Transreal Integral

This paper improves an early approach and defines an integral on transreal numbers which extends the usual integral on real numbers. The present integral works on transreal numbers and integrates all functions which are properly or improperly integrable, in the usual sense, on real numbers.


Introduction
Real numbers are used in everyday life to count and measure finite things. In some areas of mathematics and science, the real numbers are extended with a positive and a negative infinity that make sense of unboundedly large magnitudes. The transreal numbers take this further: they contain all of the real numbers, the infinities of the extended real numbers, and a new non-finite number, namely nullity. Thus the transreal numbers extend counting and measuring from the finite world of everyday experience to a hypothetical and, perhaps, actual world that contains everyday finite quantities and both infinite and nullity values at singularities. Now that we have access to transreal numbers, it seems natural to ask what operations on real numbers can be extended to transreal numbers.
In particular, what parts of real calculus -which is widely used in science and engineering to describe the workings of the physical world and to design machines and structures -can be extended to deal with infinite and nullity singularities.
In the papers [1] [3] [4] [5] we have been extending the real calculus to a transreal calculus. We have defined topology, limits, continuity, derivative and integral on transreal numbers. We aimed to extend the usual definitions on real numbers so that when any of transreal topology, limit, continuity or derivative is applied to real numbers the result is the same as when using the usual, real, definition. All real numbers are transreal numbers, so the transreal derivative, for example, can be applied to real numbers. When we take the transreal derivative of an arbitrary real function of a real variable, we get exactly the usual, real, derivative of that function. The same occurs for topology, limits and continuity. Similarly transreal topology and limits give the usual answers when applied to extended real numbers that admit infinities. However, the same does not occur for all improperly integrable functions. According to the earlier definition of the integral on transreal numbers, some improperly integrable (in the usual sense) functions are not integrable in the transreal sense. For example, the function x → sin x x is improperly integrable in [0, ∞] but is not integrable in the earlier transreal sense. In [2] we proposed an integral on extended real numbers. All properly or improperly integrable functions are integrable in the sense proposed in [2]. Here we extend that integral to the transreal numbers. In this way, the present paper introduces an integral, defined on transreal numbers, that contains the real integral, as promised in the first paragraph of the Section V of [5]. All properly or improperly integrable functions, in the usual real and extended real senses, are integrable in the transreal sense, introduced here.

Preliminaries
The set of transreal numbers, denoted R T , is formed by the real numbers and the three new elements minus infinity, infinity and nullity, which are denoted, respectively, by −∞, ∞ and Φ. Therefore R T = R∪{−∞, ∞, Φ}. Division by zero is allowed in the set of transreal numbers. Specifically −1/0 = −∞, 1/0 = ∞ and 0/0 = Φ. The arithmetic and order relation defined on R T are such that for each x, y ∈ R T it follows that: ii) The following does not hold x < Φ or Φ < x.
The consistency of transreal arithmetic is proved in [6].
For all x, y ∈ R T , we write x < y if and only if x < y does not hold and we write x > y if and only if x > y does not hold. Notice that < is not equivalent to ≥. For example Φ < 0 but Φ ≥ 0 does not hold. However, for all x, y ∈ R T \ {Φ}, it follows that x < y if and only if x ≥ y and x > y if and only if x ≤ y.
In Definition 14 in [3] and in Definition 21 in [5] we defined the supremum and infimum on transreal numbers but there is a mistake there. Those definitions must be replaced by Definition 1 below. Definition 1. Let A ⊂ R T be an arbitrary non-empty set. We say that u ∈ R T is the supremum of A and we write u = sup A if and only if one of the following conditions occurs: And we say that v ∈ R T is the infimum of A and we write v = inf A if and only if one of the following conditions occurs: iii Notice that for all a ∈ R T we have that [a, Φ] = {a, Φ} and [Φ, a] = {a, Φ}.
The transreal numbers are a topological space where the open subsets are arbitrary unions of finitely many intersections of the following four kinds of intervals: The topology of R T contains the topology of R, that is, when it is restricted to subsets of R, it coincides with the topology of R.
The definition for the convergence of a sequence is the usual in a topological space. That is a sequence, (xn) n∈N ⊂ R T , converges to x ∈ R T if and only if for each neighbourhood, V ⊂ R T of x, there is nV ∈ N such that xn ∈ V for all n ≥ nV . Notice that if (xn) n∈N ⊂ R and L ∈ R then limn→∞ xn = L in R T if and only if limn→∞ xn = L in the usual sense in R. Furthermore, (xn) n∈N diverges, in the usual sense, to negative infinity if and only if limn→∞ xn = −∞ in R T . Similarly (xn) n∈N diverges, in the usual sense, to infinity if and only if limn→∞ xn = ∞ in R T . Notice also that limn→∞ xn = Φ if and only if there is k ∈ N such that xn = Φ for all n ≥ k.
Let (xn) n∈N ⊂ R T . For each n ∈ N, we define sn := n i=1 xi. The sequence (sn) n∈N is called a series and is denoted by xn. Each sn is called a partial sum of xn and xn is called the n-th term of xn. We say that xn converges or is convergent if and only if there is the limn→∞ sn. Otherwise, xn diverges or is divergent. When xn is convergent we denote ∞ n=1 xn := limn→∞ sn. Let (xn) n∈Z ⊂ R T . For each n ∈ N we denote rn :=

Henceforth the integral defined above is called simply an integral and is denoted by
The integrable functions, in the sense of Definition 3, make a superset of the Riemann, proper or improper, integrable functions. That is, this paper's definition integrates every function which Riemann integrates properly or improperly.
) ⊂ R and f is Riemann integrable, either as a proper integral or as an improper integral.
Since a = b and Φ / ∈ [a, b] we can suppose, without loss of generality, that xn < xn+1 for all n ∈ N whatever P = (xn) n∈Z ∈ P R T f ; [a, b] . In this way, the result follows from Theorem 2.1 of [2].
The Theorem 4 shows that the integral defined in the transreal domain agrees with the usual integral when applied on real numbers.
Next, we show several cases where the integral result is nullity.
In an analogous way we can see that Therefore f is integrable and b a f = Φ. II) The proof is analogous to item I.

III) Suppose a, b /
∈ R and a = b. If a = Φ then the result is already proved above. If a = −∞ then b = −∞. For all P = (xn) n∈Z ∈ In an analogous way we can see that , since a = b, it follows that (xn) n∈Z ⊂ [a, a] = {a}, that is, xn = a for all n ∈ Z whence ∆xn = a − a ∈ {0, Φ} and since f (a) / ∈ R, it follows that mn = f (a) / ∈ R for all n ∈ Z whence mn∆xn = Φ for all n ∈ Z. Thus L f ; P = Φ for all P ∈ P R T f ; [a, b] . Thereby sup L(f ; P ); P ∈ P R T f ; [a, b] = Φ.
In an analogous way we can see that Since c ∈ (a, b) we have that a < c < b whence c ∈ R and a = Φ. For all P = (xn) n∈Z ∈ P R T f ; [a, b] it follows that (xn) n∈Z ⊂ [a, b] and limn→−∞ xn = a whence there is n0 ∈ Z such that xn ∈ [a, c) for all n ≤ n0. Thus Thereby In an analogous way we can see that Therefore f is integrable and If there is n0 ∈ Z such that xn 0 = a then xn 0 −1 = a whence ∆xn 0 = a − a ∈ {0, Φ}. Since mn 0 = f (a) = ∞, mn 0 ∆xn 0 = Φ.
Thus L f ; P = Φ for all P ∈ P R T f ; [a, b] . Thereby sup L(f ; P ); P ∈ P R T f ; [a, b] = Φ.
In an analogous way we can see that inf U (f ; P ); P ∈ P R T f ; [a, b] = Φ.
Therefore f is integrable and b a f = Φ. In an analogous way we can see that f is integrable and

Conclusion
An integral on transreal numbers was first defined in [5]. That integral is not the most general one because there are functions integrable in the usual sense which are not integrable in the sense of [5]. The present paper has taken the approach from [2] and defined an integral on transreal numbers which generalises the usual integral on real numbers. Every function integrable in the usual (Riemann) sense, properly or improperly, is integrable in the sense introduced here. In addition, several arrangements of transreal numbers make the integral results nullity.