A real-valued measure on non-Archimedean field extensions of R
aa r X i v : . [ m a t h . F A ] S e p A real-valued measure on non-Archimedean fieldextensions of R Emanuele BottazziSeptember 30, 2020
Abstract
We introduce a real-valued measure m L on non-Archimedean ordered fields ( F ,< ) that extend the field of real numbers ( R ,< ) . The definition of m L is in-spired by the Loeb measures of hyperreal fields in the framework of Robinson’sanalysis with infinitesimals. The real-valued measure m L turns out to be generalenough to obtain a canonical measurable representative in F for every Lebesguemeasurable subset of R , moreover the measure of the two sets is equal. In addi-tion, m L it is more expressive than a class of non-Archimedean uniform measures.We focus on the properties of the real-valued measure in the case where F = R , theLevi-Civita field. In particular, we compare m L with the uniform non-Archimedeanmeasure over R developed by Shamseddine and Berz, and we prove that the firstis infinitesimally close to the second, whenever the latter is defined. We also de-fine a real-valued integral for functions on the Levi-Civita field, and we prove thatevery real continuous function has an integrable representative in R . Recall thatthis result is false for the current non-Archimedean integration over R . The pa-per concludes with a discussion on the representation of the Dirac distribution bypointwise functions on non-Archimedean domains. Contents R R . . . . . . . . . . . . . . 112.2 Relation with the Loeb measure on hyperreal fields . . . . . . . . . . 142.3 The real-valued measure in higher dimension and a real-valued integral 17 m L and the non-Archimedean uniform measures 211 A real-valued integral on the Levi-Civita field 23 M -integrable representatives of real continuous functions . . . . . . . 304.6 A restricted integration by parts for the M -integral . . . . . . . . . . . 324.7 Delta-like M -integrable functions and their derivatives . . . . . . . . 33 Measure theory on non-Archimedean fields, in particular non-Archimedean extensionsof R (and not e.g. fields of p -adic numbers) holds the promise to be relevant for manyapplications. An example is the mathematical description of physical phenomena, forinstance through the representation of distributions and Young measures as pointwisefunctions over non-Archimedean fields (for more details, we refer to the discussion in[11]; some recent examples are [13, 22]). Another relevant application is differentialor algebraic geometry, as discussed for instance in [5, 27]. However, currently themeasure theory of non-Archimedean fields is limited to some particular extensions of R or to a more restricted class of sets than e.g. the σ -algebra generated by intervals.A particular class of non-Archimedean extensions of R where a sizeable measuretheory has already been developed is that of hyperreal fields of Abraham Robinson’sframework of analysis with infinitesimals [34, 35]. In this context the elementaryequivalence of ∗ R and R and the presence of the transfer principle allow for a richmeasure theory. An immediate consequence of the transfer principle applied to anyreal-valued measure µ is that ∗ µ is a hyperreal-valued set function satisfying the fol-lowing conditions • ∗ µ is non-negative; • ∗ µ is monotone; • ∗ µ is finitely additive (and, since ∗ µ is internal, it is also hyperfinitely additive ).Notice that in general the hyperreal measure ∗ µ is not σ -additive, even if µ is [20].For this reason, most of the initial works on hyperreal measure theory were focusedon hyperfinitely additive measures. For some applications, this limitation turns out notto be restrictive, since hyperfinitely additive measures are general enough to representevery real non-atomic measure, including those that are σ -additive [2, 26]. Moreover,it is possible to represent uncountably many real-valued measures with a single hyper-finitely additive measure [42], or to require further compatibility conditions betweenthe non-Archimedean measure and the real-valued measure it represents [2, 3]. In fact,these results are true even if one works with the family of hyperfinite counting mea-sures , i.e. measures of the form µ ( A ) = | A || Ω | where Ω is a hyperfinite set, A ∈ ∗ P ( Ω ) (i.e. A is an internal subset of Ω ) and | · | denotes the internal cardinality.2 novel contribution to hyperfinite measures has been introduced by Eskew in a re-cent preprint [21]. Eskew defines an ultrafilter integral of functions f : X → G , where X is an arbitrary set and G is a divisible Abelian group. The ultrafilter integral de-pends upon the choice of an ultrafilter U over the family P fin ( X ) of finite subsetsof X and takes value in a nonstandard extension of the group G (more precisely, inthe ultrapower ∏ i ∈ P fin ( X ) G / U ). By defining a standard part from this ultrapower tothe original group G , it is possible to define a G -valued integral for every function f : X → G . If G = R , this integral is general enough to represent every non-atomicmeasure, similarly to the case of hyperfinite measures [26] and numerosities [2, 3].By considering the integral of indicator functions, this technique can be used to definereal-valued measures over arbitrary sets. We believe that this technique can be suitablyrephrased as a hyperfinite sum taking values in ∗ G , thus providing an extension of theusual hyperfinite measures. For its generality, ultrafilter integration might be success-fully applied to further advance the measure theory on non-Archimedean fields.Despite the expressive power of hyperfinite measures, they lack some familiar prop-erties of measures, such as σ -additivity. The problem of determining a suitable σ -additive, real-valued measure from an internal measure has been solved by Loeb withthe introduction of the Loeb measures construction . The main idea behind the Loebmeasure construction applied to an internal measure µ consists of the following steps: • define the real-valued set function µ R by posing µ R ( A ) = ◦ µ ( A ) ; • prove that µ R is an outer measure on the algebra of internal subsets of Ω ; • use the Caratheodory’s extension theorem to extend µ R to a σ -additive measure µ L on a σ -algebra that extends the algebra of internal subsets of Ω .For more details we refer to the original paper by Loeb [30] and to other presentationsof the Loeb measure, such as [19, 20]. Since their introduction, Loeb measures haveproven to be relevant in a variety of applications. The earliest examples by Loeb dis-cuss probability theory and stochastic processes [30], but there are further applicatonsfor instance in the representation of parametrized measures [18] and in the study ofgeneralized solutions to partial differential equations [11, 12].The development of a measure theory on other non-Archimedean field extensionsof R faces significant challenges. One of the most successful projects towards this goalis the uniform measure over the Levi-Civita field defined by Shamseddine and Berz[37, 39] and further studied by other authors [13, 22, 31, 40]. The Levi-Civita field R ,introduced by Levi-Civita in [28, 29] and subsequently rediscovered by many authorsin the ’900, is the smallest non-Archimedean ordered field extension of the field R ofreal numbers that is both real closed and sequentially complete in the order topology.The main idea behind the definiton of the uniform measure by Shamseddine and Berzis that measurable sets are those that can be suitably approximated by closed intervals.Similarly, measurable functions can be suitably approximated by a family of simplefunctions. For more details on this topic, we refer to Section 4 and to [39, 13].It turns out that measurable functions on the Levi-Civita field are expressive enoughto represent some distributions [13, 22]. For instance, it is possible to define some mea-surable functions that represent the Dirac distribution, much in the spirit of the repre-sentation of distributions with functions of nonstandard analysis or other generalized3unctions (for a detailed discussion on some representations of distributions with thesetechniques we refer to [11]). Moreover, it is also possible to represent real continuousfunctions with suitable equivalence classes of weak limits of measurable functions over R [13].However, the uniform measure on the Levi-Civita field has some limitations, mainlydue to the total disconnectedness of the topology induced by the non-Archimedeanmetric. A first drawback is that the family of measurable set is not closed under com-plements and over countable unions. As a consequence, there are well-known examplesof null sets whose complement is not measurable [13, 31, 39]. In addition, measurablefunctions are only locally analytic, so in the Levi-Civita field it is not possible to obtaina measurable representative of real continuous functions [13]. Finally, in Proposition3.6 of this paper we will argue that the measurable sets in the Levi-Civita field are notexpressive enough to represent all real Lebesgue measurable sets. This is in contrastto the hyperfinitely additive measures that represent the real Lebesgue measure and forthe corresponding Loeb measures.Another approach to the definition of (real-valued or otherwise) measures over non-Archimedean fields is related to model theory. One of the earliest results is the defi-nition of a finitely additive real measure on definable sets of o-minimal extensions offields by Berarducci and Otero [5]. The measurable sets are those definable sets thatcan be suitably approximated by finite unions of rectangles (this integrability condi-tion is equivalent to the one used in the Caratheodory’s extension theorem only underthe hypothesis that the measure is finite). The measure introduced by Berarducci andOtero has also provided a starting point for the development of an Hausdorff measurefor definable sets in o-minimal structures [24]. As observed by Kaiser, these measuresare defined for bounded sets and their range is just a semiring [27].A significant contribution to the development of a non-Archimedean measure the-ory on real closed fields F with a model-theoretic approach is the work by Kaiser [27].He introduces a non-Archimedean measure for semialgebraic sets and a correspond-ing integral for semialgebraic functions over non-Archimedean real closed fields withArchimedean value groups. This measure is finitely additive, monotone and translationinvariant. In order to satisfy these properties, the measure takes values outside the field F , since some integrals of semialgebraic functions require a notion of logarithm thatis not available for arbitrary real closed fields (for more details on this limitation, werefer to the discussion in [27]).It is relevant to observe that the measure developed by Kaiser for the Levi-Civitafield is not equal to the uniform measure introduced by Shamseddine and Berz, sincethe former is defined only for semialgebraic sets, while the latter is defined also onsome countable unions of intervals, that are not semialgebraic.The problem of a non-Archimedean measure and integration has also been dis-cussed in the setting of surreal numbers by Fornasiero [23] and Costin et al. [17].However, most of the results discussed in the latter paper are negative.Taking into account the existing literature on measures on non-Archimedean fields,some authors suggest that a measure theory on non-hyperreal field extensions of R requires a tame setting. Indeed, measures in Robinson’s framework are mostly definedon internal sets , with the notable exception of the Loeb measures (the numerosities byBenci et al. [2, 3] and the related Ω -limit approach to probabilities [4, 14], on the other4and, use functions defined on the powerset of a classic set with values in a hyperrealfield. In both approaches these functions are obtained as the restriction of suitableinternal measures, as discussed for instance in [15]). So far, the notion of internal set isonly meaningful for hyperreal fields, so that the techniques of Robinson’s frameworkcannot be adapted to other non-Archimedean fields. In the more general settings ofnon-Archimedean real closed fields with Archimedean value groups, the measure isdefined only for semialgebraic sets, and the integral is defined only for semialgebraicfunctions. Finally, in the Levi-Civita field, where there is no notion of interal set andthe existing non-Archimedean measure has been developed without model-theoreticalnotions, the family of measurable sets is badly behaved: for instance, we have alreadymentioned that it is not closed under relative complements.In this paper, inspired by the success of the real-valued Loeb measure constructionand motivated from the consideration that this real-valued measure is not defined onlyon a well-behaved family of sets (namely, the internal sets), we develop a uniform, real-valued measure for non-Archimedean field extensions of ∗ R . The main idea is sharedwith the Lebesgue measure, and consists in defining an outer measure from the lengthof intervals. However, we will not consider the length of the interval of endpoints a and b to be equal to b − a , but rather to the standard part of this difference, namely thereal number closest to b − a . This will allow to define an outer measure and, via theCaratheodory’s extension theorem, a corresponding σ -additive measure.We will show that this real-valued measure shares some of the properties of theLoeb measures. For instance, the measure is defined on a σ -algebra of subsets of F that is rich enough to represent Lebesgue measurable subsets of R . It is also possible toextend the real-valued measure to F n and, consequently, to define a real-valued integralfor functions f : F → F .The real-valued measure is also compatible with some of the existing measuresdiscussed above. If F = ∗ R is a field of hyperreal numbers, then the real-valued measureagrees with the Loeb measure obtained from the nonstandard extension of the Lebesguemeasure over R (however, it is strictly weaker than the Loeb measure, since e.g. it isnot able to assign a positive finite measure to hyperfinite unions of intervals of aninfinitesimal length). If F is Cauchy complete, then the real-valued measure agreeswith the standard part of a non-Archimedean uniform measure that generalizes the onedefined by Shamseddine and Berz for the Levi-Civita field.Finally, we focus on the Levi-Civita field. By adapting the techniques developed inthe first part of the paper, we define a real-valued integral on the Levi-Civita field in away that the corresponding integrable functions are expressive enough to represent realmeasurable functions. This result improves upon the previous representation obtainedby weakly Cauchy sequences of measurable functions [13]. As an application, weimprove on previous representations of the Dirac distribution by pointwise functionson non-Archimedean domains. Section 2 contains the definition of the measure m L and of the algebra of m L -measurablesets. For a matter of convenience, we will refer to m L -measurable set as L-measurablesets . We will show that the the measure m L shares some properties with the Lebesgue5easure: it is uniform, translation invariant and homogeneous. Moreover, we willshow that m L can be interpreted as an extension to F of the real Lebesgue measure. Infact, the main result of this section is the proof that every Lebesgue measurable subsetof R has a canonical L -measurable representative in F with the same measure as theoriginal set. We also discuss the relation betwen m L and λ L , the Loeb measure obtainedfrom the Lebesgue measure, under the hypothesis that F is a field of hyperreal num-bers. As expected, the Loeb measure is more expressive than the real-valued measure,however the two measures agree on a relevant class of subsets. Finally, we extend thedefinition of the real-valued measure to the n -dimensional space F n , and from this def-inition we introduce a real-valued integral as the measure of the set under the graph ofa function. This approach is similar to the introduction of the Lebesgue integral via the n -dimensional Lebesgue measure over R n , presented for instance in [33].In Section 3 we discuss the relation between the real-valued measure m L and anon-Archimedean uniform measure m on Cauchy complete fields F . This measureis inspired by the one developed for the Levi-Civita field by Shamseddine and Berz.In fact, when F = R , then the measure defined in this paper coincides with the onedefined by Shamseddine and Berz. We prove that, if a set A ⊆ F is m -measurable, thenit is also L -measurable and m L ( A ) = ◦ m ( A ) . Moreover, we will show that the non-Archimedean measure m is significantly less expressive than the real-vaued measure m L , since the projection of m -measurable subsets of F to R can be written as a finiteunion of intervals and of a countable set.We further pursue the development of a real measure theory on the Levi-Civita fieldwith the introduction of another real-valued integral on the Levi-Civita field in Section4. The definition of this real-valued integral relies on the existing integration theory[13, 37, 39]. In analogy with the discussion in Section 3, we prove coherence with theexisting non-Archimedean integral.An application of the real-valued integral is presented in Section 4.7, where wediscuss the representation of some real distributions as pointwise functions defined onthe Levi-civita field, sharpening some of the results obtained in [13]. Throughout the paper ( F , < F ) will denote a non-Archimedean field extension of ( R , < R ) . In particular, we will suppose that R ⊂ F and that for every x , y ∈ R x < R y if andonly if x < F y . Due to this assumption, we will often write ( F , < ) instead of ( F , < F ) .A number x ∈ F is called • infinitesimal if | x | ≤ r for every r ∈ R , r > • finite if there exists r ∈ R such that | x | < r ; • appreciable if x is finite and non-infinitesimal; • infinite if | x | > r for every r ∈ R .If x ∈ F is infinitesimal, we will write x ≃
0. If x ∈ F is a nonzero infinitesimal, wewill write x ∼
0. In analogy with Robinson’s framework of analysis with infinitesimals,6f x ∈ F , we will refer to the set µ ( x ) = { y ∈ F : | x − y | ≃ } as the monad of the point x . Recall also that monads are not intervals [13].We define F fin = { x ∈ F : ∃ r ∈ R : | x | < r } , i.e. F fin is the ring of all finite elementsof F .We find it also useful to define the standard part of an element of F . Definition 1.1. if x ∈ F , we define ◦ x = inf { y ∈ R : x ≤ y } = sup { z ∈ R : z ≤ x } if | x | < r for some r ∈ R + ∞ if x > r for all r ∈ R − ∞ if x < r for all r ∈ R . The function ◦ : F → R ∪ { + ∞ , − ∞ } is well-defined and surjective. Moreover, it isa homomorphism between the rings F fin and R . Lemma 1.2.
For every x , y ∈ F fin ◦ ( x + y ) = ◦ x + ◦ y;2. ◦ ( xy ) = ◦ x ◦ y.Proof. Let x = ◦ x + ε x and y = ◦ y + ε y , with ε x and ε y infinitesimals in F . Then x + y = ◦ x + ◦ y + ε x + ε y . Since ◦ x + ◦ y ∈ R and ε x + ε y is a sum of two infinitesimals, ◦ ( x + y ) = ◦ x + ◦ y .Similarly, xy = ◦ x ◦ y + ε x ◦ y + ε y ◦ x + ε x ε y , and • ◦ x ◦ y ∈ R ; • ε x ◦ y , ε y ◦ x and ε x ε y are infinitesimals.We deduce that ◦ ( xy ) = ◦ x ◦ y .Another useful notion borrowed from Robinson’s framework is that of nearstan-dard point in a set. Definition 1.3.
Let A ⊆ F . We will say that a point x ∈ A is nearstandard in A iff ◦ x ∈ A. For every a , b ∈ F with a ≤ b we will denote by [ a , b ] F the set { x ∈ F : a ≤ x ≤ b } ,and by ( a , b ) F the set { x ∈ F : a < x < b } . The sets [ a , b ) F and ( a , b ] F are definedaccordingly. The above definitions are extended in the usual way if a = − ∞ or b = + ∞ .If F = R , we will often write [ a , b ] instead of [ a , b ] R .For all a , b ∈ F with a < b , we will denote by I ( a , b ) any of the sets ( a , b ) F , [ a , b ) F , ( a , b ] F or [ a , b ] F . We will call such sets bounded intervals of F . The length of aninterval of the form I ( a , b ) is denoted by l ( I ( a , b )) and is defined as b − a .Finally, we will denote by λ n the Lebesgue measure over R n .7 A real-valued measure on non-Archimedean exten-sions of R We begin our treatment of a real-valued measure on non-Archimedean extensions of R by introducing an outer measure over F that assumes values in the extended realnumbers R ∪ { + ∞ } . This outer measure is obtained from the standard part of thelength of an interval, in analogy with the Lebesgue outer measure. Definition 2.1.
For every a , b ∈ F , a ≤ b, define l L ( I ( a , b )) = ◦ ( b − a ) . For every A ⊆ F such that there exists a sequence of bounded intervals { I n } n ∈ N satisfying A ⊆ S n ∈ N I n ,define m L ( A ) = inf ( ∑ n ∈ N l L ( I n ) : A ⊆ [ n ∈ N I n ) . If for every sequence of bounded intervals { I n } n ∈ N we have A S n ∈ N I n , define m L ( A ) =+ ∞ . The last condition of Definition 2.1 ensures that m L is defined on the powerset of F , since e.g. F itself might not be contained in the union of any countable union ofbounded intervals. This property is essential in proving that m L is an outer measureover F . Lemma 2.2.
The function m L : P ( F ) → R ∪ { + ∞ } is an outer measure.Proof. We have already observed that m L is defined on P ( F ) .Since l L ( I ) ≥ I ⊆ F , m L ( A ) ≥ A ⊆ F . Moreover, m L ( /0 ) = m L ( A ) ≤ m L ( B ) whenever A ⊆ B , noticethat if B ⊆ S n ∈ N I n , then also A ⊆ S n ∈ N I n . As a consequence we get m L ( A ) = inf ( ∑ n ∈ N l L ( I n ) : A ⊆ [ n ∈ N I n ) ≤ inf ( ∑ n ∈ N l L ( I n ) : B ⊆ [ n ∈ N I n ) , as desired.Finally, we need to prove σ -subadditivity of m L , i.e. that if A n ⊆ F for all n ∈ N ,then m L [ n ∈ N A n ! ≤ ∑ n ∈ N m L ( A n ) . The result is trivially true if ∑ n ∈ N m L ( A n ) = + ∞ , so assume that this is not the case.Suppose then that ∑ n ∈ N m L ( A n ) ∈ R : this entails also m L ( A n ) ∈ R for every n ∈ N .Then for every ε ∈ R , ε >
0, there exists a family of sets { I ε n , k } k ∈ N such that • A n ⊆ S k ∈ N I ε n , k and • ∑ k ∈ N l L ( I ε n , k ) ≥ m L ( A n ) + ε n . 8e have also the inclusion S n ∈ N A n ⊆ S n ∈ N (cid:16) S k ∈ N I ε n , k (cid:17) . From monotonicity of theouter measure, we obtain m L [ n ∈ N A n ! ≤ ∑ n ∈ N ∑ k ∈ N l L ( I ε n , k ) ! ≤ ∑ n ∈ N (cid:16) m L ( A n ) + ε n (cid:17) = ε + ∑ n ∈ N m L ( A n ) . By the arbitrariness of the real parameter ε >
0, we conclude that m L is σ -subadditive. Remark 2.3.
From monotonicity of the outer measure m L we deduce that every setcontained in an interval of an infinitesimal length has outer measure , while if a setcontains intervals of length at least n for every n ∈ N , then its outer measure is infinite.As a consequence, m L ( F fin ) = + ∞ and m L ( A ) = + ∞ whenever A ⊃ F fin . From the outer measure m L defined over P ( F ) , it is possible to obtain a σ -algebraof measurable sets. Definition 2.4.
Given the outer measure m L on F , the following family of subsets of F is called the Caratheodory σ -algebra associated to m L : C = { A ⊆ F : m L ( B ) = m L ( B ∩ A ) + m L ( B \ A ) for all B ⊆ F } . If A ∈ C , we will say that A is L-measurable. A well known theorem of Caratheodory states that the above family C is indeed a σ -algebra, and that the restriction of m L to C , that we will denote by m L , is a completemeasure, i.e. a measure such that m L ( A ) = B ⊆ A , B ∈ C and m L ( B ) = C , we obtain the following regularity property. In the sequel,we will use it as a criterion for L -measurability. Lemma 2.5.
Let A , C ∈ C . If m L ( A ) = m L ( C ) < + ∞ , then for every B ⊆ F that satisfiesA ⊆ B ⊆ C, B ∈ C and m L ( B ) = m L ( A ) = m L ( C ) .Proof. Let A , B and C satisfy the hypotheses of the lemma. By monotonicity of theouter measure, m L ( A ) = m L ( A ) ≤ m L ( B ) ≤ m L ( C ) = m L ( C ) . Thus m L ( B ) = m L ( A ) = m L ( B ) .Since A is measurable, m L ( B ) = m L ( A ∩ B ) + m L ( B \ A ) . However, A ∩ B = A and m L ( B ) = m L ( A ) , so that m L ( B \ A ) =
0. Since C is complete, B \ A ∈ C . Thus B = A ∪ ( B \ A ) , i.e. B is the union of two L -measurable sets. Since C is a σ -algebra, henceclosed also for finite unions, B is also L -measurable.The measure m L shares some properties with the Lebesgue measure. For instance,it is translation invariant. Lemma 2.6.
If A ⊆ F is L-measurable, then for every x ∈ F the setA + x = { y : ∃ a ∈ A : y = a + x } is L-measurable and m L ( A ) = m L ( A + x ) roof. This is consequence of the two properties A ⊆ [ n ∈ N I n ⇒ A + x ⊆ [ n ∈ N ( I n + x ) and l L ( I ) = l L ( I + x ) for every interval I ⊆ F and for every x ∈ F .The proof can then be carried out as in the usual proof of translation invariance ofthe Lebesgue measure; for more details we refer e.g. to Lemma 3.15 and Theorem 3.16of [43].Notice however that m L is not positively homogeneous, and that the very samenotion of positive homogeneity needs to be adapted to the non-Archimedean setting. Proposition 2.7.
If A ⊆ F is L-measurable and if m L ( A ) < + ∞ , then for every x ∈ F fin the set xA = { y : ∃ a ∈ A : y = a + x } is L-measurable and m L ( xA ) = | ◦ x | m L ( A ) .Proof. If x =
0, the desired result is trivially satisfied, since 0 A = { } .For every x ∈ F fin , x =
0, and for every A ⊆ F with m L ( A ) < + ∞ , we have theinclusion A ⊆ [ n ∈ N I n ⇒ xA ⊆ [ n ∈ N ( xI n ) and the equality l L ( xI ) = | ◦ x | l L ( I ) for every interval I ⊆ F . This is sufficient to conclude m L ( xA ) ≤ | ◦ x | m L ( A ) for every A ⊆ F with m L ( A ) < + ∞ and for every x ∈ F fin .If x is an infinitesimal, m L ( xA ) ≤ | ◦ x | m L ( A ) = · m L ( A ) =
0. If x ∈ F fin is appre-ciable, then x − is neither infinite nor infinitesimal. Then also m L ( A ) = m L ( x − xA ) ≤ | ◦ ( x − ) | m L ( xA ) ≤ | ◦ ( x − ) || ◦ x | m L ( A ) = m L ( A ) . By combining these results, we obtain that m L ( xA ) = | ◦ x | m L ( A ) for every A ⊆ F with m L ( A ) < + ∞ and for every x ∈ F fin .The proof of L -measurability of the set xA under the hypothesis that A is L -measurablecan be obtained with an argument analogous to that of Theorem 3.18 of [43]. Example 2.8.
If A is a L-measurable set of an infinite measure, then positive ho-mogeneity fails, since for x ≃ it would lead to the indeterminate form m L ( xA ) = | ◦ x | m L ( A ) = · + ∞ . In fact, let a ∈ F be a positive infinite number and consider the setA = [ , a ] . Then m L ( xA ) can be either zero, any positive real number, or + ∞ , dependingupon the value of x. For instance, for every r ∈ F fin m L ( ra − A ) = ◦ r, m L ( ra − A ) = and m L ( ra − / A ) = + ∞ . Remark 2.9.
The measure m L shares many properties with the Lebesgue measureover R . For instance, it is uniform, positively homogeneous and translation invariantover F . Despite these similarities, m L does not satisfy other relevant properties of theLebesgue measure: for instance, it is not σ -finite, since F is not the union of countablymany sets of a finite measure. Notice however that the restriction of m L to F fin is σ -finite. In addition, the complement of a null set needs not be a dense subset of F or F fin (compare this property with the one discussed in Observation 3.7 of [43]), since F and F fin are totally disconnected with respect to the topology induced by the metric [32]. emark 2.10. The outer measure m L and the corresponding measure m L can be suit-ably rescaled. E.g. if one is interested in working with a measure that assigns length to the intervals of the form I ( , ε ) , with either ε ≪ or ε ≫ , then it is possible toassume the alternative definition l ε ( I ( a , b )) = ◦ (cid:0) b − a ε (cid:1) . The resulting measure ε m L ( A ) = inf ( ∑ n ∈ N l ε ( I n ) : A ⊆ [ n ∈ N I n ) would have the desired property ε m L ( I ( , ε )) = . Consequently, the family of sets witha finite ε m L outer measure can be interpreted as the family of sets whose measure is ofthe same magnitude as that of the intervals I ( , ε ) .Many of the properties already proved for m L and m L , such as translation invari-ance and positive homogeneity as described in Proposition 2.7, are still valid for theserescaled measures. R We will now study the relation between the measure m L over F and the Lebesguemeasure over R . Notice that Lebesgue measurable subsets of R are not in general m L -measurable in F . Consider for instance the real intervals A = [ , ] R and B = [ , ] F .Since 1 = m L ([ , ] F ) = m L ([ , ] R ) + m L ([ , ] F \ [ , ] R ) = , we conclude that [ , ] R C .However, Lebesgue measurable sets over R have a canonical L -measurable repre-sentative in F . Moreover, the measure of this representative is equal to the Lebesguemeasure of the original set.We will prove this result at first by showing that, for every Lebesgue measurableset A ⊆ R , the set st − ( A ) = { x ∈ F : ◦ x ∈ A } has outer measure equal to λ ( A ) . Thenwe will prove that if A ⊆ R is Lebesgue measurable, then st − ( A ) ∈ C . Notice thatst − ( A ) ⊆ F fin , since if x F fin , st ( x ) = ± ∞ A for every A ⊆ R . Proposition 2.11.
If A ⊆ R is Lebesgue measurable, then λ ( A ) = m L ( st − ( A )) .Proof. Consider an interval [ a , b ] R and, for all n ∈ N , define the intervals I ( a − / n , b + / n ) ⊂ F . We have st − ([ a , b ] R ) ⊂ I ( a − / n , b + / n ) and m L ( st − ([ a , b ] R )) ≤ m L (cid:18) I (cid:18) a − n , b + n (cid:19)(cid:19) = b − a + n . Since inf n ∈ N (cid:8) b − a + n (cid:9) = b − a , m L ( st − ( I ( a , b ))) ≤ b − a . Notice also that I ( a , b ) ⊂ st − ([ a , b ] R ) , I ( a , b ) ∈ C and m L ( I ( a , b )) = b − a . Then, by monotonicity of the outermeasure, m L ( st − ([ a , b ] R )) = b − a .Consider now an arbitrary Lebesgue measurable set A ⊆ R . Recall that its Lebesguemeasure λ ( A ) can be defined as λ ( A ) = inf ( ∑ n ∈ N l ( I n ) : A ⊆ [ n ∈ N I n ) .
11n the above formula, I n ⊆ R for all n ∈ N and l ( I n ) is the usual length of the realinterval I n . By definition, we have also m L ( st − ( A )) = inf ( ∑ n ∈ N l L ( J n ) : st − ( A ) ⊆ [ n ∈ N J n ) . Consider now the real intervals I n = { ◦ x : x ∈ J n } , and notice that if st − ( A ) ⊆ S n ∈ N J n ,then A ⊆ S n ∈ N I n . By definition of l L , we have also l L ( J n ) = l ( I n ) , so that ( ∑ n ∈ N l L ( J n ) : st − ( A ) ⊆ [ n ∈ N J n ) ⊆ ( ∑ n ∈ N l ( I n ) : A ⊆ [ n ∈ N I n ) . This inclusion entails the inequality λ ( A ) ≤ m L ( st − ( A )) .In order to prove that the opposite inequality is also true, let ε ∈ R , ε > { I n } n ∈ N satisfy • A ⊆ S n ∈ N I n ⊆ R and • ∑ n ∈ N l ( I n ) ≤ λ ( A ) + ε .Then st − ( A ) ⊆ S n ∈ N st − ( I n ) . By σ -subadditivty and monotonicity of m L , we deduce m L ( st − ( A )) ≤ m L [ n ∈ N st − ( I n ) ! ≤ ∑ n ∈ N m L ( st − ( I n )) . In the first part of the proof we have shown that m L ( st − ( I )) = l ( I ) for every realinterval I . From this equality we deduce m L ( st − ( A )) ≤ ∑ n ∈ N l ( I n ) ≤ λ ( A ) + ε . By thearbitrariness of the real parameter ε , we obtain m L ( st − ( A )) ≤ λ ( A ) , as desired. Proposition 2.12.
If A ⊆ R is Lebesgue measurable, then st − ( A ) ∈ C .Proof. By σ -additivity of the measure m L , it is sufficient to prove that for every boundedLebesgue measurable set A ⊆ R , st − ( A ) ∈ C . Once we have proven this result, the factthat st − ( A ) ∈ C for every Lebesgue measurable A ⊆ R can be obtained by the fact that C is closed under countable unions. For this reason, in the sequel of the proof we willsuppose that A is bounded.By definition of the outer measure m L and by Theorem 2.24 of [43], it is sufficientto prove that, if A ⊆ R is Lebesgue measurable, then ◦ ( b − a ) = m L ( I ( a , b )) = m L ( st − ( A ) ∩ I ( a , b )) + m L ( I ( a , b ) \ st − ( A )) (2.1)for every I ( a , b ) ⊆ F with a , b ∈ F , a ≤ b .Recall also that, by subadditivity of the outer measure m L , the inequality ◦ ( b − a ) ≤ m L ( st − ( A ) ∩ I ( a , b )) + m L ( I ( a , b ) \ st − ( A )) is always satisfied, so we only need to prove the opposite inequality under the additionalhypothesis that ◦ ( b − a ) < + ∞ . 12otice that [ a , b ] R is an interval, so it is Lebesgue measurable. The hypothesis that A is Lebesgue measurable ensures then that A ∩ [ a , b ] R and [ a , b ] R \ A are Lebesguemeasurable subsets of R . Applying Proposition 2.11 we obtain m L ( st − ( A ∩ [ a , b ] R ) = λ ( A ∩ [ a , b ] R ) and m L ( st − ([ a , b ] R \ A )) = λ ([ a , b ] R \ A ) . Since st − ( A ) ∩ I ( a , b ) ⊆ st − ( A ∩ [ a , b ] R ) and I ( a , b ) \ st − ( A ) ⊆ st − ([ a , b ] R \ A ) , bymonotonicity of the outer measure we have m L ( st − ( A ) ∩ I ( a , b )) ≤ m L ( st − ( A ∩ [ a , b ] R )) = λ ( A ∩ [ a , b ] R ) and m L ( I ( a , b ) \ st − ( A )) ≤ m L ( st − ([ a , b ] R \ A )) = λ ([ a , b ] R \ A ) . Putting together the two inequalities, we obtain m L ( st − ( A ) ∩ I ( a , b ))+ m L ( I ( a , b ) \ st − ( A )) ≤ λ ( A ∩ [ a , b ] R )+ λ ([ a , b ] R \ A ) = λ ([ a , b ] R ) = ◦ ( b − a ) . The above inequality is sufficient to conclude that equality (2.1) is satisfied forevery I ( a , b ) ⊆ F with a , b ∈ F , a ≤ b . As we argued in the beginning of the proof, thisis sufficient to entail that st − ( A ) ∈ C , as desired. Theorem 2.13.
If A ⊆ R is Lebesgue measurable, then the set st − ( A ) = { x ∈ F : ◦ x ∈ A } is L-measurable, and λ ( A ) = m L ( st − ( A )) .Proof. By Proposition 2.12, if A ⊆ R is Lebesgue measurable, then st − ( A ) ∈ C . As aconsequence, m L ( st − ( A )) = m L ( st − ( A )) . By Proposition 2.11, m L ( st − ( A )) = λ ( A ) ,so that also m L ( st − ( A )) = λ ( A ) .Conversely, a L -measurable subset of F fin corresponds via the standard part func-tion to a Lebesgue measurable subset of R . In other words, the standard part functionis measure-preserving. Theorem 2.14.
If A ⊆ F fin is L-measurable, then the set ◦ A = { ◦ x ∈ R : x ∈ A } isLebesgue measurable, and m L ( A ) = λ ( ◦ A ) .Proof. Notice that for every A ⊆ F fin , if A ⊆ S n ∈ N I n , then ◦ A ⊆ S n ∈ N ◦ I n , and ◦ I n isa closed interval or a singleton for every n ∈ N . Consequently, if we denote by λ theLebesgue outer measure over R , λ ( ◦ A ) ≤ m L ( A ) .As in the proof of Proposition 2.12, we will consider at first only sets included inan interval of a finite length. The desired result for arbitrary L -measurable sets can thenbe obtained by σ -additivity of the measures m L and λ .If A ⊆ [ − n , n ] F is L -measurable, then we have λ ( ◦ A ) + λ ( ◦ A c ) ≤ m L ( A ) + m L ( A c ) = n . However, countable subadditivity of the outer measure λ implies that λ ([ − n , n ] R ) ≤ λ ( ◦ A ) + λ ( ◦ A c ) . λ ( ◦ A ) + λ ( ◦ A c ) = n and, takinginto account that λ ( ◦ B ) ≤ m L ( B ) for every B ⊆ F fin , we conclude that λ ( ◦ A ) = m L ( A ) for every L -measurable set A ⊆ [ − n , n ] .In order to prove that, if A ⊆ [ − n , n ] F is L -measurable, then it is also Lebesguemeasurable, we will prove that b − a = λ ( I ( a , b )) = λ ( ◦ A ∩ I ( a , b )) + λ ( I ( a , b ) \ ◦ A ) for every I ( a , b ) ⊆ R , with a , b ∈ R , a ≤ b . Since A is L -measurable, it satisfies theCaratheodory condition ◦ ( b − a ) = m L ( A ∩ I ( a , b )) + m L ( I ( a , b ) \ A ) for every I ( a , b ) ⊆ F with a , b ∈ F , a ≤ b . Notice also that ◦ ( A ∩ I ( a , b )) = ◦ A ∩ [ ◦ a , ◦ b ] R and ◦ ( I ( a , b ) \ A ) ⊇ [ ◦ a , ◦ b ] \ ◦ A . By the previous part of the proof, if a , b ∈ R we have λ ( ◦ A ∩ [ a , b ] R ) = m L ( A ∩ [ a , b ] F ) and λ ([ a , b ] R \ ◦ A ) ≤ m L ([ a , b ] F \ A ) . Taking into account that A satisfies the Caratheodory measurability condition over F ,we obtain λ ( ◦ A ∩ I ( a , b )) + λ ( I ( a , b ) \ ◦ A ) ≤ m L ( A ∩ [ a , b ] F ) + m L ([ a , b ] F \ A ) = b − a , as desired.Thus we have proved that for every bounded L -measurable set A ⊆ F fin , ◦ A isLebesgue measurable and m L ( A ) = λ ( ◦ A ) . For an arbitrary L -measurable set A ⊆ F fin ,we have already argued that the desired result can be obtained from σ -additivity of themeasures m L and λ . If F is a sufficiently saturated field of hyperreal numbers of Robinson’s frameworkof analysis with infinitesimals, so that λ L , the Loeb measure associated to the realLebesgue measure, can be defined, then it is possible to study relation between m L and λ L . From Theorem 2.13, we can already conclude that both measures agree on thepreimage of Lebesgue measurable subsets of R via the standard map function.However the two measures are different: consider for instance an infinite hyper-natural number N and the set A = S Nn = [ n , n + N − ] ∗ R . Then λ L ( A ) = N · N − = A ∗ R fin and that A is a hyperfinite union of intervals of an infinitesi-mal length. Recall that hyperfinite subsets in Robinson’s framework of analysis withinfinitesimals have uncountable external cardinality (while, by definition, they have fi-nite internal cardinality, since they can be put in an internal bijection with an internalinitial segment of ∗ N . For more details on the distinction between internal and externalcardinality, we refer to [25]). As a consequence, every countable sequence of intervals { I n } n ∈ N satisfying A ⊆ S n ∈ N I n must include at least one interval of an infinite length,so that m L ( A ) = + ∞ . We deduce that either A C or m L ( A ) = + ∞ .Despite these differences, the measure m L is compatible with the Loeb measure λ L over ∗ R fin in the sense that if a subset of ∗ R fin is L -measurable, then it is also λ L -measurable and the two measures coincide.14 heorem 2.15. Let F = ∗ R , a sufficiently saturated field of hyperreal numbers, anddenote by L the σ -algebra of Loeb measurable subsets of ∗ R . Then for every A ⊆ ∗ R fin ,if A ∈ C then also A ∈ L . Moreover, m L ( A ) = λ L ( A ) .Proof. Recall that, by the Caratheodory’s extension theorem, C is the smallest σ -algebra containing the family C = ( [ n ∈ N I n : { I n } n ∈ N is a sequence of pairwise disjoint bounded intervals of ∗ R ) . By definition of l L , for every interval I ⊆ ∗ R we have l L ( I ) = λ L ( I ) , so that forevery sequence of bounded intervals { I n } n ∈ N we have the equality ∑ n ∈ N l L ( I n ) = ∑ n ∈ N λ L ( I n ) . Moreover, S n ∈ N I n ∈ L and λ L [ n ∈ N I n ! = ∑ n ∈ N l L ( I n ) = m L [ n ∈ N I n ! . (2.2)For every n ∈ N , define now the following families of subsets of ∗ R fin . • C n = C ∩ P ([ − n , n ] ∗ R ) ; • C fin = C ∩ P ( ∗ R fin ) ; • C n = C ∩ P ([ − n , n ] ∗ R ) ; • C fin = C ∩ P ( ∗ R fin ) ; • E n = { A ∈ C ∩ C : A ⊆ [ − n , n ] ∗ R and λ L ( A ) = m L ( A ) } ; and • E fin = (cid:8) A ∈ C ∩ C : A ⊆ ∗ R fin and λ L ( A ) = m L ( A ) (cid:9) .Notice that we are not assuming that the members of any of the above families must beinternal. By equation (2.2), λ L and m L assume the same values on elements of C fin . Inaddition, we have the inclusions C n ⊆ E n for every n ∈ N .In order to prove that for every A ∈ C fin then also A ∈ E fin , i.e. that A is Loebmeasurable and m L ( A ) = λ L ( A ) , we will prove that C n ⊆ E n for all n ∈ N by usingDinkyn’s π – λ theorem. The desired result can then be obtained by noticing that, by σ -additivity, λ L ( A ) = ∑ n ∈ N λ L ( A ∩ ([ − n − , − n ] ∪ [ n , n + ])) and m L ( A ) = ∑ n ∈ N m L ( A ∩ ([ − n − , − n ] ∪ [ n , n + ])) , and that the inclusion C n ⊆ E n for all n ∈ N entails that λ L ( A ∩ ([ − n − , − n ] ∪ [ n , n + ])) = m L ( A ∩ ([ − n − , − n ] ∪ [ n , n + ])) n ∈ N , so that also λ L ( A ) = m L ( A ) .Recall that a π -system over a set X is a family of subsets of X closed under finiteintersections, and a λ -system over X is a family of subsets of X that1. contains the empty set;2. is closed under complements;3. is closed under countable disjoint unions.It is easy to see that C n is a π -system for every n ∈ N . We now want to prove that E n = (cid:8) A ⊆ ∗ R fin : λ L ( A ) = m L ( A ) (cid:9) is a λ -system for every n ∈ N .1. Clearly /0 ∈ E n , since /0 ⊆ [ − n , n ] ∗ R and λ L ( /0 ) = m L ( /0 ) = λ L ( A ) = m L ( A ) for some A ∈ C ∩ C , A ⊆ [ − n , n ] ∗ R , and let A c = [ − n , n ] ∗ R \ A . Taking into account that [ − n , n ] ∈ C ∩ C , that λ L ([ − n , n ] ∗ R ) = m L ([ − n , n ] ∗ R ) = n for every n ∈ N and that A and A c are disjoint, λ L ( A ) + λ L ( A c ) = m L ( A ) + m L ( A c ) = n . Since we have assumed that λ L ( A ) = m L ( A ) ,we have also λ L ( A c ) = m L ( A c ) , i.e. A c ∈ E n , as desired.3. Suppose that { A m } m ∈ N is a sequence of pairwise disjoint sets in E n . By σ -additivity of the measures λ L and m L , then we have λ L [ m ∈ N A m ! = ∑ m ∈ N λ L ( A m ) and m L [ m ∈ N A m ! = ∑ m ∈ N m L ( A m ) . Since we have assumed that λ L ( A m ) = m L ( A m ) for all m ∈ N , then also S m ∈ N A m ∈ E , as desired.We have verified that for every n ∈ N C n is a π -system, E n is a λ -system, and C n ⊆ E n for every n ∈ N . Then Dinkyn’s π – λ theorem ensures that the σ -algebragenerated by C n is a subset of E n for every n ∈ N . However, the σ -algebra generatedby C n is C n , so that C n ⊆ E n for every n ∈ N , as desired.By translation invariance of the measure m L , a similar result applies also to subsetsof the translates x + ∗ R fin , x ∈ ∗ R . However, the above result cannot be extended oversupersets of ∗ R fin (or to supersets of its translates x + ∗ R fin , x ∈ ∗ R ), since the Dinkyn’s π – λ theorem can only be applied to finite or σ -finite measurable sets.The difference between the two measures can be explained in terms of the model-theoretic notions used in their definitions. In fact, the Loeb measure λ L relies heavilyon the properties of star transform, on the notion of internal sets and on the transferprinciple of Robinson’s framework. Instead, the uniform measure m L is defined fromfirst principles and does not exploit the strength of these notions. This difference ex-plains the greater versatility of the Loeb measures and their applicability to a varietyof mathematical problems. On the other hand, an advantage of the measure m L is thatit can be defined even for those field extensions of R where there is no analogous of astar transform, of a transfer principle or of a notion of internal sets.16 .3 The real-valued measure in higher dimension and a real-valuedintegral In this section we generalize the definition of the real-valued measure m L to F n for all n ∈ N . Definition 2.16.
We say that a bounded rectangle in F n is the product of n bounded in-tervals in F . If R = I , . . . , I n is a bounded rectangle, define m nL ( R ) = ◦ ( l ( I ) · . . . · l ( I n )) .For every A ⊆ F n such that there exists a sequence of bounded rectangles { R n } n ∈ N satisfying A ⊆ S n ∈ N R n , definem nL ( A ) = inf ( ∑ n ∈ N m nL ( R n ) : A ⊆ [ n ∈ N R n ) . If for every sequence of bounded rectangles { R n } n ∈ N we have A S n ∈ N R n , definem nL ( A ) = + ∞ . As with the one-dimensional set function m L , m nL is an outer measure for all n ∈ N .Consequently, one can define the σ -algebra of measurable subsets of F n . The definitionis analogous to that of the Lebesgue integral in dimension n from the Lebesgue measurein dimension n +
1, as exposed for instance in [33].
Definition 2.17.
Given the outer measure m L on F n , the following family is called the Caratheodory σ -algebra associated to m L : C ( F n ) = { A ⊆ F : m nL ( B ) = m nL ( B ∩ A ) + m nL ( B \ A ) for all B ⊆ F n } . If A ∈ C ( F n ) , we will say that A is L-measurable. The family C ( F n ) is a σ -algebra, and that the restriction of m nL to C ( F n ) , that wewill denote by m nL , is a complete measure. As we have seen for the one-dimensionalmeasure, the real-valued measures m nL are translation invariant and positively homoge-neous. Moreover, by adapting the proof of Theorem 2.13, we obtain that if A ⊆ R n is a Lebesgue measurable set, then st − ( A ) ⊆ F n is L -measurable and m nL ( st − ( A )) = λ n ( A ) .The n -dimensional measures can also be used to define a real-valued integral forfunctions over F . Definition 2.18.
Let A ⊆ F n , be a L-measurable set and let f : A → F be a non-negativefunction. We say that f is L-integrable iff U ( f ) = { ( x , . . . , x n , x n + ) ∈ F n + : 0 ≤ x n + ≤ f ( x , . . . , x n ) } is L-measurable and m n + L ( U ( f )) < + ∞ . If f : A → F is a non-negative m nL -measurablefunction, we define Z A f dm nL = m n + L ( U ( f )) . We say that f : A → F is L-integrable iff f + and f − are. If f : A → F is a L-integrable function, we define Z A f dm nL = Z A f + dm nL − Z A f − dm nL . m n + L , the integral is R -linear. Moreover, the linearity property can be extended in the same spirit as positivehomogeneity (see Proposition 2.7). Proposition 2.19.
If A ⊂ F n is a L-measurable set, then for every L-integrable func-tions f and g over A and for every x , y ∈ F fin , Z A ( x f + yg ) dm nL = ◦ x Z A f dm nL + ◦ y Z A g dm nL . Proof.
Once we prove that the set U ( g ) has the same measure as the set { ( x , . . . , x n , x n + ) ∈ F n + : f ( x , . . . , x n ) ≤ x n + ≤ g ( x , . . . , x n ) } , linearity is a consequence of the definitionof the integral and of positive homogeneity of the measure m n + L .The proof that the two sets have the same measure can be obtained by adaptingthe proof of Theorem 16 (g) of Chapter 6, Section 4 of [33]. If f and g are bothstep functions, i.e. if both are defined over an interval and they are piecewise con-stant over subintervals of their domain, then the desired assertion is a consequence oftranslation invariance and positive homogeneity of the measure m n + L . If f and g arearbitrary, the proof relies on the possibility to approximate the sets U ( f ) , U ( g ) and { ( x , . . . , x n , x n + ) ∈ F n + : f ( x , . . . , x n ) ≤ x n + ≤ g ( x , . . . , x n ) } by rectangles up to anarbitrary precision.The integral of a function defined on a domain of a finite measure is invariant byinfinitesimal perturbations of the function. Proposition 2.20.
Let A ⊆ F n be a L-measurable set with m nL ( A ) < + ∞ . If f : A → F is L-integrable and if g : A → F satisfies f ( x ) ≃ g ( x ) for every x ∈ A. Then Z A f dm nL = Z A g dm nL . Proof.
By hypothesis over f and g , U ( f − g ) ⊂ A × [ − / n , / n ] F for every n ∈ N . Asa consequence, m n + L ( U ( f − g )) ≤ n m nL ( A ) . Since m nL ( A ) < + ∞ , U ( f − g ) is a null setin F n + . This and linearity of the integral is sufficient to entail the desired result.Notice that the above result relies in an essential way upon the hypothesis that m nL ( A ) < + ∞ . A counterexample that does not satisfy this hypothesis is defined asfollows. Let ε ∈ F , ε > ε ∼
0. Define A = [ , ε − ] F , f ( x ) = x ∈ A and g ( x ) = ε for all x ∈ A . Then m L ( A ) = + ∞ , R A f dm L = R A g dm L = F fin is not affected also byinfinitesimal perturbations of the domain. Proposition 2.21.
Let A ⊆ F n be a L-measurable set and let B be a L-measurable setsatisfying m L ( A ∆ B ) = . If f : A ∪ B → F fin is L-integrable, then Z A f dm nL = Z B f dm nL . Proof.
Denote by g the restriction of f to A ∆ B . The hypotheses over f , A and B entailthat U ( g ) ⊂ ( A ∆ B ) × [ − ω , ω ] F for every infinite ω ∈ F . This and the hypothesis that m L ( A ∆ B ) = R A ∆ B f dm nL = . The desired equality is a consequence of thisresult and of linearity of the integral. 18imilarly to Proposition 2.20, the hypotheses over f are necessary. A counterexam-ple that does not satisfy this hypothesis is defined as follows. Let ε ∈ F , ε > ε ∼ A = [ , ] F , f ( x ) = ε − for x ∈ [ , ε ] F and f ( x ) = B = [ ε , ] F ,then A ∆ B = [ , ε ] F , so that m L ( A ∆ B ) =
0. However, R A f dm L = R A ∆ B f dm L = R B f dm L = L -integrable. Moreover, the integrals of the two functions assumethe same value. Theorem 2.22.
If f : A ⊆ R n → R is Lebesgue integrable, then f : st − ( A ) → R defined by f ( x ) = f ( ◦ x ) is L-integrable and R st − ( A ) f dm L = R A f dx.Proof. Suppose at first that λ n ( A ) < + ∞ . Recall that R A f dx = λ n + ( U ( f )) and that,by an argument analogous to that of Theorem 2.13, λ n + ( U ( f )) = m n + L (cid:0) st − U ( f ) (cid:1) < + ∞ . Thanks to the hypothesis that λ n ( A ) < + ∞ , we can adapt the proofs of Proposi-tions 2.20 and 2.21 to obtain that m n + L (cid:0) st − U ( f ) (cid:1) = m n + L (cid:0) U (cid:0) f (cid:1)(cid:1) , so that Z A f dx = λ n + ( U ( f )) = m n + L (cid:0) st − U ( f ) (cid:1) = m n + L (cid:0) U (cid:0) f (cid:1)(cid:1) = Z st − ( A ) f dm n + L , as desired.If λ n ( A ) = + ∞ , we obtain the desired result by σ -additivity of the measures λ n and m nL and by σ -finiteness of λ n .Thanks to Propositions 2.20 and 2.21, it is possible to sharpen Theorem 2.22. Corollary 2.23.
If A ⊆ R n is a Lebesgue measurable set with λ n ( A ) < + ∞ , if f : A → R is Lebesgue integrable and if B ⊆ F nfin is a L-measurable set that satisfies ◦ B = A, thenf : B → F is L-integrable and R B f dm L = R A f dx.Moreover, if g : B → F is L-integrable and ◦ g ( x ) = f ( ◦ x ) for every x ∈ B, then R B g dm L = R A f dx. In this section we compare the real-valued measure m L to a class of uniform mea-sures that generalize the uniform measure developed by Shamseddine and Berz for theLevi-Civita field to arbitrary Cauchy complete non-Archimedean extensions of the realnumbers. In analogy with the Lebesgue measure theory and following [13, 37, 39], we say thata subset of F is m -measurable if it can be approximated with arbitrary precision by acountable sequence of intervals. 19 efinition 3.1. A set A ⊆ F is m-measurable if and only if for every ε ∈ F there existtwo sequences of mutually disjoint intervals { I n } n ∈ N and { J n } n ∈ N such that1. S n ∈ N I n ⊆ A ⊆ S n ∈ N J n ;2. ∑ n ∈ N l ( I n ) and ∑ n ∈ N l ( J n ) converge in F ;3. ∑ n ∈ N l ( J n ) − ∑ n ∈ N l ( I n ) ≤ ε . Notice that, according to the above definition, F is not m -measurable, since if F ⊆ S n ∈ N J n , then ∑ n ∈ N l ( J n ) does not converge in F . Moreover, the family of m -measurablesets is not an algebra, since it is not closed under complements. In addition, due to theproperties of convergence in non-Archimedean field extensions of R , it is also notclosed under countable unions. For further discussion on the family of measurable setson the Levi-Civita field R , we refer to [13, 31, 39].Nevertheless, it is possible to define a F -valued function, that we will still call a measure , according to the convention established in [39], on the family of m -measurablesets under the additional hypothesis that F is Cauchy complete in the order topology. Lemma 3.2.
Suppose that F is Cauchy complete in the order topology. Then for everym-measurable set A ⊂ F m ( A ) = sup ( ∑ n ∈ N l ( I n ) : { I n } n ∈ N is a sequence of mutually disjoint intervals with [ n ∈ N I n ⊆ A ) andm ( A ) inf ( ∑ n ∈ N l ( J n ) : { J n } n ∈ N is a sequence of mutually disjoint intervals with A ⊆ [ n ∈ N J n ) are well-defined. Moreover, m ( A ) = m ( A ) .Proof. The proof can be obtained from the argument in Section 2 and Proposition 2.2of [39]. Notice that this argument only depends upon the hypothesis that F is Cauchycomplete in the order topology and does not rely on other properties of the Levi-Civitafield.By exploiting the above Lemma, it is possible to define a measure for any m -measurable set A . Definition 3.3.
Suppose that F is is Cauchy complete in the order topology. If A ⊂ F is a m-measurable set, then the measure of A, denoted by m ( A ) , is defined asm ( A ) = m ( A ) = m ( A ) . Remark 3.4.
If F = R , the Levi-Civita field, then the measure m of Definition 3.3 isthe uniform measure developed by Shamseddine and Berz. Its properties are discussedin detail in [13, 37, 39]. .2 The relation between m L and the non-Archimedean uniformmeasures The main result of this section is that the real-valued measure m L is compatible withthe non-Archimedean unform measure m on every Cauchy complete non-Archimedeanextension of the real numbers. Namely, m -measurable sets are also L -measurable, andthe real-valued measure is equal to the standard part of the non-Archimedean one. Theorem 3.5.
Suppose that F is Cauchy complete in the order topology. If A ⊂ F ism-measurable, then A ∈ C and ◦ m ( A ) = m L ( A ) .Proof. Let { I n } n ∈ N and { J n } n ∈ N be two families of mutually disjoint intervals sat-isfying S n ∈ N I n ⊆ A ⊆ S n ∈ N J n and ∑ n ∈ N l ( J n ) − ∑ n ∈ N l ( I n ) ≃
0. As a consequence, ◦ m ( A ) = ◦ m ( S n ∈ N I n ) = ◦ m ( S n ∈ N J n ) .Notice that S n ∈ N I n and S n ∈ N J n are L -measurable, since they are a countable unionof L -measurable sets. Since ∑ n ∈ N l ( I n ) converges in F , there exists i ∈ N such that ◦ ( l ( I n )) = l L ( I n ) = n > i . We deduce that ◦ m [ n ∈ N I n ! = ◦ ∑ n ∈ N l ( I n ) ! = ◦ ∑ n ≤ i l ( I n ) ! + ◦ ∑ n > i l ( I n ) ! = ◦ ∑ n ≤ i l ( I n ) ! = ∑ n ≤ i ◦ l ( I n ) . and m L [ n ∈ N I n ! = ∑ n ∈ N l L ( I n ) = ∑ n ≤ i l L ( I n ) = ∑ n ≤ i ◦ l ( I n ) . From the previous equalities we conclude m L [ n ∈ N I n ! = ◦ m [ n ∈ N I n ! and, with a similar argument, we obtain also m L [ n ∈ N J n ! = ◦ m [ n ∈ N J n ! . Thus m L ( S n ∈ N I n ) = ◦ m ( S n ∈ N I n ) = ◦ m ( A ) = ◦ m ( S n ∈ N J n ) = m L ( S n ∈ N J n ) . If m L ( S n ∈ N I n ) = m L ( S n ∈ N J n ) is finite (and possibly equal to 0), by Lemma 2.5,we conclude that A ∈ C and m L ( A ) = m L ( S n ∈ N I n ) = m L ( S n ∈ N J n ) = ◦ m ( A ) .If m L ( S n ∈ N I n ) = m L ( S n ∈ N J n ) = + ∞ , we only need to prove that A is L -measurable.Let i ∈ N such that ◦ ( l ( I n )) = l L ( I n ) = n > i . Then A = (cid:0) S in = I n (cid:1) ∪ N , where N ⊂ R is a m -measurable set with m ( N ) ≃
0. By the first part of the proof, N is a L -nullset, and in particular it is measurable. We have written A as a finite union of intervalsand of a L -null set. Since intervals are L -measurable and since C is a σ -algebra, henceclosed also for finite unions, we deduce that A is also L -measurable, as desired.The previous result allows also to gauge the expressive power of the uniform mea-sure m . In order to do so, we will exploit the standard part function ◦ in order to projectsubsets of F fin to subsets of R . It turns out that m -measurable subsets of F fin are pro-jected to the union of a set that is at most countable and of a finite union of intervals.21 roposition 3.6. Suppose that F is Cauchy complete in the order topology. If A ⊂ F fin is measurable, then there exists a finite (possibly empty) or countable set C, a naturalnumber n ∈ N and m intervals K , . . . , K m such that ◦ A = { ◦ x : x ∈ A } = C ∪ [ j ≤ m K j ! . Proof.
Since A ⊆ F fin is measurable, we deduce that m ( A ) is a finite number. More-over, there are two families of mutually disjoint intervals { I n } n ∈ N and { J n } n ∈ N satisfy-ing • S n ∈ N I n ⊆ A ⊆ S n ∈ N J n , • ∑ n ∈ N l ( J n ) − ∑ n ∈ N l ( I n ) ≃ • m ( A ) − ∑ n ∈ N l ( I n ) ≃ m ( A ) we deduce that l ( I n ) ∈ R fin and that l ( J n ) ∈ R fin for all n ∈ N .Since each of the sums ∑ n ∈ N l ( I n ) and ∑ n ∈ N l ( J n ) converges in F , let i = sup { n ∈ N : l ( I n ) } and i = sup { n ∈ N : l ( J n ) } . Then for every n ≤ i there exists a n , b n ∈ R , a n < b n such that ◦ I n = [ a n , b n ] R . On the other hand, if n > i then there exists c n ∈ R such that ◦ I n = { c n } . Similarly, for every n ≤ i there exists a n , b n ∈ R , a n < b n suchthat ◦ J n = [ a n , b n ] R and for every n > i then there exists c n ∈ R such that ◦ I n = { c n } .We obtain the inclusions { c n : n > i } ∪ [ n ≤ i [ a n , b n ] R ⊆ ◦ A ⊆ { c n : n > i } ∪ [ n ≤ i [ a n , b n ] R . If we prove that S n ≤ i [ a n , b n ] R = S n ≤ i [ a n , b n ] R , then we obtain the desired result.Notice that ∑ n ∈ N l ( J n ) − ∑ n ∈ N l ( I n ) = ∑ n ≤ i l ( J n ) − ∑ n ≤ i l ( I n ) ! + ∑ n > i l ( J n ) − ∑ n > i l ( I n ) ! . Since l ( J n ) ≃ n > i and l ( I n ) ≃ n > i , Lemma 2.11 of [13]implies that ∑ n > i l ( J n ) − ∑ n > i l ( I n ) ≃
0. This result and the hypothesis that ∑ n ∈ N l ( J n ) − ∑ n ∈ N l ( I n ) ≃ ∑ n ≤ i l ( J n ) = ∑ n ≤ i l ( I n ) . This can only happen if S n ≤ i [ a n , b n ] R = S n ≤ i [ a n , b n ] R , as desired.As a consequence, we can choose e.g. m = i , K j = [ a j , b j ] for every j ≤ m and C = S n ≥ i { c n } ∩ A .The above result and Proposition 3.6 entail that the real-valued measure m L allowsfor more measurable sets than the non-Archimedean measure m .22 A real-valued integral on the Levi-Civita field
In this section we exploit the same ideas used in the construction of the real-valuedmeasure and introduce a real-valued integral on functions defined on the Levi-Civitafield. This real-valued integral is obtained from the non-Archimedean integral definedby Shamseddine and Berz in [39] and further developed in [13, 37]. Recently, theintegration on the Levi-Civita field has also been extended in dimension 2 and 3 byFlynn and Shamseddine [22, 40], but in our paper we focus only on the integration indimension 1.We start by recalling the basic definitions and properties of the Levi-Civita fieldand of its non-Archimedean integral.
Definition 4.1.
A set F ⊂ Q is called left-finite if and only if for every q ∈ Q the set { x ∈ F : x ≤ q } is finite. The Levi-Civita field is the set R = { x : Q → R : { q : x ( q ) = } is left-finite } , with the pointwise sum and the product defined by the formula ( x · y )( q ) = ∑ q + q = q x ( q ) · y ( q ) . For a review of the algebraic and topological properties of R , we refer for instanceto [6, 8, 9, 36] and references therein.In the Levi-Civita field there are two notions of convergence: the one induced bythe metric, analogous to the usual definition of limit for real-valued sequences, usuallycalled strong convergence, and the weak convergence. Definition 4.2.
A sequence { a n } n ∈ N of elements of R strongly converges to l ∈ R ifand only if ∀ ε ∈ R , ε > , ∃ n ∈ N : ∀ m > n | c m − l | < ε . A sequence { a n } n ∈ N of elements of R weakly converges to l ∈ R if and only if ∀ ε ∈ R , ε > , ∃ n ∈ N : ∀ m > n max q ∈ Q , q ≤ ε − | ( c m − l )( q ) | < ε . We will denote weak convergence with the expression w-lim n → ∞ a n = l. If a n ∈ R for all n ∈ N and if x ∈ R , we assume that the expression ∑ n ∈ N a n ( x − x ) n denotes the weak limit w-lim k → ∞ ∑ n ≤ k a n ( x − x ) n . Definition 4.3.
We denote by P ( I ( a , b )) the algebra of all power series that weaklyconverge for every x ∈ I ( a , b ) . Since power series with real coefficients weakly converge also in R (we refer to[13] for more details), it is possible to use them to define several extensions of realcontinuous functions to functions defined on the Levi-Civita field. These extensionsare obtained from the Taylor series expansion of a function at a point.23 efinition 4.4. Let f ∈ C ∞ ([ a , b ]) . The analytic extension of f is defined asf ∞ ( r + ε ) = ∞ ∑ i = f i ( r ) ε i i ! . for all r ∈ [ a , b ] and for all ε ∈ M o such that r + ε ∈ [ a , b ] R (i.e. on the nearstandardpoints of [ a , b ] R ). If f is analytic, then f ∞ will be called the canonical extension of f .The only exceptions are the canonical extensions of the exponential function and of thetrigonometric functions sine and cosine, still denoted by e x , sin ( x ) and cos ( x ) for allx ∈ R with λ ( x ) ≥ .Let f ∈ C n ([ a , b ]) , possibly with n = ∞ . The order k extension of f , with ≤ k ≤ n,is defined by f k ( r + ε ) = k ∑ i = f i ( r ) ε i i ! . for all r ∈ [ a , b ] and for all ε ∈ M o such that r + ε ∈ [ a , b ] R . It is possible to prove that each of the extensions introduced in Definition 4.4 isunique and well-defined. The functions f k extend the corresponding real function f ∈ C n ([ a , b ]) in the sense that for every x ∈ [ a , b ] f ( x ) = f k for all k ≤ n , possibly with n = ∞ . For more properties of the continuations of real functions to the Levi-Civitafield, we refer to [6, 8, 9, 10, 13, 36]. We briefly recall the basic notions of the non-Archimedean integration on the Levi-Civita field, following the approach of [13], that differs from the one of [37, 39] in thedefinition of the simple and measurable functions.As with the Lebesgue measure, the family of measurable functions is obtained froma family of simple functions. For the remainder of the paper, we will work with thefamily of simple functions P = S a , b ∈ R , a < b P ( I ( a , b )) : as a consequence, a function f is simple iff there exists an interval I such that supp f = I and f is a power series thatconverges for every x ∈ I . Proposition 4.5.
If f is simple on I ( a , b ) , then there exists a unique simple functiong : [ a , b ] R → R such that g | I ( a , b ) = f .Proof. See [13].With a slight abuse of notation, if f is simple on I ( a , b ) , we will still denote by f the simple function defined on [ a , b ] R that coincides with f on I ( a , b ) .From the algebra of simple functions it is possible to define the family of measur-able functions. Following [13], we do not require that measurable functions must bebounded. Definition 4.6.
Let A ⊂ R be measurable and let f : A → R . The function f is mea-surable iff for all ε ∈ R , ε > , there exists a sequence of mutually disjoint intervals { I n } n ∈ N such that . S n ∈ N I n ⊂ A;2. ∑ n ∈ N l ( I n ) strongly converges in R ;3. m ( A ) − ∑ n ∈ N l ( I n ) < ε ;4. for all n ∈ N , f is simple on I n .We will denote by M ( A ) the set of measurable functions on A. Since every simple function has an antiderivative, it is possible to define the integralof a simple function over an interval by imposing the validity of the fundamental theo-rem of calculus. The integral of a measurable function over a measurable set can thenbe obtained as a limit of the integrals of simple functions over a sequence of intervalssatisfying Definition 4.6.
Definition 4.7.
If f is a simple function over I ( a , b ) whose antiderivative is F, then Z I ( a , b ) f ( x ) = lim x → b F ( x ) − lim x → a F ( x ) . Notice that the two limits in the previous equality are well-defined, since F is simpleon I ( a , b ) and, thanks to Proposition 4.5, F can be extended to a simple function on [ a , b ] R .If A ⊂ R is a measurable set and f : A → R is a measurable function, then defineF A = ( { I n } n ∈ N : [ n ∈ N I n ⊆ A , I n are mutually disjoint and ∀ n ∈ N f is simple on I n ) . The integral of f over A is defined as Z A f ( x ) = lim { I n } n ∈ N ∈ F A , ∑ n ∈ N I n → m ( A ) ∑ n ∈ N Z I n f ( x ) ! whenever the limit on the right side of the equality is defined (and possibly equal to ± ∞ whenever the sequence k ∑ n ≤ k R I n f ( x ) diverges), and it is undefined otherwise. The integral on the Levi-Civita field is coherent with the Lebesgue integral.
Lemma 4.8.
If a , b ∈ R and f ∈ C ω ([ a , b ]) , then f ∞ is measurable and Z [ a , b ] f ∞ = Z [ a , b ] f ( x ) dx . Moreover, if I ( c , d ) ⊆ [ a , b ] R , then Z I ( c , d ) f ∞ ≈ Z [ c [ ] , d [ ]] f ( x ) dx . However, notice that extensions of the form f k , with k ∈ N , are not measurable oversets of a non-infinitesimal measures, since they are not simple over such sets.In analogy with the real measure theory, it is possible to introduce spaces of mea-surable functions whose p -th power has a well-defined integral.25 efinition 4.9. Let A ⊂ R be a measurable set. If ≤ p < ∞ , define L p ( A ) = (cid:26) [ f ] : f and f p are measurable , Z A | f | p is defined and Z A | f | p < + ∞ (cid:27) . If f ∈ L p ( A ) , then define k f k p = ( R A | f | p ) / p . The properties of the L p spaces are studied in detail in [13]. We now introduce a real-valued integral on the Levi-Civita field by exploiting a similaridea as the one used in Section 2 for the introduction of the real-valued measure m L . Inorder to avoid confusion with the L -integral introduced in Section 2.3, we will denotethis new integral as M -integral. The letter M suggests dependence of the integral uponthe family of measurable functions over the Levi-Civita field introduced in Definition4.6. Definition 4.10.
Let A ⊂ R be a measurable set and let f : A → R be a boundedfunction. f is M-integrable iff sup (cid:26) ◦ (cid:18) Z A g (cid:19) : g ∈ L ( A ) and g ( x ) ≤ f ( x ) ∀ x ∈ A (cid:27) = inf (cid:26) ◦ (cid:18) Z A g (cid:19) : g ∈ L ( A ) and g ( x ) ≥ f ( x ) ∀ x ∈ A (cid:27) . If f is bounded and M-integrable, we define Z MA f = sup (cid:26) ◦ (cid:18) Z A g (cid:19) : g ∈ L ( A ) and g ( x ) ≤ f ( x ) ∀ x ∈ A (cid:27) = inf (cid:26) ◦ (cid:18) Z A g (cid:19) : g ∈ L ( A ) and g ( x ) ≥ f ( x ) ∀ x ∈ A (cid:27) . Through this section, we will prove that measurable functions in L are also M -integrable, and the standard part of the non-Archimedean integral is equal to the real-valued integral. Currently, we can only prove this result for bounded measurable func-tions. Proposition 4.11.
Let A ⊂ R be a measurable set. If f ∈ L ( A ) is bounded, then f isM-integrable over A. Moreover, ◦ ( R A f ) = R MA f .Proof. If f ∈ L ( A ) , then f ∈ (cid:8) g ∈ L ( A ) : g ( x ) ≤ f ( x ) ∀ x ∈ A (cid:9) ∩ (cid:8) g ∈ L ( A ) : g ( x ) ≥ f ( x ) ∀ x ∈ A (cid:9) . Thus ◦ ( R A f ) ≤ R MA f ≤ ◦ ( R A f ) , i.e. ◦ ( R A f ) = R MA f .26 orollary 4.12. Let A ⊂ R be a measurable set. Then, for all ≤ p < + ∞ , if f ∈ L p ( A ) is bounded, f p is M-integrable and R MA f p = ◦ (cid:16) R MA f p (cid:17) .Proof. If f ∈ L p ( A ) is bounded, then | f | p ∈ L ( A ) is also bounded. By Corollary4.12 of [39], | R A f p | ≤ R A | f p | . Since f ∈ L p ( A ) , R A | f p | is defined and it is not equalto + ∞ . Thus f p ∈ L ( A ) . By Proposition 4.11 we conclude that R MA f p = ◦ (cid:16) R MA f p (cid:17) ,as desired.The definition of M -integrable functions can be extended to unbounded functionsin the usual way. Definition 4.13.
Let A ⊂ R be a measurable set and let f : A → R be an unbounded,non-negative function. We say that f is M-integrable iff sup (cid:26) ◦ (cid:18) Z A g (cid:19) : g ∈ L ( A ) and g ( x ) ≤ f ( x ) ∀ x ∈ A (cid:27) < + ∞ If f is unbounded, non-negative and M-integrable, we define Z MA f = sup (cid:26) ◦ (cid:18) Z A g (cid:19) : g ∈ L ( A ) and g ( x ) ≤ f ( x ) ∀ x ∈ A (cid:27) . If f is unbounded and non-positive, we say that f is M-integrable iff f − is, and wedefine R MA f = − R MA f − .If f is unbounded, we say that f is M-integrable iff f + and f − are M-integrable. Iff is unbounded and M-integrable, we define R MA f = R MA f + − R MA f − . The real-valud integral is linear.
Proposition 4.14.
If A ⊂ R is a measurable set, then for every M-integrable functionsf and g over A and for every x , y ∈ R fin , Z MA ( x f + yg ) = ◦ x Z MA f + ◦ y Z MA g . (4.1) Proof. If f and g are non-negative and if x and y are non-negative, then linearity of thenon-Archimedean integral over the Levi-Civita field and the properties of the standardpart entail the equalitiessup (cid:26) ◦ (cid:18) Z A h (cid:19) : h ∈ L ( A ) and h ( x ) ≤ x f ( x ) + yg ( x ) ∀ x ∈ A (cid:27) = ◦ x sup (cid:26) ◦ (cid:18) Z A h (cid:19) : h ∈ L ( A ) and h ( x ) ≥ f ( x ) ∀ x ∈ A (cid:27) + ◦ y sup (cid:26) ◦ (cid:18) Z A h (cid:19) : h ∈ L ( A ) and h ( x ) ≥ g ( x ) ∀ x ∈ A (cid:27) . This is sufficient to deduce that equation (4.1) is satisfied.For arbitrary f , g , x , y , the desired property can be obtained by applying the previouspart of the proof to ( x f ) + , ( x f ) − , ( yg ) + and ( yg ) − .27inally, we are ready prove that all m -measurable functions in L , even those thatare not bouunded, are M -integrable, and that their M -integral is equal to the standardpart of the integral of Definition 4.7. Proposition 4.15.
Let A ⊂ R be a measurable set. If f ∈ L ( A ) and if R A | f | ∈ R fin ,then f is M-integrable and Z MA f = ◦ (cid:18) Z A f (cid:19) . Proof. If f ∈ L ( A ) is bounded, then the desired result is entailed by Proposition 4.11.Suppose then that f is unbounded and non-negative, and that λ ( R A | f | ) ≥
0. Asa consequence of the latter hypothesis we have also ◦ ( R A f ) < + ∞ , so that f is M -integrable. Moreover, f ∈ (cid:8) g ∈ L ( A ) : g ( x ) ≤ f ( x ) ∀ x ∈ A (cid:9) so that ◦ ( R A f ) ≤ R MA f .Let now ε ∈ R , ε > < λ ( ε · m ( A )) < + ∞ . Define also a function˜ f : A → R by posing ˜ f ( x ) = f ( x ) + ε . Then ˜ f (cid:8) g ∈ L ( A ) : g ( x ) ≤ f ( x ) ∀ x ∈ A (cid:9) .As a consequence, Z MA f ≤ ◦ (cid:18) Z A ˜ f (cid:19) = ◦ (cid:18) Z A f + Z A ε (cid:19) = ◦ (cid:18) Z A f (cid:19) + ◦ ( ε · m ( A )) . Since 0 < λ ( ε · m ( A )) < + ∞ , ◦ ( ε · m ( A )) = R MA f ≤ ◦ ( R A f ) .We conclude ◦ ( R A f ) ≤ R MA f ≤ ◦ ( R A f ) , as desired.If f is unbounded, non-positive and with λ ( R A | f | ) ≥
0, we can apply the aboveargument to − f .For an arbitrary f ∈ L ( A ) , the previous arguments entail that R MA f ± = ◦ ( R A f ± ) ,so that Z MA f = Z MA f + − Z MA f − = ◦ (cid:18) Z A f + (cid:19) − ◦ (cid:18) Z A f − (cid:19) = ◦ (cid:18) Z A f + − Z A f − (cid:19) = ◦ (cid:18) Z A f (cid:19) , as desired.As a consequence of the coherence between the real-valued integral and the integralin R , we obtain the following result, analogous to Corollary 4.12. Corollary 4.16.
Let A ⊂ R be a measurable set. If f ∈ L p ( A ) , then f p is M-integrableand R MA f p = ◦ ( R A f p ) . It is also possible to define a real-valued integral of functions defined upon some non-measurable sets A ⊆ R . For a relevant class of non-measurable sets, the definition isanalogous to that of the Riemann integral over unbounded sets.28 efinition 4.17. For every q ∈ Q , define A ( q ) = { x ∈ R : λ ( x ) ≥ q } and B ( q ) = { x ∈ R : λ ( x ) > q } .We say that a function f : A ( q ) → R is M-integrable if lim t → ∞ Z M [ − td q , td q ] fexists finite in R . If f is M-integrable over A ( q ) , we define R MA ( q ) f ( x ) = lim t → ∞ R M [ − td q , td q ] f ( x ) .Similarly, we say that a function f : B ( q ) → R is M-integrable if the real limit lim t → q + Z M [ − d t , d t ] fexists finite in R . If f is M-integrable over B ( q ) , we define R MB ( q ) f ( x ) = lim t → q + R M [ − d t , d t ] f ( x ) .Finally, we say that a function f : R → R is M-integrable if the real limit lim t →− ∞ Z M [ − d t , d t ] fexists finite in R . If f is M-integrable over R , we define R M R f ( x ) = lim t →− ∞ R M [ − d t , d t ] f ( x ) .These definitions are extended in the expected way to sets of the form A ( q ) ∩ [ a , b ] R ,B ( q ) ∩ [ a , b ] R , [ a , + ∞ ] R and [ − ∞ , b ] R . These integrals over non-measurable subsets of R are linear, as a consequence oflinearity of the integral over measurable sets. Proposition 4.18.
If A ⊂ R is a measurable set or if it is a set of the form A ( q ) or B ( q ) for some q ∈ Q , then every M-integrable functions f and g over A and every x , y ∈ R fin satisfy equality (4.1) .Proof. We have proven this result under the hypothesis that A is a measurable set inProposition 4.14.If A = A ( q ) or A = B ( q ) for some q ∈ Q , then this is a consequence of the linearity ofthe integral over measurable sets and of the linearity of the real limit of functions. Example 4.19.
Let a ∈ R , a = − , and consider the function f ( x ) = x a defined on theset { x ∈ R fin : x ≥ } = A ( ) ∩ [ , d − ] R . These functions are analytic over intervalsof the form [ , n ] R , with n ∈ N . However, recall that they are not integrable overthe non-measurable set { x ∈ R fin : x ≥ } . Nevertheless, Proposition 4.11 entailsthe equality R M [ , n ] R x a = R [ , n ] R x a for every n ∈ N . Moreover, by Lemma 4.10 of [13], R [ , n ] R x a = R [ , n ] x a dx = n a + a + − a + .A similar argument applies to the function x x − . In this case, we obtain that R M [ , n ] R x − = R [ , n ] R x − = R [ , n ] x − dx = ln ( n ) for every n ∈ N .As a consequence, f ( x ) = x a is M-integrable over { x ∈ R fin : x ≥ } if and only ifa < − , in analogy to what happens for the corresponding real functions. .5 M -integrable representatives of real continuous functions We will now establish an analogous of Theorem 2.22 for the M -integral. Recall that,for the non-Archimedean integral, the only analogue to Theorem 2.22 is Lemma 4.8,that is only valid for real analytic functions of a bounded domain. Instead, for the M -integral we will prove that every canonical or non-canonical extension of a continuousfunction f ∈ C k ([ a , b ]) is M -integrable, and the M -integral is equal to the Lebesgueintegral of f over [ a , b ] . Theorem 4.20.
Let f ∈ C k ([ a , b ]) . Then f h is M-integrable for every ≤ h ≤ k and Z M [ a , b ] R f h = Z [ a , b ] f ( x ) dx . Proof.
Recall that a function s : [ a , b ] → R is a step function if and only if there existsome n ∈ N , a partition of [ a , b ] into n intervals I , . . . , I n and n real numbers s , . . . , s n ,such that s = ∑ i ≤ n s i χ I i . Step functions have a well-defined Riemann integral equal to ∑ i ≤ n s i l ( I i ) .If I , . . . , I n is a partition of [ a , b ] , define J , . . . , J n as the partition of [ a , b ] R satisfy-ing the following properties: • if I i = [ a i , b i ] , then J i = [ a i , b i ] R ; • if I i = ( a i , b i ] , then J i = ( a i , b i ] R ; • if I i = [ a i , b i ) , then J i = [ a i , b i ) R ; • if I i = ( a i , b i ) , then J i = ( a i , b i ) R ;If s : [ a , b ] → R is a step function, then it can be extended to a function ˜ s : [ a , b ] R → R by posing ˜ s = ∑ i ≤ n s i J i . The function ˜ s is trivially m -measurable, moreover Z [ a , b ] R ˜ s = Z M [ a , b ] R ˜ s = Z [ a , b ] s ( x ) dx . (4.2)Let now f ∈ C k ([ a , b ]) and let 0 ≤ h ≤ k . For every ε ∈ R , ε > s + ε be astep function satifying s + ε ( x ) > f ( x ) for every x ∈ [ a , b ] and Z [ a , b ] s + ε ( x ) dx − Z [ a , b ] f ( x ) ≤ ε . (4.3)Similarly, let s − ε be a step function satifying s − ε ( x ) < f ( x ) for every x ∈ [ a , b ] and Z [ a , b ] f ( x ) dx − Z [ a , b ] s − ε ( x ) ≤ ε . (4.4)30otice that, by definition, we have ˜ s + ε ( x ) > f h ( x ) for every x ∈ [ a , b ] R and ˜ s − ε ( x ) < f h ( x ) for every x ∈ [ a , b ] R . Moreover, equations (4.2), (4.3) and (4.4) entail that R [ a , b ] R ˜ s + ε − R [ a , b ] R ˜ s − ε < ε .Since ε is an arbitrary positive real number, we deduce thatsup (cid:26) ◦ (cid:18) Z [ a , b ] R g (cid:19) : g ( x ) ≤ f h ( x ) ∀ x ∈ [ a , b ] R (cid:27) = inf (cid:26) ◦ (cid:18) Z [ a , b ] R g (cid:19) : g ( x ) ≥ f h ( x ) ∀ x ∈ [ a , b ] R (cid:27) = Z [ a , b ] f ( x ) dx i.e. that f h is M -integrable and R M [ a , b ] R f h = R [ a , b ] f ( x ) dx .A similar result applies also to continuous and integrable functions defined overopen intervals or over R . Corollary 4.21.
If f ∈ C k (( a , b )) and R ( a , b ) | f ( x ) | dx < + ∞ , then f h is M-integrable forevery ≤ h ≤ k and Z M ( a , b ) f h = Z ( a , b ) f ( x ) dx . Proof. If f is bounded, it is sufficient to apply the same argument of Theorem 4.20.If f is not bounded and non-negative, it is sufficient to notice thatsup (cid:26) Z A s ( x ) dx : s is a step function and s ( x ) < f ( x ) ∀ x ∈ ( a , b ) (cid:27) = Z ( a , b ) f ( x ) dx and that, with a similar argument as the one used in the proof of Theorem 4.20,sup (cid:26) ◦ (cid:18) Z ( a , b ) R g (cid:19) : g ( x ) ≤ f h ( x ) ∀ x ∈ ( a , b ) R (cid:27) = sup (cid:26) Z ( a , b ) R s ( x ) dx : s is a step function and s ( x ) < f ( x ) ∀ x ∈ ( a , b ) (cid:27) . For arbitrary functions f ∈ C k (( a , b )) , it is sufficient to apply the previous part of theproof to f + and to f − . Notice also that the hypothesis R ( a , b ) | f ( x ) | dx < + ∞ is sufficientto entail that both R ( a , b ) f + ( x ) dx < + ∞ and R ( a , b ) f − ( x ) dx < + ∞ . Corollary 4.22.
If f ∈ C k ( R ) and R R | f ( x ) | dx < + ∞ , then f h is M-integrable over R fin for every ≤ h ≤ k and Z M R fin f h = Z R f ( x ) dx . Proof.
It is a consequence of the definition of R M R fin f h and of the property that R [ a , b ] R f h = R [ a , b ] f ( x ) dx for every a , b ∈ R . 31 .6 A restricted integration by parts for the M -integral Notice that the M -integral does not satisfy the fundamental theorem of calculus. Acounterexample is obtained as follows: define a function F : R → R by posing F ( x ) = (cid:26) x < x ∈ µ ( ) x > x µ ( ) . (4.5)Then F is continuous, F ′ ( x ) = x ∈ R , and F is not m -measurable over anybounded interval that includes µ ( ) . However, F is M -integrable on every interval [ a , b ] R ⊂ R fin . Consequently, the fundamental theorem of calculus fails, since e.g. if a < b > F ( b ) − F ( a ) = = = Z M [ a , b ] R F ′ . Consequently, an integration by parts formula fails for the real-valued integral.These drawbacks occur once a sufficiently large class of functions are integrable.In fact, it is well-known that in non-Archimedean field extensions of R derivable func-tions with null derivative need not be constant, and that functions with positive deriva-tive need not be increasing. For more details on this issue where F = R , we refer forinstance to [8, 39] and references therein. Notice also that in fields of hyperreal num-bers this problem is overcome by only working with internal functions, and that thehyperreal counterpart of function (4.5) is external.Despite these limitations, we can still establish a reduced version of the fundamen-tal theorem of calculus and of an integration by parts formula. As expected, theseresults can only be obtained for measurable functions in the sense of Definition 4.6. Proposition 4.23.
If a , b ∈ R fin , a < b and if f , g ∈ L ([ a , b ] R ) , then for every F ∈ L ([ a , b ] R ) such that F ′ ( x ) = f ( x ) for all x ∈ [ a , b ] R , Z M [ a , b ] R f = ◦ ( F ( b ) − F ( a )) . Proof.
As a consequence of Proposition 3.24 of [13], for f and F satisfying the hy-potheses we have Z [ a , b ] R f = F ( b ) − F ( a ) . The desired equality can then be obtained as a consequence of Proposition 4.15.As a consequence of the above result, we also get a restricted integration by partsfor the L -integral. Proposition 4.24.
If a , b ∈ R fin , a < b and if f , g ∈ L ([ a , b ] R ) , then ◦ ( f ( b ) g ( b ) − f ( a ) g ( a )) = Z MA f ′ g + Z MA f g ′ . roof. Recall that, if f and g are analytic over the Levi-Civita field, then ( f g ) ′ = f ′ g + g ′ f (see e.g. Theorem 5.2 of [7]). Thanks to Proposition 4.23, we deduce that ◦ ( f ( b ) g ( b ) − f ( a ) g ( a )) = Z M [ a , b ] R f ′ g + Z M [ a , b ] R g ′ f , as desired. M -integrable functions and their derivatives Flynn and Shamseddine recently showed that it is possible to represent the Dirac dis-tribution with some measurable functions defined over the Levi-Civita field [22]. Theirresults are mostly concerned with the duality between the so-called delta functions andanalytic functions over R . A duality with real continuous functions is developed in[13], where the discussion is extended also to the derivatives of the Dirac distribution.However, the proposed approach is somewhat cumbersome, since it is not possible torepresent real continuous functions that are not analytic with measurable functions overthe Levi-Civita field (see Proposition 3.16 and Lemma 3.17 of [13]).This drawback is overcome by using the M -integral defined in Section 4. As anexample, we now discuss the representation of the Dirac distribution and of its deriva-tives with M -integrable functions on R . It is relevant to compare the treatment with thereal-valued integral with the one obtained with weak limits of m -measurable functions,discussed in detail in [13].We extend the definition of Dirac-like measurable functions given in [13] to that ofDirac-like M -integrable functions. Definition 4.25.
A M-integrable function δ r : R fin → R is Dirac-like at r ∈ A iff1. δ r ( x ) ≥ for all x ∈ A;2. there exists h ∈ M o , h > such that supp δ r ⊆ [ r − h , r + h ] R ⊆ A;3. R M R fin δ r = . Notice that, as a consequence of Proposition 4.15, Dirac-like measurable functionsare also Dirac-like M -integrable functions.With this definition and thanks to Theorem 4.20, it is possible to show that Dirac-like M -integrable functions are good representatives of the real Dirac distribution. Proposition 4.26.
Let r ∈ R . For all Dirac-like M-integrable functions δ r and for allf ∈ C ( R ) , Z M R fin (cid:0) δ r · f (cid:1) = f ( r ) . Proof.
The integral is well-defined thanks to Corollary 4.22. Notice that f is constantover µ ( r ) . Then we have Z M R fin (cid:0) δ r · f (cid:1) = f ( r ) · Z M R fin δ r = f ( r ) , as desired. 33n a similar way it is possible to represent the derivatives of the Dirac distribution. Proposition 4.27.
Let r ∈ R . For all Dirac-like M-integrable functions δ r , if δ r ∈ L ( R ) ∩ C k ( R ) , then for all f ∈ C n ( R ) with n ≥ k, and for all k ≤ j ≤ n, Z M R fin (cid:16) δ ( k ) r · f j (cid:17) = ( − ) k f ( k ) ( r ) . Proof.
The integral is well-defined thanks to Corollary 4.22.Moreover, if h ∈ M o , h > δ r ⊂ [ r − h , r + h ] R , then f j ∈ L ([ r − h , r + h ] R ) . In addition, δ ( m ) r ( r ± h ) = ≤ m ≤ k . Taking into account theseequalities, the restricted integration by parts formula established in Proposition 4.24ensures that Z M R fin (cid:16) δ ( k ) r · f j (cid:17) = Z M [ r − h , r + h ] R (cid:16) δ ( k ) r · f j (cid:17) = ( − ) k Z M [ r − h , r + h ] R (cid:16) δ r · f ( k ) j (cid:17) for every 0 ≤ m ≤ k . By Proposition 4.26, ( − ) k Z M [ r − h , r + h ] R (cid:16) δ r · f ( k ) j (cid:17) ≃ ( − ) k · ◦ (cid:16) f ( k ) j ( r ) (cid:17) . Our hypotheses over j and Theorem 89 of [6] entail that ◦ (cid:16) f ( k ) j ( r ) (cid:17) = f ( k ) ( r ) , so theproof is concluded.Notice that the above proposition is false if j < k . Under this hypothesis, f j islocally a polynomial of degree at most j , so that f ( k ) j ( x ) = x ∈ R fin .A comparison of propositions 4.26 and 4.27 with their counterparts in [13] showsthe advantage of the use of the real-valued integral for the representation of real distri-butions in the Levi-Civita field. Significantly, in [13] we were not able to define a C ∞ structure over the spaces L p ∩ C ∞ ( R ) with the duality given by the non-Archimedeanintegral. However, the results discussed in this section suggest that it is possible to doso by using the duality induced by the real-valued integral. Acknowledgements
Alessandro Berarducci and Mauro Di Nasso provided insightful comments to someideas presented in Section 2.
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