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General Countable Product Measures

A new construction of product measures is given for an arbitrary sequence of measure spaces via outer measure techniques giving a coherent extension of the classical theory of finite product measures to countable many. Moreover, the Lebesgue spaces of this measures are simplified in terms of finite product measures. This decomposition simplifies all the considerations regarding infinite dimensional integration and gives to it a computational framework.


Introduction
The classical theory of product measures deals with two measure spaces (X, Σ X , µ X ) and (Y, Σ Y , µ Y ) in order to construct the product measure space (X×Y, Σ X ⊗Σ Y , µ X ⊗µ Y ), where Σ X ⊗ Σ Y is the σ-algebra generated by R X×Y := {A × B : A ∈ Σ X , B ∈ Σ Y } and µ X ⊗ µ Y is a measure on Σ X ⊗ Σ Y satisfying the identity The most common method to prove the existence of this measure is through the celebrated Caratheodory extension theorem as follows. Denote by U(R X×Y ) the family of finite unions of elements of R X×Y , then it is easy to verify that U(R X×Y ) is an algebra of subsets of X × Y and that every element of U(R X×Y ) can be written as a finite union of pairwise disjoint members of R X×Y . Define the set function . It is classical (see e.g. [3,6]), that the set function µ 0 is well defined and σ-additive on U(R X×Y ). Hence by Caratheodory extension theorem, there exists a measure µ on the σ-algebra Σ X ⊗ Σ Y that extends µ 0 . Therefore, µ satisfies identity (1) for each A ∈ Σ X and B ∈ Σ Y .
Consider now a sequence {(Ω i , Σ i , µ i )} i∈N of measure spaces. Our aim is to generalize the classical product measure space to countable many, i.e., to construct the infinite prod- Ω i : C i ∈ Σ i , ∀i ∈ {1, 2, ..., m} and m ∈ N and i∈N µ i is a measure satisfying an analogue of identity (1) for this general setting. For instance, if the finiteness condition i∈N µ i (Ω i ) ∈ [0, +∞) holds, the measure i∈N µ i must satisfy the identity i=m+1 Ω i ∈ R(Σ i ) i∈N . The first first attempt to address this problem was for the particular case of probability spaces. In 1933, A. Kolmogoroff proved in [12] the existence of a probability measure i∈N m [0,1] on the measurable space [0, 1] N , i∈N B [0,1] , where B [0,1] and m [0,1] are the Borel σ-algebra of [0, 1] and the Lebesgue measure of [0, 1] respectively, such that identity (2) holds for every Kolmogoroff's proof was based on the compactness of the product space [0, 1] N . More general cases were discussed by Z. Lomnicki and S. Ulam in 1934 on the reference [14]. In 1943, S. Kakutani generalized for general probability spaces the results of Kolmogoroff, Lomnicki and Ulam proving in [10] the next celebrated result. Theorem 1.1. Given {(Ω i , Σ i , µ i )} i∈N a family of probability spaces, there exists an unique probability measure i∈N µ i on the measurable space ( × i∈N Ω i , i∈N Σ i ) satisfying for Kakutani's proof of this result has become standard in Probability and Measure Theory.
The key tool of the proof relies on a result of E. Hopf (cf. [8], [19,Theorem 3.2]). In 1996, S. Saeki gives in [17] a new proof of Theorem 1.1, proving it in a more natural terms without the use of Hopf's result. If the measure spaces {(Ω i , Σ i , µ i )} i∈N are not of probability but satisfy the finiteness condition i∈N µ i (Ω j ) ∈ [0, +∞), then normalizing each measure space, it can be also proven as a rather direct consequence of Theorem 1.1, the existence of a measure on ( × i∈N Ω i , i∈N Σ i ) satisfying identity (2).
On other hand, if i∈N µ i (Ω i ) = +∞, identity (2) is no longer useful for the construction of infinite product measures since its value is always infinite. Nevertheless, we can ask the measure to verify the new identity Therefore, the natural extension of the classical theory for the nonfinite case is the product The first purpose of this article is to provide a construction of this product space for an arbitrary family of measure spaces {(Ω i , Σ i , µ i )} i∈N . This matter has not been ignored in the last century and several attempts have been made trying to formalize it.
In 1963, E.O. Elliott and A.P. Morse published a paper [5] constructing this kind of product spaces through a reformulation of the classical infinite product called plus product. Let a = (a n ) n∈N be a sequence of positive real numbers and A (a) = {n ∈ N : a n > 1}, then they defined the plus product of the sequence (a n ) n∈N by n∈N + a n := n∈A (a) a n · n / ∈A (a) a n , with the convention 0 · ∞ = ∞ · 0 = 0 and setting the value of the empty product to 1. The main purpose of defining this concept lies in the fact that the plus product exists for every positive sequence (a n ) n∈N of real numbers, which does not happen with the classic product (take for instance the sequence defined by a n := 2 if n ≡ 0 mod 2 1 2 if n ≡ 1 mod 2. for each n ∈ N). The classical and the plus product do not, in general, coincide, but if the condition n∈A (a) a n < +∞ is satisfied, they coincide. We define the set of finite plus rectangles by Elliot and Morse proved that given a family of measure spaces It must be observed that if a finite plus rectangle C = × i∈N C i ∈ F + (Σ i , µ i ) i∈N satisfies + i µ i (C i ) = 0, then its plus product must coincide with the classical product i µ i (C i ). However, if + i µ i (C i ) = 0, the products does not, in general, coincide. In consequence, this result does not stablish the existence of the required product measure since there are substantial sequence satisfying + i µ i (C i ) = 0 with nonzero classical product (take for instance the sequence a n = exp((−1) n+1 /n) for each n ∈ N).
In 2004 R. Baker, proved in [2, Theorem I], the existence of the required product measure for the particular case in which the involved spaces {(X i , d i )} i∈N are locally compact metric and satisfy the property D: (D) For every i ∈ N and δ > 0, there exists a sequence {A j } j∈N ⊂ B X i such that d i (A j ) < δ for each j ∈ N and where d i (A j ) stands for the diameter of A j in X i and B X i for the borel σ-algebra of X i .
He proved that given a sequence {(X i , d i )} i∈N satisfying the previous properties and a corresponding sequence of regular Borel measure spaces {(X i , B X i , µ X i )} i∈N , there exists a measure λ B on the measurable space ( × i∈N X i , i∈N B X i ) satisfying for every C = × i∈N C i ∈ F (B X i , µ X i ) i∈N , the following identity Finally, in 2011 G.R. Pantsulaia presented in [15,Theorem 3.10] the best generalization of product measures to countable many till the date. He proved the following.
This result provides a standard prove of the existence of the product measure without imposing any topological assumption over the involved spaces. However, the measure spaces (Ω i , Σ i , µ i ) must be σ-finite for every i ∈ N and the measure λ P is not constructed via outer measure techniques, that it is not a necessary condition but it always facilitates several computations like Fubini theorem's proof.
This paper is organized as follows. In section one, we give a construction of the product measure i∈N µ i satisfying identity (3), via outer measure techniques without imposing any condition on the underlying measure spaces. This result will extend Theorem 1.2 establishing the best extension of the classical theory till the date. In section three and four we simplify the structure of Lebesgue spaces of this measure in terms of finite product measures via a theorem due to B. Jessen [9]. Roughly speaking, we decompose This decomposition allows to consider infinite dimensional functions as a sequence of finite dimensional ones and stabilises a computational method to compute the integral of this kind of maps. Finally, thanks to the previous considerations, in section five we construct a measure for each infinite dimensional separable Banach space X following the work of T. Gill, A. Kirtadze, G. Pantsulaia and A. Plichko [7] and we state an analogue of the cited decomposition for Lebesgue spaces of this measure.

Existence of the Product Measure
In this section, given an arbitrary sequence of measure spaces {(Ω i , Σ i , µ i )} i∈N , we will prove the existence of a measure i∈N µ i on the measurable space ( × i∈N Ω i , i∈N Σ i ) satisfying the identity A key fact we will use is the existence of this measure for the particular case in which the involved measure spaces {(Ω i , Σ i , µ i )} i∈N satisfies the finiteness condition i µ i (Ω i ) ∈ [0, +∞). More precisely, there exists a measure i∈N µ i The existence of this measure is a rather direct consequence of Theorem 1.1. It is straightforward to verify that for this measure, identity (5) also holds.
Let {(Ω i , Σ i , µ i )} i∈N be an arbitrary family of measure spaces. We define the volume map vol : The principal tool that is going to be used in the construction of the required product measure will be the next result, that establishes that the vol map is a good choice for the extension of the classical volume formula.
Before stating the proof of Theorem 2.1, we will give two lemmas.
Let (a n ) n∈N , (b n ) n∈N be two real positive sequences such that (1) a n ≤ b n for each n ∈ N.
(2) n∈N b n ∈ [0, +∞) then Moreover, Proof. If n∈N b n = 0, the result is evident. Suppose n∈N b n ∈ (0, +∞). Then, since (a n /b n ) n∈N ⊂ [0, 1], the partial products are monotone and bounded by 1. In consequence n∈N an bn ∈ [0, +∞). Using elementary properties of the limit This concludes the proof of (8). For (9) just note that the partial products satisfies n i=1 a n ≤ n i=1 b n for each n ∈ N.
there exists a measure i∈N µ C i on the measurable space (C , i∈N Σ C i ) satisfying for It must be observed that the set map vol is well defined over Let us begin proving item (1) and (2). In this cases C n ⊂ C for each n ∈ N and Since i∈N µ C i is a measure on (C , i∈N Σ C i ) and it coincides with the map vol over F (Σ C i , µ C i ) i∈N , we deduce identities (6) and (7). Finally, suppose the hypothesis of case (3) holds. It is apparent that By Lemma 2.3, it becomes apparent that C ∩C n ∈ F (Σ i , µ i ) i∈N and vol(C ∩C n ) ≤ vol(C n ) for each n ∈ N. Therefore, by (11) and (12), we deduce This concludes the proof.
For the prove of the existence of a product measure for arbitrary family of measure spaces for every A ∈ P( × i∈N Ω i ). If no cover exists, we set µ * (A) := +∞. We will prove that this outer measure is, in fact, a measure on the σ-algebra i∈N Σ i and that it satisfies identity (5) for each C ∈ F (Σ i , µ i ) i∈N . We start stating some useful lemmas.
Proof. It is clear that the union of the right hand of (14) is disjoint.
⊆ Let (c n ) n∈N ∈ C c , then (c n ) n∈N / ∈ C and there exists n 0 ∈ N such that c n 0 / ∈ C n 0 . Thus c n 0 ∈ C c n 0 . Define the subset I (cn)n = {n ∈ N : c n / ∈ C n } = {n ∈ N : c n ∈ C c n } ⊂ N. By the last argument, I (cn)n = ∅. Let N 0 = min I (cn)n , then clearly This concludes the proof.
Proof. First, denote therefore it is apparent that Theorem 2.6 (Measurability). The family of subsets R(Σ i ) i∈N is µ * -measurable.
Proof. Take C ∈ R(Σ i ) i∈N and B ∈ P( × i∈N Ω i ). Consider ε > 0 and a family {B n } n∈N ⊂ If the cover {B n } n∈N does not exist, then necessarily µ * (B) = +∞ and the inequality follows. Suppose such cover exists, then by Lemma 2.4, there exists {C i } i∈N ⊂ R(Σ i ) i∈N pairwise disjoint of the form By Lemma 2.5, it follows that B n ∩ C , B n ∩ C i ∈ F (Σ i , µ i ) i∈N and therefore, by Theorem 2.1 item (1), we arrive to the identity Hence, by equation (15) and the definition of the outer measure µ * , (13), it becomes apparent that The last step follows from the following inclusion and the definition of infumum Taking ε 0, inequality (16) holds for every B ∈ P( × i∈N Ω i ) and therefore C is µ *measurable.
In Theorem 2.6 we have proved that R(Σ i ) i∈N is a subfamily of the Caratheodory σalgebra C and therefore by Caratheodory extension theorem, µ * defines a measure on Finally, we will prove that the outer measure µ * satisfies identity (5) for each C ∈ F (Σ i , µ i ) i∈N . Proposition 2.7 (Volume). For each C ∈ F (Σ i , µ i ) i∈N , the following identity holds By Theorem 2.1 item (3), we deduce the following inequality Then, taking the infimum over all covers, we stablish that vol(C ) ≤ µ * (C ). Finally, considering the particular cover {C n } n∈N ⊂ F (Σ i , µ i ) i∈N defined by which implies vol(C ) = µ * (C ). This finished the proof.
In conclusion, we have proved the next existence theorem. The last statements of this result are proven analogously to those of [2, Theorem I].
Theorem 2.8. Let {(Ω i , Σ i , µ i )} i∈N be a family of measure spaces, then there exists a measure

Decomposition Theorem for Finite Measures
In this section L p of infinite product measures are studied and simplified in terms of L p of finite product ones, for the particular case in which the finiteness condition is satisfied. Let {(C i , Σ i , µ i )} i∈N be a family of measure spaces satisfying the finiteness condition i µ i (C i ) ∈ (0, ∞). We use the notation C i instead of Ω i to emphasize that the involved measures satisfies the required condition. Consider the measure space C , i∈N Σ i , i∈N µ i , where C = × i∈N C i and i∈N µ i is the measure constructed in the last section in terms of the outer measure (13). In this section we will deal with the space To simplify the notation, throughout this paper we will use unstintingly the notations L p (C ) and Moreover, for each n ∈ N, we will use the notation C n to denote × n i=1 C i . Consider for 1 ≤ p < ∞, the vector space Moreover, in the definition of lim n L p (C n ), we are identifying for every n < m, S m n (f ) ≡ f , where S m n is the linear embedding S m n : L p (C n ) ֒ L p (C m ) defined by S m n (f ) : C m − R (ω 1 , ω 2 , · · · , ω m ) f (ω 1 , ω 2 , · · · , ω n ) for each f ∈ L p (C n ). Hereinafter, we will use this identifications. If we consider the functional lim n ∞ f n Lp(C n ) then, for each 1 ≤ p < ∞, the pair (lim n L p (C n ) , · lim Lp ) defines a normed space. The main result of this section states that the next spaces are isometrically isomorphic for 1 ≤ p < ∞ and consequently we would have the embedding that allows us to consider functions defined in spaces of infinite dimensions as a sequence of finite dimensional ones.
3.1. Preliminary Lemmas. We will give some lemmas concerning some dense subspaces of L p (C ) and lim n L p (C n ) that will be useful for the proof of (17). Throughout this section, we will identify for each n ∈ N, the spaces Proof. By definition, we have Consequently since R(Σ i ) i∈N is an algebra of subsets, by Lemma 3.4.6 of [4], the space Span χ R : R ∈ R(Σ i ) i∈N is dense in L p (C ). Finally, due to the inclusion the result follows.
, ∀n ≥ N , then for each 1 ≤ p < ∞, the subspace N ∈N F N p is dense in lim n L p (C n ). Proof. Let (f n ) n∈N ∈ lim n L p (C n ) and consider the sequence (F m ) m∈N ⊂ lim n L p (C n ) defined by and since (f n ) n∈N ∈ lim n L p (C n ), it is apparent that Thus we have proved Consequently since (F m ) m∈N ⊂ N ∈N F N p , we conclude that the subspace N ∈N F N p is dense in lim n L p (C n ).
Another key result will be the Jessen's theorem. This theorem was proved by B. Jessen in 1934 in his PhD Thesis [9] and nowadays is proved by martingales techniques [18,Theorem 7.16]. However, Jessen proved the result only for probability spaces. We need a version for the general case in which the involved spaces (C i , Σ i , µ i ) are not necessarily of probability. This version is given in the next result which proof can be adapted mutatis mutandis from the proofs of Theorems II, III and IV of Baker's paper [2]. It must be observed that the extra conditions imposed by Baker on the measure spaces are not used in the proofs of Theorems II, III and IV.
x ∈ C and also with convergence in L p (C ).
x ∈ C and also with convergence in L p (C ).

Decomposition Theorem.
Once the necessary machinery has been presented, we proceed to prove the identification (17).
Theorem 3.4. Let 1 ≤ p < ∞ and consider the operators Then T is an isometric isomorphism with S = T −1 . In particular, L p (C ) is isometrically isomorphic to lim n L p (C n ), Proof. Firstly, note that given (f n ) n∈N ∈ lim n L p (C n ), the sequence is Cauchy in L p (C ) since by Fubini's theorem Thus T is well defined and it is linear. On the other hand, by Theorem 3.
By Theorem 3.3, the next sequence converges and consequently for n < m, we have We have proved that or equivalently S(f ) ∈ lim n L p (C n ). This establishes that S is well defined and that it is linear. Once we have proved the consistency of the linear operators T and S, we will prove that they are isometries. Take (f n ) n∈N ∈ lim n L p (C n ), then Thus, T is an isometry. On the other hand, if f ∈ L p (C ), we compute where the last step follows from Theorem 3.3 item (3). Therefore, S is also an isometry. Finally we will see that S = T −1 . Consider (f n ) n∈N ∈ N ∈N F N p and let g ∈ L p (C M ) such that f n = g n i=M +1 µ i (C i ) for n ≥ M, then if we denote g n = g(x n−1 , ω n−1 ) for each n ∈ N, we have The last steps are justified by the equivalence relation defined on lim n L p (C n ). Thus, we Since N ∈N F N p is dense in lim n L p (C n ) and S • T is continuous (since it is an isometry), it is apparent that On the other hand, consider f ∈ n∈N L p (C n ). Then f ∈ L p (C M ) for some M ∈ N and if we denote f n = f (x n−1 , ω n−1 ) for each n ∈ N, we have Consequently, if f ∈ n∈N L p (C n ), it is apparent that Since n∈N L p (C n ) is dense in L p (C ) and T • S is continuous, we conclude that . This finishes the proof.

Decomposition Theorem for Infinite Measures
In this section, we will give the analogue of Theorem 3.4 for the case of the sequence of measure spaces {(R, B R , m R )} i∈N where B R is the borel σ-algebra of R and m R is the Lebesgue measure of R. Observe that this sequence does not satisfy the finiteness condition since i m R (R) = +∞ and therefore Theorem 3.4 cannot be applied directly. Consider the corresponding product measure space Let us denote R N := × i∈N R. It should be observed that i∈N B R = B R N where we are considering on R N the product topology. Moreover, it becomes apparent that the measure i∈N m R is, in fact, the Baker measure λ B , constructed in [2, Theorem I]. Thus, we will denote the measure i∈N m R by λ B . For the sake of notation, hereinafter, we will also denote for each a = (a n ) n∈N ∈ N and m ∈ N [a n , a n + 1), where N := Z N . The next result simplifies considerably the integration issues on the measure space (R N , B R N , λ B ).
Proof. Suppose that O f is uncountable and let F be a finite subset of O f . Therefore, since Thus, recalling the definition of uncountable sum, we arrive at for each a ∈ O f , by Proposition 0.20 of [6], and this gets a contradiction since f ∈ L 1 (R N , B R N , λ B ). Therefore O f is countable. Now, we prove identity (18).
e. for each a ∈ N \O f . Hence, since This concludes the proof.
Define for each a = (a n ) n∈N ∈ N , the translation consisting in all sequence f = (f a ) a satisfying (1) f a ∈ L p ([0, 1) N , λ B ) for each a ∈ N .
(2) f a = 0 λ B -a.e. for each a ∈ N \O f for some countable subset O f ⊂ N .
Moreover, given f ∈ L p (R N , B R N , λ B ), there exists a countable subset O f ⊂ N such that Proof. Fix 1 ≤ p < ∞ and define On the other hand, for each a ∈ N , by the change of variable formula for measures and identity (19) Therefore, f • T −1 a ∈ L p ([0, 1) N , λ B ) for each a ∈ N and f • T −1 a = 0 for each a ∈ N \O f . Moreover Thus P is well define and it is an isometry. Moreover, (22) proves identity (21). Finally, we will see that P is onto. Let f = (f a ) a ∈ a∈N L p ([0, 1) N , λ B ) and define By the monotone convergence theorem it follows that and again by the change of variable formulâ Therefore f ∈ L p (R N , B R N , λ B ) and it is straightforward to verify that a a = f. This concludes the proof.

Measures on Banach Spaces
Let (X, · ) be an infinite dimensional separable Banach space and let X = (x n , x * n ) n∈N be an absolutely convergent Markushevich basis (M-basis in short). Any infinite dimensional separable Banach space admits an absolutely convergent M-basis (see for instance [16]), therefore all the considerations given in this section are valid for any infinite dimensional separable Banach space. The aim of this section is to construct a normalized translation invariant borel measure for any Banach space clarifying the structure of its corresponding Lebesgue spaces. We start considering the linear map for each B ∈ B X , where B X stands for the borel σ-algebra of X. It becomes apparent that µ X is a measure on the measurable space (X, B X ). Moreover, by (23) and Theorem 2.8, the measure µ X is translation invariant and if we define Q := {x ∈ X : |x * n (x)| ≤ 1/2, ∀n ∈ N}, then, an straightforward computation yields µ X (Q) = λ B (B(Q)) = λ B × n∈N − 1 2 , 1 2 = 1.
It is convenient to remark that the proof of the existence of a measure on (X, B X ) satisfying this properties is not new since it was constructed by T. Gill, A. Kirtadze, G. Pantsulaia and A. Plichko in [7,Theorem 4.3]. Here, this type of measure has been constructed differently, via outer measure techniques developed in section I but we follow the philosophy of [7]. The next result establishes the structure of Lebesgue spaces of (X, B X , µ X ) and establishes an analogue of Theorem 4.2 for the category of Banach spaces. In the following result, given X, Y two Banach spaces, the notation X ֒ Y means that there exists a linear isometry from X to Y .
Theorem 5.1. Let (X, B X , µ X ) be the constructed measure space, then for 1 ≤ p < +∞ Moreover, given f ∈ L p (X, B X , µ X ), there exists a countable subset O f ⊂ N such that Proof. The operator B induces the map This operator is an isometric embedding, since given f ∈ L p (X, B X , µ X ), by the change of variable formula for measures Therefore, C is well define and it defines an isometry. Consequently Hence, an straightforward application of Theorem 4.2 yields embedding (24). Finally, we will prove identity (25). Let f ∈ L p (X, B X , µ X ), then (f • B −1 )χ B(X) ∈ L p (R N , B R N , λ B ) and by (21), there exists a countable subset O f ⊂ N such that This concludes the proof.
We finalize this section and the article summarizing the presented results in the following theorem.
Theorem 5.2. Let (X, · ) be a infinite dimensional separable Banach space and let X = (x n , x * n ) n∈N be an absolutely convergent M-basis. Then, there exists a translation invariant borel measure µ X on the measurable space (X, B X ) such that µ X (Q) = 1, where Q := {x ∈ X : |x * n (x)| ≤ 1/2, ∀n ∈ N}. Moreover, for 1 ≤ p < +∞ L p (X, B X , µ X ) ֒ a∈N L p ([0, 1) N , λ B ), and given f ∈ L p (X, B X , µ X ), there exists a countable subset O f ⊂ N such that