (cid:1) Using (28) and the T -invariance of it is easy to check, thatst − (st( T − (st − ( B )) ∩ A )) ∩ A = T − (st − ( B )) ∩ A ) . So, ν Y ( τ − T ( B )) = µ L ( T − (st − ( B )) ∩ A )) = µ L ( T − (st − ( B )) = µ L (st − ( B )) = ν Y ( B ) . In the last chain of equalities we used the facts that µ L ( A ) = 1 and that T being a permutationpreserves the Loeb measure. ✷ Proposition 17.
1) In conditions of Proposition 16 for any a > and for any y ∈ Y the followingpositive functional l a ( · , T, y ) on C ( X ) is defined: l a ( f, T, y ) = ◦ A K ( ∗ f, T, y ) , where ◦ (cid:0) KM (cid:1) = a and f ∈ C ( X )
2) If ∀ K, L ∈ ∗ N ∞ ( KM ≈ LM ≈ ⇒ A K ( ∗ f, T, y ) ≈ A L ( ∗ f, T, y ) ), then l ( f, T, y ) is defined bythe same formula as in 1). In this case l ( f, T, y ) = e f ( y )
3) If T : Y → Y is S -continuous, then the functional l ( · , T, y ) is τ T -invariant for all y ∈ Y . roof . The correctness of the definition in 1) follows from Theorem 2. The statement 2)follows from Proposition 7. To prove statement 3) notice that if T is S -continuous on Y , then τ T is continuous on X and, thus, ∗ ( f ◦ τ T ) ↾ Y is a lifting of f ◦ τ T . So, ∀ y ∀ K ∈ ∗ N ∗ ( f ◦ τ T )( T K ( y )) ≈ f ( τ T ( ◦ T K ( y ))) = f ( ◦ T K +1 ( y )) ≈ ∗ f ( T K +1 y )These equivalences allows to prove that A K ( ∗ ( f ◦ τ ) , T, y ) ≈ A K ( ∗ f, T, y ) (see the proof of Propo-sition 5). ✷ Recall that a continuous transformation τ : X → X is said to be uniquely ergodic if there existsonly one τ -invariant Borel measure on X . Theorem 9. If τ is a uniquely ergodic transformation of a compact metric space X , Y ⊆ ∗ X is a hyperfinite set such that st ( Y ) = X , and T : Y → Y is an internal permutation such that ∀ y ∈ Y st ( T ( y )) = τ ( st ( y )) , then for every y ∈ Y such that the τ -orbit of st ( y ) is dense in X , forevery N ∈ ∗ N ∞ and for every f ∈ C ( X ) one has A N ( ∗ f ↾ Y, T, y ) ≈ Z X f dν, (29) where ν is the τ -invariant measure. Proof . Let y ∈ Y satisfy conditions of the theorem. For a number K ∈ ∗ N we denote theinitial K -segment of the T -orbit of Y by S ( K, y ). Then for any K ∈ ∗ N ∞ one has st( S ( K, y )) = Y ,since the closed set st( S ( K, y )) contains the τ -orbit of st( y ). Let K be a T -period of y . Then K ∈ ∗ N ∞ . Otherwise, the τ -orbit of st( y ) would be finite, while we assume X to be infinite. It iseasy to see that it is enough to prove the theorem for every N ∈ ∗ N ∞ such that N ≤ K . Underthis assumption all elements of the set Y = { y, T y, ..., T N − y } are distinct. Since st( Y ) = X , theset Y defines the Borel measure ν Y on X . Let T : Y → Y be the permutation of Y that differsfrom T only for one element T N − y : T ( T N − y ) = y . Set A = Y \ { T N − y } . Then X , τ , Y , T ,and A satisfy conditions of Proposition 16: µ L ( A ) = 1, ∀ z ∈ A st( T z ) = τ (st( z )), i.e. τ T = τ and T is S -continuous on A , since τ is a continuous map. By Proposition 16 the measure ν Y is τ -invariant. Thus, ν Y = ν due to the unique ergodicity of the map τ . If f ∈ C ( X ), then obviously ∗ f ↾ Y is an S -integrable lifting of f . This proves the equality (29). ✷ Example 4 . Let Y = { , m , M , . . . , M − M } and T : Y → Y is defined by the formula T (cid:0) kM (cid:1) = k ⊕ M , where ⊕ is the addition modulo M . Then ( Y, st) is a hyperfinite approximation of the segment[0 ,
1] with the Lebesgue measure, and the map T is a h.a. of the identical map id : [0 , → [0 , Y, µ L , T ) is isomorphic to the one, considered in Examples 1 - 3up to the trivial isomorphism k kM . The function F ( k ) of Example 3 is a lifting of the function f ( x ) = x , which is the function f of Example 3. Obviously ∀ n, x A n ( f, id, dx ) = f ( x ). So, in thisexample ∀ N ∈ ∗ N ∞ ( NM ≈ ⇒ A N ( F, T, y ) ≈ e F ( y )) µ L -a.e. Example 5 . Let α ∈ R . Denote by S ( α ) the shift of the interval [0 ,
1] by α , i.e. the dynamicalsystem D = ([0 , , τ, dx ), where τ ( x ) = x ⊕ α and ⊕ is the addition modulo 1. In this example weconstruct two distinct h. a. of the dynamical system S ( ). Similar considerations are appropriatefor an arbitrary rational shift of the unit interval.Let Y be the same as in the previous example and M = 2 N and let T : Y → Y be given by theformula: T ( y ) = y ⊕ NM . It is easy to see that ∀ y ∈ Y T ( y ) = τ ( y ) and if y , then τ ( ◦ y ) = ◦ τ ( y ). Krylov-Bogoljubov theorem states the existence of at least one τ -invariant measure. µ L ( Y ∩ M( )) = 0, the dynamical system ( Y, µ L , T ) is a h.a. of the system S ( ). It is easyto see that for an arbitrary S -integrable function F on Y , for every N ∈ ∗ N ∞ and for every y ∈ Y one has A N ( F, T, y ) ≈ ( F ( y ) + F ( y ⊕ )) (=, if N is an even number). In particular, if F is an S -integrable lifting of a function f ∈ L (0 ,
1) then for almost all y ∈ Y and for all N ∈ ∗ N ∞ onehas A N ( F, T, y ) ≈
12 ( f ( ◦ y ) + f ( ◦ y + 12 )) = lim n →∞ A n ( f, τ, ◦ y ) = e f ( ◦ y ) . (30)In this case every point y ∈ Y is periodic with the period 2. By Theorem 8 there exists atransitive h.a. of S ( ). In this case it can be easily constructed explicitly.Let ( Y, µ L , T ) be the same as above, but M = 2 N + 1. Then it is easy to see that if y = kM , then T ( y ) = k + N ( mod M ) M . So, T is isomorphic to a permutation of { , , ..., M − } given by the formula k k + N ( mod M ). Since gcd( N, M ) = 1, this permutation is transitive. It is easy to see thatthe transformation T is a h.a. of τ . Indeed, consider the set B = [0 , \ { } . Then τ is continuouson B . Let A = st − ( B ) = Y \ (M(0) ∪ M(1) ∪ M( )). Then µ L ( A ) = 1 and A is invariant for T .An easy calculation show that ∀ y ∈ A ∗ τ ( y ) − T ( y ) = M . Since ∗ [0 , \ (M(0) ∪ M(1) ∪ M( )) isinvariant for ∗ τ , one has T ( y ) ≈ τ ( y ) ≈ τ ( ◦ y ). Moreover, it is easy to see that, if y = kM ∈ A , thenfor every K ∈ ∗ N such that k − KM ∈ A one has ∗ τ K ( y ) − T K ( y ) = K M (31)So, if KM ≈
0, then ∀ n ≤ K the set A is invariant for T n and ∀ y ∈ Y ◦ T n ( y ) ≈ τ n ( ◦ y ) µ L -a.e.Let f : [0 , → R be a standard continuous function, then obviously ∗ f ↾ Y is an S -integrablelifting of f . Assume first that f satisfies the Lipschitz condition. Then, using the equality (31), oneimmediately obtains that if KM ≈
0, then ∀ y ∈ A A K ( ∗ f, T, y ) ≈ A K ( ∗ f, τ, y ) ≈ e f ( ◦ y ) (see formula(30)). For any K ∈ ∗ N denote l K ( · , T, y ) the positive functional on C ( X ) given by the formula l K ( f, T, y ) = ◦ A K ( ∗ f, T, y ). Since in the case of KM ≈ f satisfying the Lipschitzcondition l K ( f, T, y ) = e f ( ◦ y ) and such functions are dense in C ( X ), we see that the functional l ( · , T, y ) of Proposition 17 is defined. The formula (30) shows that l ( · , T, y ) = 12 ( δ ◦ y + δ ◦ y ⊕ . )Using formula (31) one can easily obtain explicit expressions for l a ( · , T, y ) for positive a ∈ R .These expressions depend on relations between a and ◦ y . For example, if 0 . a < ◦ y < .
5, then l a ( f, T, y ) = 1 a ◦ y Z ◦ y − . a f ( t ) dt + ◦ y +0 . Z ◦ y +0 . − . a f ( t ) dt . We see that l a → λ as a →
0. It easy to see also that for all a > y ∈ A the functional l a is τ -invariant, though τ is not continuous everywhere on [0 , Example 6 . Let τ be a shift of [0 ,
1] by an irrational number α > τ is uniquely ergodic. Consider M, N ∈ ∗ N ∞ such that gcd( M, N ) = 1 and NM ≈ α . Considerthe same finite space Y as in Example 5. Let T : Y → Y be given by the formula T ( y ) = y + NM (mod 1). By the same reasons as in Example 6 the permutation T is a cycle of length M and T is a h.a. of τ . Thus, for every S -continuous function F : Y → ∗ R , K ∈ ∗ N ∞ and y ∈ Y one has A K ( F, T, Y ) ≈ Av ( F ). Compare with Example 5.26 xample 7 . (Approximations of Bernoulli shifts). Let Σ m = { , , . . . m − } . Consider thecompact space X = Σ Z m with the Tychonoff topology. Let a be a function, such that dom( a ) ⊂ Z is finite, and range( a ) ⊆ Σ m . Let S a = { f ∈ X | f ↾ dom( a ) = a } . Then the family of all such S a form a base of neighborhoods of the compact space X . For g ∈ ∗ X set f = g ↾ Z , then f ∈ X andit is easy to see that f = st( g ).The continuous transformation τ : X → X defined by the formula τ ( f )( n ) = f ( n + 1) where f ∈ X and n ∈ Z is an invertible Bernoulli shift. Every probability distribution { p , . . . , p m − } ( p i > , m − P i =0 p i = 1) on Σ m defines a Borel measure on X that is obviously invariant with respectto τ . It is well-known that τ is ergodic for each of these measures. So, the transformation τ is notuniquely ergodic. Here we restrict ourselves only to the case of the uniform distribution on Σ m ,i.e. to the case of p = · · · = p m − = m . The corresponding Borel measure on X is denoted by ν .Obviously, ν ( S a ) = m −| dom ( a ) | .We construct here two hyperfinite approximations of the dynamical system ( X, ν, τ ). Firstwe consider the straightforward approximation by a hyperfinite shift. Fix N ∈ ∗ N ∞ and set Y = Σ {− N,...,N } m . Then M = | Y | = m N +1 . Define λ : Y → ∗ X as follows. For y ∈ Σ {− N,...,N } m set λ ( y )( n ) = (cid:26) y ( n ) , | n | ≤ N , | n | > N (32)Then st ◦ λ ( y ) = y ↾ Z . Thus, for every standard neighborhood S a defined above one has ( st ◦ λ ) − ( ∗ S a ) = { y ∈ Y | y ↾ dom( a ) = a } . So, µ L ((st ◦ λ ) − ( ∗ S a )) = ν ( S a ) = m − dom ( a ) . This provesthat ( Y, λ ) is a h.a. of (
X, ν ).Certainly, an arbitrary internal map from Y to Σ ∗ Z \{− N,...,N } m can be used to define the values λ ( y )( n ) for | n | > N and y ∈ Y in the definition of λ (32)In what follows we use notations y ≈ y and st( y ) for λ ( y ) ≈ λ ( y ) and st( λ ( y )) respectively.Define the map S → S by the formula S ( y )( n ) = y ( n + 1( mod 2 N + 1)) for any y ∈ Y and n ∈ {− N, . . . , N } . Then obviously τ (st( y )) = st( S ( y )) for all y ∈ Y . So, ( Y, λ, S ) is a h.a. of thedynamical system (
X, ν, τ ).Since every point y ∈ Y is (2 N + 1)-periodic with respect to S the permutation S is nottransitive. Though the existence of a transitive h.a. of τ is proved in Theorem 8, it is not easy toconstruct such an approximation explicitly.To do this we reproduce here the construction of de Bruijn sequences. Definition 5. An ( m, n ) -de Bruijn sequence on the alphabet Σ m is a sequence s = ( s , s , . . . , s L − ) of L = m n elements s i ∈ Σ m such that all consecutive subsequences ( s i , s i ⊕ , . . . , s i ⊕ n − ) of length n are distinct.Here and below in this example the symbols ⊕ and ⊖ denote + and - modulo L , so that thesequence s is considered as a sequence of symbols from Σ n placed on a circle. It was proved [4, 5] that there exist ( m !) m n − · m − n ( m, n )-de Bruijn sequences. See also [6] fora simple algorithm for de Bruijn sequences and more recent references.To construct a transitive h.a. T : Y → Y of τ fix arbitrary ( m, N + 1) de Bruijn sequence s = ( s , s , . . . , s M − ) here L = M . Let y = ( y − N , . . . , y − N ) ∈ Y . Then there exists the uniqueconsecutive subsequence σ ( y ) = ( s i , s i ⊕ , . . . , s i ⊕ N ) such that y j = s j ⊕ i ⊕ N ). Set P ( σ ( y )) =( s i ⊕ , . . . , s i ⊕ N ⊕ ) and T ( y ) = σ − ( P ( σ ( y ))). Notice that if i < M − N , then for all j ≤ N one27as s j ⊕ i ⊕ N = s j + i + N . So, T ( y ) j = y j +1 for all j < N and, thus, for all standard j . This lastequality implies that st( T ( y )) = τ (st( y )) for all y ∈ Y such that the first entry of the sequence σ ( y ) is the i -s term of the initial de Bruijn sequence for i ≤ L − N −
1. So, µ L ( { y | st( T ( y )) = τ (st( y )) } ) ≥ M − NM ≈
1. This proves that T is a h.a. of τ . We call T a de Bruijn approximation of τ . It is interesting to study the behavior of ergodic means of described approximations. Thisproblem will be discussed in another paper. We confine ourselves with two simple remarks.1. If σ ( y ) = h s i , . . . , s i +2 N and i < M − N , then A n ( F, T, y ) = A n ( F, S, y ) for all n < N .2. Let S = { f ∈ X | f (0) = 1 } , so that ν ( S ) = and let χ be a characteristic function of S . For y ∈ Y let f = st( y ). Set A ( y ) = f − ( { } ) ∩ N ). Recall that the density of A ( y ) is given bythe formula d ( A ( y )) = lim m →∞ | A ( y ) ∩ { , . . . , m − }| m It is easy to see that for m < N one has A m ( ∗ χ , T, y ) = | A ( y ) ∩ { , . . . , m − }| m . So, for all y ∈ Y such that the density d ( A ( y )) exists one has ∃ K ∈ ∗ N ∞ ∀ m ∈ ∗ N ∞ ( m ≤ K = ⇒ A m ( χ , T, y ) ≈ d ( A ( y )). Due to Proposition 15 there exist K ∈ ∗ N ∞ such that for µ L -almost all y ∈ Y one has A m ( ∗ χ , T, y ) ≈ . Standard version of the notion of a hyperfinite approximation of a dynamical system.
We use the same approach as above to formulate a sequence version of the notion of a hyperfiniteapproximation of a dynamical system.
Definition 6.
Let ( X, ρ ) be a compact metric space, ν be a Borel measure on X , τ : X → X be ameasure preserving transformation of X , { Y n ⊆ X | n ∈ N } be a sequence of finite approximation ofthe measure space ( X, ν ) in the sense of Definition 3 and T n : Y n → Y n be a sequence of permutationsof Y n . We say that a sequence T n is an approximating sequence of the transformation τ if for every N ∈ ∗ N ∞ the internal permutation T N : Y N → Y N is a h.a. of τ in the sense of Definition 4. In thiscase we say that the sequence of finite dynamical systems ( Y n , µ n , T n ) approximates the dynamicalsystem ( X, ν, τ ) . Here µ n is a uniform probability measure on Y n . The reformulation of this definition in full generality in standard mathematical terms is prac-tically unreadable. However, it is easy to reformulate it for the case of an almost everywherecontinuous transformation τ . This case covers a lot of important applications.Denote the set of all points of continuity of the map τ : X → X by D τ . Lemma 6.
Suppose that ν ( D τ ) = 1 and let Y ⊆ X be a h.a. of the measure space ( X, ν ) . Thena permutation T : Y → Y is a h.a. of the transformation τ if and only if for every positive ε ∈ R one has | Y | ( |{ y ∈ Y | ρ ( T ( y ) , ∗ τ ( y )) > ε }| ) ≈ . (33)28 roof (= ⇒ ) Let A = { y ∈ Y | ◦ y ∈ D τ } = st − ( D τ ), B = { y ∈ Y | T ( y ) ≈ τ ( ◦ y ) } . Then, µ L ( A ) = 1 since Y is a h.a. of the measure space ( X, ν ) and ν ( D τ ) = 1. Since T is a h.a. of τ , onehas µ L ( B ) = 1. Thus, µ L ( A ∩ B ) = 1. Since τ is continuous on ∆ τ one has ∀ x ∈ X ◦ x ∈ D τ = ⇒ ∗ τ ( x ) ≈ τ ( ◦ x ) . (34)So, ∀ y ∈ A ∩ B ∗ τ ( y ) ≈ τ ( ◦ y ) and thus, ∀ y ∈ A ∩ B ∗ τ ( y ) ≈ T ( y ). So, for every positive ε ∈ R one has { y ∈ Y | ρ ( T ( y ) , ∗ τ ( y )) > ε } ⊆ Y \ ( A ∩ B ). This proves (33).( ⇐ =) Suppose that (33) holds for every positive ε ∈ R . Then obviously µ L ( { y ∈ Y | T ( y ) ≈ ∗ τ ( y ) } ) = 1. On the other hand, by (34) one has µ L ( { y ∈ Y | ∗ τ ( y ) ≈ τ ( ◦ y ) } ) = 1. Thus, µ L ( { y ∈ Y | T ( y ) ≈ τ ( ◦ y ) } ) = 1, i.e. T is a h.a. of τ ✷ Lemma 6 implies immediately the following
Proposition 18 (Standard version of Definition 6) . In conditions of Definition 6 and Lemma 6the sequence of permutations T n : Y n → Y n is an approximating sequence of the transformation τ if and only if for every positive ε ∈ R one has lim n →∞ | Y n | ( |{ y ∈ Y n | ρ ( T n ( y ) , τ ( y )) > ε }| ) = 0 . ✷ (35) Theorem 10.
Let ( X, ρ ) be a compact metric space and ν be a Borel measure on X such that themeasure space ( X, ν ) satisfies the conditions of Proposition 14. Then for every measure preservingtransformation τ : X → X with ν ( D τ ) = 1 there exist a sequence of finite sets Y n ⊆ X and asequence of permutations T n : Y n → Y n such that the sequence of finite dynamical systems ( Y n , T n ) approximates the dynamical system ( X, ν, τ ) in the sense of Definition 6. Moreover, one can choosetransitive permutations T n . Proof.
Let Y n ⊆ X be a sequence that approximates the measure space ( X, ν ) in the sense ofDefinition 3. Such sequence exists by Proposition 12. Then by Proposition 13 for any N ∈ ∗ N ∞ the set Y N is a h.a. of the measure space ( X, ν ) in the sense of Definition 2. By Theorem 8 thereexists a (transitive) permutation T N : Y N → Y N that is a h.a. of the transformation τ . By Lemma6, since ν ( D τ ) = 1, this means that ( Y N , T N ) satisfies (33) for every standard positive ε . In thisproof the letter T maybe with lower indexes always denotes a (transitive) permutation.For every numbers n, m ∈ N define the set A n,m = (cid:26) k ∈ N | ∃ T : Y k → Y k (cid:18) | Y k | · (cid:12)(cid:12)(cid:12)(cid:12) { y ∈ Y k | ρ ( T ( y ) , τ ( y )) > n } (cid:12)(cid:12)(cid:12)(cid:12) < m (cid:19)(cid:27) . Since ∀ N ∈ ∗ N ∞ N ∈ ∗ A n,m , there exists a standard function N ( n, m ) such that ∀ k >N ( n, m ) k ∈ A n,m . By the definition of sets A m,n , there exists a standard function T ( k, n, m ) : Y k → Y k with the domain { ( n, m, k ) ∈ N | k > N ( n, m ) } such that1 | Y k | · (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) y ∈ Y k | ρ ( T k ( y ) , τ ( y )) > n (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) < m . Now it is easy to see that if r = N ( n, n ) + n , then the sequence ( Y r , T r ) satisfies the conditionsof Proposition 18 ✷ eferences [1] S. Albeverio, J.E. Fenstad, R. Hoeg-Krohn, T. Lindstrom, Nonstandard Methods in StochasticAnalysis and Mathematical Physics,
Academeic Press, Orlando, 1986.[2] M. Brin, G. Stuck,
Itroduction to Dynamical Systems,
Cambridge University Press, 2002.[3] I.P. Cornfeld, S.V. Fomin, and Ya.G. Sinai,
Ergodic Theory
Springer-Verlag, New York, 1982.[4] N.G. de Bruijn, A combinatorial problem,
Koninklijke Nederlandse Akademie Wetenschappen ,1946, , pp. 758 - 764.[5] T. van Aardenne-Ehrenfest, N.G. de Bruijn, Circuits and trees in oriented linear graphs, Bull.Belgium Math. Soc. - Simon Stevin , 1951, , pp. 203 - 217.[6] A.M. Alhakim, A Simple Combinatorial Algorithm for de Bruijn sequences, American Math.Monthly , 2010, , pp. 728 - 732.[7] E.I. Gordon,
Nonstandard Methods in Commutative Harmonic Analysis , American Mathe-matical Society, Providence, RI, 1977.[8] E.I. Gordon, C.W. Henson, P.A. Loeb. On representation of dynamical systems on Lebesguespaces by hyperfinite dynamical systems.
Abstarcts of Papers Presented to the AMS , 2004, , p. 62.[9] E.I. Gordon, A.G. Kusraev and S.S. Kutateladze, Infinitesimal Analysis , Kluwer AcademicPublishers, Dordrecht-Boston-London, 2002.[10] K. Hrbacek, Axiomatic foundations for nonstandard analysis,
Fundamenta Mathematicae , pp. 1 - 19.[11] C.W. Henson, D.Ross, Analytic mappings of hyperfinite sets, Proc. Amer. Math. Soc. , 1993, , pp. 587 - 595.[12] A.G. Kachurovskii, Fluctuations of means in the Birkhoff-Khinchin ergodic theorem,
SiberianAdvances in Math. , pp. 103 - 144.[13] A.G. Kachurovskii, Rates of convergence ub ergodic theorems, Ruusian Math Surveys , pp. 650 - 703[14] T. Kamae, A simple proof of the ergodic theorem using nonstandard anlysis, Isr. J. Math. , pp. 284 - 290.[15] V. Kanovei, M. Reeken, Nonstandard Analysis, Axiomatically , Springer-Verlag, Berlin-Heidelberg, 2004[16] Y. Katznelson, B. Weiss, A simple proof of some ergodic theorems,
Isr. J. Math. ,pp. 291 - 296.[17] H.J. Keisler, K. Kunen, A. Miller, and S. Leth, Descriptive set theory over hyperfinite sets, J. Symbolic, Logic 1989, pp. 1167 - 1180.3018] P.A. Loeb and M.P.H. Wolff, Nonstandard Analysis for the Working Mathematician , KluwerAcademic Publishers, Dordrecht-Boston-London, 2000.[19] E. Nelson,
Radically Elementary Probability Theory , Annals of Mathematics Studies, Prince-ton University Press, 1987.[20] E. Nelson, Internal set theory: a new approach to nonstandard analysis,
Bull. Amer. Math.Soc.83