A foundational theory of an extended Cauchy integral for generalized regulated real-valued functions on

Abstract

In this manuscript, we provide a generalization of the Cauchy integral to appropriate families of real-valued functions defined in ${\mathbb{R}}^n$. The results are motivated by a generalization of regulated functions. The foundational theory for this extended Cauchy integral is developed in detail in this self-contained work. We derive some properties for appropriate families of functions and introduce the Cauchy integral for appropriate step functions. Using these results, the integral for generalized regulated functions is presented next together with some basic properties of the generalized Cauchy integral, including the linearity, the non-negativity and the monotonicity properties. Various illustrative examples on the applicability of our theoretical results are provided.

Keywords:

• Appropriate families
• extended Cauchy integral
• generalized regulated functions

2010 Mathematics Subject Classification:

• Primary 26Axx

1. Introduction

Nowadays, the Riemann–Darboux integral is still a popular tool in science and engineering in view of its simplicity and intuitiveness [1]. Moreover, since the fundamental theorem of calculus holds, this integral has become a popular topic for developing operative mathematical skills at the undergraduate level [2,3]. On the other hand, however, the theory of the Riemann–Darboux integral presents a number of limitations from a mathematical point of view. As an example, the exchange of the derivative (or the limit) and the integral operators requires strong assumptions on the family of functions involved [4]. Among other important results, the dominated convergence theorem and the monotone convergence theorem for the Riemann–Darboux integral need to suppose that the convergence of the functions is uniform. This is a very restrictive hypothesis, especially when only point-wise convergence is required for those results to be valid under Lebesgues' theory of integration.

Despite having great historical and theoretical importance, another important shortcoming in the Riemann–Darboux theory is that functions which are integrable cannot be completely characterized within this theory. Indeed, the characterization of these functions is carried out employing arguments from Lebesgues' theory of measure and integration [5]. In fact, it is well known that a function is integrable exactly when its collection of discontinuities has a Lebuesgue measure equal to zero. On the other hand, the Cauchy integral provides a theory in which the integrable functions are completely characterized without a need to appeal to Lebegues' theory [6]. This is obviously a theoretical advantage over the Riemann–Darboux integral, in spite of the fact that Cauchy's integral theory is less popular and known.

In the present manuscript, we will extend the definition of the Cauchy integral given in [7]. In that work, the integral is introduced as the countable limit of integrals of step functions that converge uniformly to a regulated function. Our approach will hinge on generalizing the concept of regulated functions following ideas and results derived in [8]. It is important to mention here that Cauchy's integral was defined in [9] using primitives of regulated functions. However, in the present work, we believe that it is more natural and convenient to extend the definition given in [7] and avoid the use of primitives. To that end, Section 2 of this work will study appropriate families, that is, families of functions on which generalized regulated functions are defined. We will establish then the appropriate family of interest in this work. In Section 3, we will define the generalized Cauchy integral for appropriate step functions. Then we will define the generalized Cauchy integral for generalized regulated functions in Section 4, and demonstrate some properties, like the linearity, the non-negativity and the monotonicity. Some illustrative examples are provided in Section 5. Finally, this work closes with a section of concluding remarks and discussions.

2. Preliminaries

Generalized regulated functions were introduced in 1979 by T. Davison [8], by using the so-called ‘appropriate families’. The present section is devoted to recall those families and establish the generic appropriate family of $\mathbb{R}$ which will be used throughout this work.

Definition 2.1

In general, assume that X is some fixed topological space, and let $\mathcal{F}$ be a collection of sets in X. The class $\mathcal{F}$ is an appropriate family when the next three requirements hold:

1. Under the operations of unions and intersections of sets, $\mathcal{F}$ is a lattice [10].

2. If C and D are sets in $\mathcal{F}$ and $D\subseteq C$, it follows that $C\setminus D\in \mathcal{F}$.

3. A basis for the topology of X is the collection of all open sets of X in $\mathcal{F}$.

Generalized regulated functions will be introduced through appropriate families using the following definition.

Definition 2.2

Let $\mathcal{F}$ be an appropriate family on the topological space X, and assume that $x\in X$. Then $f:X\to \mathbb{R}$ will be called a regulated function at x if for each $\epsilon >0$ we can find $N\in \mathcal{F}$, such that:

1. N is neighbourhood of x, and

2. there are sets $A_1,\dots A_k\in \mathcal{F}$ such that $N=A_1\cup \cdots \cup A_k$, with the property that $|f(t)-f(s)|\le \epsilon$, for all $i=1,\dots ,k$ and $s,t\in A_i$.

For the remainder of this manuscript, we will let X be ${\mathbb{R}}^n$ with the standard topology and $n\in \mathbb{N}$. Additionally, we will let R represent the collection of all regulated functions.

Definition 2.3

As usual, intervals are sets of the following forms: $\begin{array}{rl}(a,b)& :=\{x\in \mathbb{R}:a<x<b\},\\ [a,b]& :=\{x\in \mathbb{R}:a\le x\le b\},\\ [a,b)& :=\{x\in \mathbb{R}:a\le x<b\},\\ (a,b]& :=\{x\in \mathbb{R}:a<x\le b\}.\end{array}$Here, $a\le b$ are real numbers. Intervals of the first type are called open, while those of the second type are called closed. A box in ${\mathbb{R}}^n$ is a set of the form $B=I_1\times \cdots \times I_n$, where the sets $I_1,\dots ,I_n$ are intervals in the real numbers. When the intervals $I_1,\dots ,I_n$ are open and have the same length, then we say that B is an open box.

In what follows, we let $\mathcal{F}$ be the collection of finite unions of boxes in ${\mathbb{R}}^n$. We claim that $\mathcal{F}$ is an appropriate family.

Lemma 2.1

Under unions and intersections of sets, the family $\mathcal{F}$ is a lattice.

Proof.

Clearly, $(\mathcal{F},\subseteq )$ is a partial order. Let $U,V\in \mathcal{F}$. The definition of $\mathcal{F}$ assures that (1) $\begin{array}{rl}U& =\bigcup _{i=1}^{m_1}{U}_i,\\ V& =\bigcup _{j=1}^{m_2}{V}_j,\end{array}$(1) with $U_i$ and $V_j$ boxes, for each $i=1,\dots ,m_1$, $j=1,\dots m_2$. It is obvious that the supremum (or least upper bound) of U and V is the set $sup\{U,V\}=\left(\bigcup _{i=1}^{m_1}{U}_i\right)\bigcup \left(\bigcup _{j=1}^{m_2}{V}_j\right)=\bigcup _{k=1}^{m_1+m_2}{R}_k,$where $R_k=U_k$ when $k=1,\dots ,m_1$, and $R_k=V_{k-m_1}$ if $k=m_1+1,\dots ,m_1+m_2$. On the other hand, the infimum (or greatest lower bound) of U and V is (2) $\begin{array}{rl}inf\{U,V\}& =\left(\bigcup _{i=1}^{m_1}{U}_i\right)\bigcap \left(\bigcup _{j=1}^{m_2}{V}_j\right)\\ & =\bigcup _{i=1}^{m_1}\left[\bigcup _{j=1}^{m_2}(U_i\cap V_j)\right]=\bigcup _{k=1}^{m_1{m}_2}{S}_k,\end{array}$(2) where each $S_k$ is a set of the form $U_i\cap V_j$. Since each $U_i$ and $V_j$ are boxes, it follows that $R_k$ is also a box. As a consequence, we conclude that $\mathcal{F}$ is a lattice.

Lemma 2.2

The condition $U\setminus V\in \mathcal{F}$ holds if $V\subseteq U$ and $U,V\in \mathcal{F}$.

Proof.

Assume that $U,V\in \mathcal{F}$ are of the form (1(1) $\begin{array}{rl}U& =\bigcup _{i=1}^{m_1}{U}_i,\\ V& =\bigcup _{j=1}^{m_2}{V}_j,\end{array}$(1) ), and notice that $U\setminus V=\bigcup _{i=1}^{m_1}\left[\bigcap _{j=1}^{m_2}(U_i\setminus V_j)\right].$We claim that $U_i\setminus V_j$ is a finite union of boxes. Indeed, the case when $U_i\setminus V_j=\mathrm{\varnothing }$ is trivial, so let us suppose that $U_i\setminus V_j\ne \mathrm{\varnothing }$. By definition, $U_i$ and $V_j$ have the forms $\begin{array}{rl}{U}_i& =I_1\times \cdots \times I_n,\\ V_j& =J_1\times \cdots \times J_n,\end{array}$where $I_k,J_k$ are intervals, for each $k\in \{1,\dots ,n\}$. Notice that $I_k\setminus J_k$ is equal to an interval $R_{k_0}$ or the union of two disjoint intervals $R_{k_1}$ and $R_{k_2}$. Let us consider the cross-products of all the possible combinations of these intervals, that is, the boxes of the form (3) $R_{1_r}\times \cdots \times R_{n_r},$(3) where $k_r$ is equal to $k_0$ if $I_k\setminus J_k$ is a single interval, or is equal to the values $k_1$ and $k_2$ if $I_k\setminus J_k$ is the union of two intervals. Obviously, these boxes are disjoint. In this way, $U_i\setminus V_j$ is equal to the finite disjoint union of the boxes (3(3) $R_{1_r}\times \cdots \times R_{n_r},$(3) ), and we can denote this fact as follows: $U_i\setminus V_j=\bigcup _{k=1}^{m_{\mathrm{ij}}}{R}_k^{\mathrm{ij}}.$As a consequence, $U\setminus V=\bigcup _{i=1}^{m_1}\left[\bigcap _{j=1}^{m_2}(U_i\setminus V_j)\right]=\bigcup _{i=1}^{m_1}\left[\bigcap _{j=1}^{m_2}\left(\bigcup _{k=1}^{m_{\mathrm{ij}}}{R}_k^{\mathrm{ij}}\right)\right].$Finally, using induction, we will check now that $\bigcap _{j=1}^{m_2}(\bigcup _{k=1}^{m_{\mathrm{ij}}}{R}_k^{\mathrm{ij}})$ is a finite union of boxes. If $m_2=2$, then we can mimic the argument to derive (2(2) $\begin{array}{rl}inf\{U,V\}& =\left(\bigcup _{i=1}^{m_1}{U}_i\right)\bigcap \left(\bigcup _{j=1}^{m_2}{V}_j\right)\\ & =\bigcup _{i=1}^{m_1}\left[\bigcup _{j=1}^{m_2}(U_i\cap V_j)\right]=\bigcup _{k=1}^{m_1{m}_2}{S}_k,\end{array}$(2) ). Let us suppose that the claim is true for $m_2=r>2$. Notice then that $\begin{array}{rl}\bigcap _{j=1}^{r+1}\left(\bigcup _{k=1}^{m_{\mathrm{ij}}}{R}_k^{\mathrm{ij}}\right)& =\left[\bigcap _{j=1}^r\left(\bigcup _{k=1}^{m_{\mathrm{ij}}}{R}_k^{\mathrm{ij}}\right)\right]\bigcap \left(\bigcup _{k=1}^{m_{i,r+1}}{R}_k^{i,r+1}\right)\\ & =\left(\bigcup _{k=1}^p\widehat{R_k}\right)\bigcap \left(\bigcup _{k=1}^{m_{i,r+1}}{R}_k^{i,r+1}\right)=\bigcup _{k=1}^l{R}_k.\end{array}$The last equality was obtained as in the proof of (2(2) $\begin{array}{rl}inf\{U,V\}& =\left(\bigcup _{i=1}^{m_1}{U}_i\right)\bigcap \left(\bigcup _{j=1}^{m_2}{V}_j\right)\\ & =\bigcup _{i=1}^{m_1}\left[\bigcup _{j=1}^{m_2}(U_i\cap V_j)\right]=\bigcup _{k=1}^{m_1{m}_2}{S}_k,\end{array}$(2) ). This result follows now by mathematical induction.

Lemma 2.3

A basis for the topology of X is the collection of sets in $\mathcal{F}$ which are open.

Proof.

Let ${\mathcal{B}}^{\prime }$ denote the set of open boxes in ${\mathbb{R}}^n$ (which is a basis for the standard topology on ${\mathbb{R}}^n$), and denote by $\tau ({\mathcal{B}}^{\prime })$ the topology generated by ${\mathcal{B}}^{\prime }$. Define then the collection $\mathcal{B}=\mathcal{F}\cap \tau ({\mathcal{B}}^{\prime })$. It will be shown firstly that $\mathcal{B}$ is the basis for a topology in X. More precisely, the following conditions will be established [11]:

1. There are basis elements that contain x, for each $x\in X$.

2. Suppose that $x\in B_1\cap B_2$ and $B_1,B_2\in \mathcal{B}$. Then $x\in B\subseteq B_1\cap B_2$ for some $B\in \mathcal{B}$.

To check that the first condition is satisfied, let $x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$, and define the box $R_x\in \mathcal{F}$ by $R_x=(x_1-1,x_1+1)\times \cdots \times (x_n-1,x_n+1)$. Clearly, $x\in R_x$ and $R_x\in \mathcal{B}$, which show that (i) is true. To establish property (ii), let $B_1,B_2\in \mathcal{B}$ and suppose $x=(x_1,\dots ,x_n)\in B_1\cap B_2$. Notice that $\begin{array}{rl}{B}_1& =\bigcup _{i=1}^{m_1}{R}_{1i},\\ B_2& =\bigcup _{j=1}^{m_2}{R}_{2j}.\end{array}$Since $x\in B_1\cap B_2$, then $x\in R_{1i}$ and $x\in R_{2j}$, for some indexes i and j. On the other hand, each intersection $R_{1i}\cap R_{2j}$ is a box, which means that it is a cross-product of intervals $I_k$, each one of them with endpoints $a_k$ and $b_k$, where $k=1,\dots ,n$. Let us define it now (4) $\begin{array}{rl}\delta & =\underset{k=1,\dots ,n}{min}\{|b_k-x_k|,|x_k-a_k|\},\end{array}$(4) (5) $\begin{array}{rl}B& =(x_1-\delta ,x_1+\delta )\times \cdots \times (x_n-\delta ,x_n+\delta )\subseteq R_{1i}\\ & \cap R_{2j}.\end{array}$(5) Clearly, $x\in B$ and $B\subseteq B_1\cap B_2$ and $B\in \mathcal{B}$. Conclude that $\mathcal{B}$ is a basis for a topology.

Assume that $\tau (\mathcal{B})$ represents the smallest topology containing $\mathcal{B}$. The following discussion will establish that $\tau (\mathcal{B})=\tau ({\mathcal{B}}^{\prime })$. To prove that $\tau ({\mathcal{B}}^{\prime })\subseteq \tau (\mathcal{B})$, let $B^{\prime }\in {\mathcal{B}}^{\prime }$ and $x\in {\mathbb{R}}^n$ satisfy $x\in B^{\prime }$. Due to the fact that $B^{\prime }$ is an open box, then $B^{\prime }\in \mathcal{B}$ and $x\in B\subseteq B^{\prime }$. Conversely, to prove that $\tau (\mathcal{B})\subseteq \tau ({\mathcal{B}}^{\prime })$, let $x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$ and $B\in \mathcal{B}$ satisfy $x\in B$. By definition, B has the form $B=\bigcup _{i=1}^m{R}_i,$where $R_i$ is a box, for each $i=1,\dots m$. Since $x\in B$, then $x\in R_i$ for some $i\in \{1,\dots ,m\}$. We know that $R_i$ is a cross-products of intervals $I_k$, each one with ends $a_k$ and $b_k$, where $k=1,\dots ,n$. Let us define δ as (4(4) $\begin{array}{rl}\delta & =\underset{k=1,\dots ,n}{min}\{|b_k-x_k|,|x_k-a_k|\},\end{array}$(4) ), and B as $B^{\prime }$ in (5(5) $\begin{array}{rl}B& =(x_1-\delta ,x_1+\delta )\times \cdots \times (x_n-\delta ,x_n+\delta )\subseteq R_{1i}\\ & \cap R_{2j}.\end{array}$(5) ). Evidently, $B^{\prime }\in {\mathcal{B}}^{\prime }$ and $x\in B^{\prime }\subseteq B$, which implies that $\tau (\mathcal{B})\subseteq \tau ({\mathcal{B}}^{\prime })$. The claim that $\tau ({\mathcal{B}}^{\prime })=\tau (\mathcal{B})$ has been proved now.

Theorem 2.1

The collection $\mathcal{F}$ is an appropriate family.

Proof.

The conclusion follows from Lemmas 2.1–2.3.

3. The integral for step functions

In this stage, we will extend Cauchy's integral for a suitable family of step functions. More precisely, we will define the integral for appropriate step functions. In a first step, we will recall some standard (though important) concepts.

Definition 3.1

Define the length of any of the intervals in Definition 2.3 as $|I|=b-a$. If $B=I_1\times \cdots \times I_n$ is a box, we define the volume of B through $|B|=|I_1|\times |I_2|\times \cdots \times |I_n|$. Given $R=\bigcup _{i=1}^m{B}_i\in \mathcal{F}$ where the sets $B_i$ are disjoint, we define the volume of R as $|R|=|B_1|+\cdots +|B_m|$. In general, we let $|\mathrm{\varnothing }|=0$ for the sake of convenience.

By definition, it is clear that the length and volume are non-negative quantities.

Lemma 3.1

If $R\in \mathcal{F}$, then

1. R is a disjoint finite union of boxes.

2. Whenever R can be represented as a finite disjoint union $R_1\cup \cdots \cup R_k$ of boxes, then $|R|$ is independent of the partition. More precisely, if $R_1^{\prime }\cup \cdots \cup R_{k^{\prime }}^{\prime }$ is another partition of R then $|R_1|+\cdots +|R_k|=|R_1^{\prime }|+\cdots +|R_{k^{\prime }}^{\prime }|$.

Proof.

See [12].

Corollary 3.1

If $B\subseteq R$ and $R,B\in \mathcal{F}$ are satisfied, then $|B|\le |R|$.

Proof.

It is obvious that $R=(R\setminus B)\cup B$. From 3.1 we obtain $|R|=|R\setminus B|+|B|$. The non-negativity of the volume assures now that $|B|\le |R|$.

We are now in a position to extend Cauchy's integral to appropriate step functions. To that end, let ${\chi }_S$ denote the characteristic function on the set S, for each $S\in \mathcal{F}$. It is easy to see that ${\chi }_S$ is a regular function [8]. Assume that E represents the subspace of R spanned by the set $\{{\chi }_S:S\in \mathcal{F}\}$ as a vector space over $\mathbb{R}$. The appropriate step functions will be those which are elements of E (see [8]).

Lemma 3.2

If $\phi \in E$, then there exist distinct $a_1,\dots a_m\in \mathbb{R}\setminus \{0\}$ and nonempty disjoint sets $S_1,\dots ,S_m\in \mathcal{F}$, with the property that (6) $\phi =\sum _{i=1}^m{a}_i{\chi }_{S_i}.$(6) This representation is unique and is known as the standard representation of φ.

Proof.

Since $\phi \in E$, then there are $b_1,\dots b_p\in \mathbb{R}$ and $G_1,\dots ,G_p\in \mathcal{F}$ such that $\phi =\sum _{j=1}^p{b}_j{\chi }_{G_j}.$In a first step, we will show that there exist disjoint sets $H_1,\dots ,H_q\in \mathcal{F}$ and constants $c_1,\dots ,c_q\in \mathbb{R}$, such that $\sum _{j=1}^p{b}_j{\chi }_{G_j}=\sum _{i=1}^q{c}_i{\chi }_{H_i}.$To that end, we proceed by induction. If p = 2, then we have that $\sum _{j=1}^2{b}_j{\chi }_{G_j}=b_1{\chi }_{G_1\setminus G_2}+(b_1+b_2){\chi }_{G_1\cup G_2}+b_2{\chi }_{G_2\setminus G_1}.$In this way, we have represented φ as a sum of characteristic functions (each one multiplied by a constant) on disjoint sets in $\mathcal{F}$. Let us suppose the claim is true when p = k. Then $\begin{array}{rl}\sum _{j=1}^{k+1}{b}_j{\chi }_{G_j}& =\sum _{j=1}^k{b}_j{\chi }_{G_j}+b_{k+1}{\chi }_{G_{k+1}}\\ & =\sum _{i=1}^l{c}_i{\chi }_{H_i}+b_{k+1}{\chi }_{G_{k+1}}\\ & =\sum _{i=1}^l{c}_i{\chi }_{H_i\setminus G_{k+1}}+\sum _{i=1}^l(c_i+b_{k+1}){\chi }_{H_i\cap G_{k+1}}\\ & +b_{k+1}{\chi }_{G_{k+1}\setminus \bigcup _{i=1}^l{H}_i},\end{array}$where $H_1,\dots ,H_l\in \mathcal{F}$ are disjoint sets. This proves our claim. As a consequence, φ has a representation of the form $\phi =\sum _{j=1}^p{b}_j{\chi }_{G_j}=\sum _{r=1}^q{c}_r{\chi }_{H_r}.$where $H_1,\dots ,H_q\in \mathcal{F}$ are disjoint sets, and $c_1,\dots ,c_q\in \mathbb{R}$. Notice that we may suppose these sets are nonempty, otherwise we can omit the terms with characteristic functions on the empty set, and the equality would still hold.

Denote the different nonzero members of the collection $\{c_r:r\in \{1,\dots ,q\}\}$ by $a_i$, for $i=1,\dots ,m$, and let $S_i$ be the union of the sets $H_r$ satisfying $a_i=c_r$. Clearly $\phi =\sum _{r=1}^q{c}_r{\chi }_{H_r}=\sum _{i=1}^m{a}_i{\chi }_{S_i},$where $a_1,\dots a_m\in \mathbb{R}$ are distinct nonzero numbers and $S_1,\dots ,S_m\in \mathcal{F}$ are nonempty disjoint sets.

We will prove next that the standard representation is unique. To that end, suppose that there exist different nonzero numbers $d_1,\dots d_l\in \mathbb{R}$ and nonempty disjoint sets $D_1,\dots ,D_l\in \mathcal{F}$, with the property that $\phi =\sum _{j=1}^l{d}_j{\chi }_{D_j}.$Firstly, we claim that $\bigcup _{i=1}^m{S}_i=\bigcup _{j=1}^m{D}_j$. Indeed, if $t\in \bigcup _{i=1}^m{S}_i$ then $t\in S_r$ for some $r\in \{1,\dots ,m\}$. Thus, $0\ne a_r{\chi }_{S_r}(t)=\sum _{i=1}^m{a}_i{\chi }_{S_i}(t)=\sum _{j=1}^l{d}_j{\chi }_{D_j}(t).$As a consequence of this, there is some $k\in \{1,\dots ,l\}$ with the property that ${\chi }_{D_k}(t)\ne 0$, so $t\in D_k$. Therefore, $\bigcup _{i=1}^m{S}_i\subset \bigcup _{j=1}^m{D}_j$, while the other inclusion is demonstrated analogously. Let $r\in \{1,\dots ,m\}$ and $t\in S_r$. Then there exists some $k\in \{1,\dots ,l\}$ such that $t\in D_k$. This implies that $\begin{array}{rl}{a}_r& =a_r{\chi }_{S_r}(t)=\sum _{i=1}^m{a}_i{\chi }_{S_i}(t)\\ & =\sum _{j=1}^l{d}_j{\chi }_{D_j}(t)=d_k{\chi }_{D_k}(t)=d_k.\end{array}$Therefore, for each $r\in \{1,\dots ,m\}$, there exists some $k\in \{1,\dots ,l\}$, such that $c_r=d_k$. In the same way, it is possible to check that for each $k\in \{1,\dots ,l\}$, there is some $r\in \{1,\dots ,m\}$ such that $c_r=d_k$. Thus m = l and the coefficients of the representations are the same. Finally, if $r\in \{1,\dots ,m\}$ and $t\in S_r$, then $f(t)=\sum _{i=1}^m{a}_i{\chi }_{S_i}(t)=a_r=d_k,$which yields $t\in G_k$. This means that $S_r\subseteq G_k$, In the same way, it is possible to prove that $G_k\subseteq S_r$. Thus the sets associated to the characteristic functions of both representations are the same, whence the uniqueness readily follows.

Definition 3.2

Let $\phi \in E$ be written in the standard representation (6(6) $\phi =\sum _{i=1}^m{a}_i{\chi }_{S_i}.$(6) ). We define the integral of φ as $\int \phi (t)\mathrm{d}t=\sum _{i=1}^m{a}_i|S_i|.$

Clearly, the integral is well defined since the standard representation is unique.

Lemma 3.3

Suppose that $\phi \in E$ is given in the standard representation (6(6) $\phi =\sum _{i=1}^m{a}_i{\chi }_{S_i}.$(6) ). Suppose that $b_1\dots ,b_p\in \mathbb{R}$ and $G_1,\dots ,G_p\in \mathcal{F}$ are such that (7) $\phi =\sum _{j=1}^p{b}_j{\chi }_{G_j}.$(7) Then $\sum _{i=1}^m{a}_i|S_i|=\sum _{j=1}^p{b}_j|G_j|.$

Proof.

We may suppose that all the coefficients $b_j$ are nonzero since, otherwise, can omit the terms with zero coefficient from (7(7) $\phi =\sum _{j=1}^p{b}_j{\chi }_{G_j}.$(7) ). We consider two cases for the proof.

Case 1. The sets $G_j$ are disjoint. Using arguments similar to those employed in Lemma 3.2, it is easy to prove that $\bigcup _{i=1}^m{S}_i=\bigcup _{j=1}^l{G}_j$. Then $\begin{array}{rl}{S}_i& =\bigcup _{j=1}^p(S_i\cap G_j),\\ G_j& =\bigcup _{i=1}^m(S_i\cap G_j).\end{array}$Using now Lemma 3.1 we obtain that $\begin{array}{rl}|S_i|& =\sum _{j=1}^p|S_i\cap G_j|,\\ |G_j|& =\sum _{i=1}^m|S_i\cap G_j|.\end{array}$Moreover, if $r\in \{1,\dots ,m\}$ and $k\in \{1,\dots ,p\}$ satisfy that $S_r\cap G_k$ is nonempty, then for each $t\in S_r\cap G_k$ we have that $a_r=\sum _{i=1}^m{a}_i{\chi }_{S_i}(t)=\sum _{j=1}^l{b}_j{\chi }_{G_j}(t)=b_k.$As a consequence, $a_r=b_k$ when $S_r\cap G_k\ne \mathrm{\varnothing }$. From this, $\begin{array}{rl}\sum _{i=1}^m{a}_i|S_i|& =\sum _{i=1}^m\sum _{\begin{array}{c}j=1\\ S_i\cap G_j\ne \mathrm{\varnothing }\end{array}}^p{a}_i|S_i\cap G_j|\\ & =\sum _{i=1}^m\sum _{\begin{array}{c}j=1\\ S_i\cap G_j\ne \mathrm{\varnothing }\end{array}}^p{b}_j|S_i\cap G_j|=\sum _{j=1}^p{b}_j|G_j|.\end{array}$Case 2. The sets $G_j$ are not disjoint. Proceeding by induction and letting p = 2, $\begin{array}{rl}\sum _{j=1}^2{b}_j|G_j|& =b_1|G_1\setminus G_2|+(b_1+b_2)|G_1\cap G_2|\\ & +b_2|G_2\setminus G_1|.\end{array}$The right end is a summation of volumes of disjoint sets (multiplied by some constant). Therefore, we can apply the first case and obtain $\sum _{j=1}^2{b}_j|G_j|=\sum _{i=1}^m{a}_i|S_i|.$Suppose now that the conclusion is true when p = k. Therefore, $\begin{array}{rl}\sum _{j=1}^{k+1}{b}_j|G_j|& =\sum _{j=1}^k{b}_j|G_j|+b_{k+1}|G_{k+1}|\\ & =\sum _{i=1}^l{c}_i|H_i|+b_{k+1}|G_{k+1}|\end{array}$where $H_1,\dots ,H_l\in \mathcal{F}$ are nonempty and pairwise disjoint and the constants $c_1,\dots ,c_l\in \mathbb{R}$ are different and nonzero by the induction hypothesis. Thus, $\begin{array}{rl}& \sum _{j=1}^{k+1}{b}_j|G_j|\\ & =\sum _{i=1}^l{c}_i(|H_i\setminus G_{k+1}|+|H_i\cap G_{k+1}|)\\ & +b_{k+1}(|G_{k+1}\setminus \bigcup _{i=1}^l{H}_i|+\sum _{i=1}^l|G_{k+1}\cap H_i)|)\\ & =\sum _{i=1}^l{c}_i|H_i\setminus G_{k+1}|+\sum _{i=1}^l(c_i+b_{k+1})|H_i\cap G_{k+1}|\\ & +b_{k+1}|G_{k+1}\setminus \bigcup _{i=1}^l{H}_i|.\end{array}$The right-hand side is a sum of volumes of disjoint sets (multiplied by a constant), which means that we can apply the first case to reach $\begin{array}{rl}& \sum _{i=1}^l{c}_i|H_i\setminus G_{k+1}|+\sum _{i=1}^l(c_i+b_{k+1})|H_i\cap G_{k+1}|\\ & +b_{k+1}|G_{k+1}\setminus \bigcup _{i=1}^l{H}_i|=\sum _{j=1}^m{a}_i|S_i|.\end{array}$As a consequence, $\sum _{j=1}^{k+1}{b}_j|G_j|=\sum _{j=1}^m{a}_i|S_i|$which is what we wanted to prove.

Lemma 3.4

If $\phi ,\psi \in E$ and $c\in \mathbb{R}$, then

(i)	$\int \mathrm{c\phi }(t)\mathrm{d}t=c\int \phi (t)\mathrm{d}t$, and
(ii)	$\int (\phi +\psi )(t)\mathrm{d}t=\int \phi (t)\mathrm{d}t+\int \psi (t)\mathrm{d}t$.

Proof.

Let φ and ψ be given in the standard representation, say, $\phi =\sum _{i=1}^m{a}_i{\chi }_{S_i},\psi =\sum _{j=1}^p{b}_j{\chi }_{G_j}$To prove (i), notice that $\int \mathrm{c\phi }(t)\mathrm{d}t=\sum _{i=1}^{m}c{a}_i|S_i|=c\sum _{i=1}^m{a}_i|S_i|=c\int \phi (t)\mathrm{d}t.$To establish (ii) now, observe that $\begin{array}{rl}\phi +\psi & =\sum _{i=1}^m{a}_i{\chi }_{S_i\setminus \bigcup _{j=1}^p{G}_j}+\sum _{i=1}^m\sum _{j=1}^p(a_i+b_j){\chi }_{S_i\cap G_j}\\ & +\sum _{j=1}^p{b}_j{\chi }_{G_j\setminus \bigcup _{i=1}^m{S}_i}.\end{array}$Lemma 3.3 guarantees now that $\begin{array}{rl}& \int (\phi +\psi )(t)\mathrm{d}t\\ & =\sum _{i=1}^m{a}_i|S_i\setminus \bigcup _{j=1}^p{G}_j|+\sum _{i=1}^m\sum _{j=1}^p(a_i+b_j)|S_i\cap G_j|\\ & +\sum _{j=1}^p{b}_j|G_j\setminus \bigcup _{i=1}^m{S}_i|\\ & =\sum _{i=1}^m{a}_i|S_i\setminus \bigcup _{j=1}^p{G}_j|+\sum _{i=1}^m{a}_i\sum _{j=1}^p|S_i\cap G_j|\\ & +\sum _{j=1}^p{b}_j\sum _{i=1}^m|S_i\cap G_j|\\ & +\sum _{j=1}^p{b}_j|G_j\setminus \bigcup _{i=1}^m{S}_i|\\ & =\sum _{i=1}^m{a}_i[|S_i\setminus \bigcup _{j=1}^p{G}_j|+|S_i\cap (\bigcup _{j=1}^p{G}_j)|]\\ & +\sum _{j=1}^p{b}_j[|G_j\setminus \bigcup _{i=1}^m{S}_i|\\ & +|G_j(\bigcup _{i=1}^m{S}_i)|]\\ & =\sum _{i=1}^m{a}_i|S_i|+\sum _{j=1}^p{b}_j|G_j|=\int \phi (t)\mathrm{d}t+\int \psi (t)\mathrm{d}t\end{array}$which is what we wanted to prove.

4. The integral for regulated functions

In this section, we will define Cauchy's integral for generalized regulated functions. To that end, we will present some useful properties of Cauchy's integral for appropriate step functions.

Definition 4.1

Let $B\in \mathcal{F}$ and $\phi \in E$. The integral of φ over B is defined by ${\int }_B\phi (t)\mathrm{d}t=\int \phi (t){\chi }_B(t)\mathrm{d}t.$

Clearly, this concept is well defined. Indeed, if $B\subseteq \mathcal{F}$ and $\phi \in E$ have standard representation given by (6(6) $\phi =\sum _{i=1}^m{a}_i{\chi }_{S_i}.$(6) ), then $\phi {\chi }_B=\left(\sum _{i=1}^m{a}_i{\chi }_{S_i}\right){\chi }_B=\sum _{i=1}^m{a}_i{\chi }_B{\chi }_{S_i}=\sum _{i=1}^m{a}_i{\chi }_{B\cap S_i}.$This means that $\phi {\chi }_B\in E$.

The next result is similar to the last theorem proved in [8]. However, we needed to apply different hypotheses to reach the conclusion. The proof requires to be modified in the way.

Lemma 4.1

Let $B\in \mathcal{F}$, and assume that f is some real-valued function defined in ${\mathbb{R}}^n$. Then f is regulated on $\overline{B}$ exactly in the case that there is some sequence $({\phi }_n)\subseteq E$ that converges uniformly on $\overline{B}$ to f.

Proof.

Let us suppose f is regulated on B, and let $\epsilon >0$. We will construct a function ${\phi }_{\epsilon }\in E$ satisfying $|f(x)-{\phi }_{\epsilon }(x)|<\epsilon$ for each $x\in B$. When $x\in \overline{B}$, there is some neighbourhood $N_x=A_1^x\cup \cdots \cup A_{k_x}^x$ of x, satisfying $|f(s)-f(t)|\le \epsilon$, for each $s,t\in A_i^x$. Here, $A_1^x,\dots ,A_{k_x}^x\in \mathcal{F}$. Since $N_x$ is a neighbourhood of x, then the interior of $N_x$ contains x. If we let $S_x=\mathrm{int}{N}_x$, then $B=\bigcup _{i=1}^m{R}_i\subseteq \bigcup _{i=1}^m\overline{R_i}\subseteq \bigcup _{x\in \overline{B}}{S}_x,$where $R_1,\dots ,R_m$ are boxes from the definition of $B\in \mathcal{F}$. Compactness of the set $\bigcup _{i=1}^m\overline{R_i}$ implies that the open covering $\{S_x:x\in \overline{B}\}$ has a finite subcovering, say, $\{S_{x_1},\dots ,S_{x_r}\}$. As a consequence, $B\subseteq \bigcup _{i=1}^r{S}_{x_i}\subseteq \bigcup _{i=1}^r\bigcup _{j=1}^{k_{x_i}}{A}_j^{x_i}.$Relabeling we have that $B\subseteq \bigcup _{i=1}^p{V}_i$, and $|f(t)-f(s)|<\epsilon$ for each $s,t\in V_i$. Using the second property of appropriate families, we can suppose that the sets $V_i$ are disjoint. Let $y_i\in V_i$ be arbitrary, and define ${\phi }_{\epsilon }=\sum _{i=1}^{p}f(y_i){\chi }_{V_i}.$If $x\in \overline{B}$, then there is some $j\in \{1,\dots ,p\}$ such that $x\in V_j$. Hence $\begin{array}{rl}|f(x)-{\phi }_{\epsilon }(x)|& =|f(x)-\sum _{i=1}^{p}f(y_i){\chi }_{V_i}(x)|\\ & =|f(x)-f(y_j)|<\epsilon .\end{array}$The uniform convergence of $({\phi }_{\frac{1}{m}}{)}_{m\in \mathbb{N}}\subseteq E$ to f on B readily follows.

Conversely, suppose that the sequence $({\phi }_n)\subseteq E$ converges uniformly on $\overline{B}$ to f, and fix $\epsilon >0$. By hypothesis, there exists some $n\in N$, with the property that, for each $x\in \overline{B}$, it follows that $|f(x)-{\phi }_n(x)|<\frac{\epsilon }{3}$. Let $x\in \overline{B}$ and recall that ${\phi }_n\in R$. Then there exists a neighbourhood $N=A_1\cup \cdots \cup A_k$ of x with each $A_i\in \mathcal{F}$, such that $|{\phi }_n(s)-{\phi }_n(t)|\le \frac{\epsilon }{3}$, for each $s,t\in A_i$. This means that if $s,t\in A_i$, then $\begin{array}{rl}|f(s)-f(t)|& \le |f(s)-{\phi }_n(s)|+|{\phi }_n(s)-{\phi }_n(t)|\\ & +|{\phi }_n(t)-f(t)|<\epsilon .\end{array}$As a consequence f is regulated on $\overline{B}$, as desired.

Lemma 4.2

Let $B\in \mathcal{F}$, assume f is regulated on $\overline{B}$, and suppose $({\phi }_m),({\psi }_m)\subseteq E$ are two sequences that converge uniformly on $\overline{B}$ to f. Then

1. $({\int }_B{\phi }_m(t)\mathrm{d}t)$ is a Cauchy sequence.

2. $\underset{m\to \mathrm{\infty }}{lim}{\int }_B{\phi }_m(t)\mathrm{d}t=\underset{m\to \mathrm{\infty }}{lim}{\int }_B{\psi }_m(t)\mathrm{d}t$.

Proof.

Observe firstly that the sequences $({\chi }_B{\phi }_m),({\chi }_B{\psi }_m)\subseteq E$ converge uniformly to ${\chi }_{B}f$. Let $\begin{array}{rl}{\chi }_B{\phi }_m& =\sum _{i=1}^{h_m}{a}_{\mathrm{im}}{\chi }_{S_{\mathrm{im}}},\\ {\chi }_B{\psi }_m& =\sum _{j=1}^{l_m}{b}_{\mathrm{jm}}{\chi }_{G_{\mathrm{jm}}}\end{array}$be the standard representations for ${\chi }_B\phi$ and ${\chi }_B\psi$, respectively.

To prove (i), fix $\epsilon >0$, and notice that the absolute convergence of $({\chi }_B{\phi }_m)$ to ${\chi }_{B}f$ assures the existence of some $M\in \mathbb{N}$, such that if $m\ge M$, then $|{\chi }_B{\phi }_m(t)-{\chi }_{B}f(t)|<{\epsilon }^{\prime }$, for each $t\in {\mathbb{R}}^n$. Here, ${\epsilon }^{\prime }=\frac{\epsilon }{2|B|}$. Let $m,k\ge M$, and assume that $i\in \{1,\dots ,h_m\}$ and $j\in \{1,\dots ,h_k\}$. Then the following inequalities are satisfied:

1. If $S_{\mathrm{im}}\cap S_{\mathrm{jk}}\ne \mathrm{\varnothing }$ then $|a_{\mathrm{im}}-a_{\mathrm{jk}}|<2{\epsilon }^{\prime }$.

2. If $S_{\mathrm{im}}\setminus (\bigcup _{j=1}^{\mathrm{hk}}{S}_{\mathrm{jk}})\ne \mathrm{\varnothing }$ then $|a_{\mathrm{im}}|<{\epsilon }^{\prime }$.

3. ] If $S_{\mathrm{jk}}\setminus (\bigcup _{i=1}^{\mathrm{hm}}{S}_{\mathrm{im}})\ne \mathrm{\varnothing }$ then $|a_{\mathrm{jk}}|<{\epsilon }^{\prime }$.

Additionally, the facts that $\bigcup _{i=1}^{\mathrm{hm}}{S}_{\mathrm{im}}\subseteq B$ and $\bigcup _{j=1}^{\mathrm{hk}}{S}_{\mathrm{jk}}\subseteq B$, from Corollary 3.1 yield the following inequality: $|\left(\bigcup _{i=1}^{\mathrm{hm}}{S}_{\mathrm{im}}\right)\cup \left(\bigcup _{j=1}^{\mathrm{hk}}{S}_{\mathrm{jk}}\right)|\le |B|.$As a consequence, $\begin{array}{rl}& {\int }_B{\phi }_m-{\int }_B{\phi }_k\\ & =\sum _{i=1}^{h_m}{a}_{\mathrm{im}}|S_{\mathrm{im}}|-\sum _{j=1}^{h_k}{a}_{\mathrm{jk}}|S_{\mathrm{jk}}|\\ & =\sum _{i=1}^{h_m}{a}_{\mathrm{im}}|S_{\mathrm{im}}\setminus \bigcup _{j=1}^{\mathrm{hk}}{S}_{\mathrm{jk}}|+\sum _{i=1}^{h_m}{a}_{\mathrm{im}}|S_{\mathrm{im}}\cap \left[\bigcup _{j=1}^{\mathrm{hk}}{S}_{\mathrm{jk}}\right]|\\ & -\sum _{j=1}^{h_k}{a}_{\mathrm{jk}}|S_{\mathrm{jk}}\setminus \bigcup _{i=1}^{\mathrm{hm}}{S}_{\mathrm{im}}|-\sum _{j=1}^{l_m}{a}_{\mathrm{jk}}|S_{\mathrm{jk}}\cap \left[\bigcup _{i=1}^{\mathrm{hm}}{S}_{\mathrm{im}}\right]|\\ & =\sum _{i=1}^{h_m}{a}_{\mathrm{im}}|S_{\mathrm{im}}\setminus \bigcup _{j=1}^{\mathrm{hk}}{S}_{\mathrm{jk}}|+\sum _{i=1}^{h_m}{a}_{\mathrm{im}}\sum _{j=1}^{\mathrm{hk}}|S_{\mathrm{im}}\cap S_{\mathrm{jk}}|\\ & -\sum _{j=1}^{h_k}{a}_{\mathrm{jk}}|S_{\mathrm{jk}}\setminus \bigcup _{i=1}^{\mathrm{hm}}{S}_{\mathrm{im}}|-\sum _{j=1}^{h_k}{a}_{\mathrm{jk}}\sum _{i=1}^{\mathrm{hm}}|S_{\mathrm{im}}\cap S_{\mathrm{jk}}|\\ & <2{\epsilon }^{\prime }(\sum _{i=1}^{h_m}|S_{\mathrm{im}}\setminus \bigcup _{j=1}^{\mathrm{hk}}{S}_{\mathrm{jk}}|+\sum _{i=1}^{h_m}\sum _{j=1}^{\mathrm{hk}}|S_{\mathrm{im}}\cap S_{\mathrm{jk}}|\\ & +\sum _{j=1}^{h_k}|S_{\mathrm{jk}}\setminus \bigcup _{i=1}^{\mathrm{hm}}{S}_{\mathrm{im}}|)\\ & =2{\epsilon }^{\prime }|\left(\bigcup _{i=1}^{\mathrm{hm}}{S}_{\mathrm{im}}\right)\cup \left(\bigcup _{j=1}^{\mathrm{hk}}{S}_{\mathrm{jk}}\right)|\le 2{\epsilon }^{\prime }|B|=\epsilon .\end{array}$We conclude $({\int }_B{\phi }_m(t)\mathrm{d}t)$ is a Cauchy sequence.

To establish (ii) now, let $\epsilon >0$. Uniform convergence of the sequences $({\chi }_B{\phi }_m)$ and $({\chi }_B{\psi }_m)$ to ${\chi }_{B}f$ imply that there exists $M\in \mathbb{N}$ such that, if $m\ge M$, then

(c)	$|{\chi }_B{\phi }_m(t)-{\chi }_{B}f(t)|<\epsilon$, for each $t\in {\mathbb{R}}^n$, and
(d)	$|{\chi }_B{\psi }_m(t)-{\chi }_{B}f(t)|<\epsilon$, for each $t\in {\mathbb{R}}^n$.

Here, ${\epsilon }^{\prime }=\frac{\epsilon }{2|B|}$. If we proceed now as in the proof of (i), we can readily obtain the conclusion of part (ii).

The following is our definition of the generalized Cauchy integral. Our definition will hinge on the definition for the one-dimensional case presented in [7].

Definition 4.2

Let $B\in \mathcal{F}$, and suppose that f is a regulated function on $\overline{B}$. We define the integral of f over B as ${\int }_{B}f(t)\mathrm{d}t=\underset{m\to \mathrm{\infty }}{lim}{\int }_B{\phi }_m(t)\mathrm{d}t$where $({\phi }_m)\subseteq E$ is any sequence which converges uniformly to f on $\overline{B}$.

Notice that this concept is well defined in light of Lemmas 4.1 and 4.2.

Theorem 4.1

Let $B\in \mathcal{F}$ and $c\in \mathbb{R}$. If f and g are regulated functions on $\overline{B}$, then

1. ${\int }_B\mathrm{cf}(t)\mathrm{d}t=c{\int }_{B}f(t)\mathrm{d}t$.

2. ${\int }_B(f+g)(t)\mathrm{d}t={\int }_{B}f(t)\mathrm{d}t+{\int }_{B}g(t)\mathrm{d}t$.

Proof.

Let $({\phi }_m),({\psi }_m)\subseteq E$ be sequences that converge uniformly to f and g in $\overline{B}$, respectively. Property (i) follows from Lemma 3.4 and the following identities: $\begin{array}{rl}{\int }_B\mathrm{cf}(t)\mathrm{d}t& =\underset{m\to \mathrm{\infty }}{lim}c{\int }_E{\phi }_m(t)\mathrm{d}t=c\underset{m\to \mathrm{\infty }}{lim}{\int }_E{\phi }_m(t)\mathrm{d}t\\ & =c{\int }_{B}f(t)\mathrm{d}t.\end{array}$On the other hand, also follows from Lemma 3.4 and the identities $\begin{array}{rl}{\int }_B(f+g)(t)\mathrm{d}t& =\underset{m\to \mathrm{\infty }}{lim}({\int }_E{\phi }_m(t)\mathrm{d}t+{\int }_E{\psi }_m(t)\mathrm{d}t)\\ & =\underset{m\to \mathrm{\infty }}{lim}{\int }_E{\phi }_m(t)\mathrm{d}t+\underset{m\to \mathrm{\infty }}{lim}{\int }_E{\psi }_m(t)\mathrm{d}t\\ & ={\int }_{B}f(t)\mathrm{d}t+{\int }_{B}g(t)\mathrm{d}t.\end{array}$This proves the result.

Theorem 4.2

Let $B\in \mathcal{F}$, and assume that f is a regulated function on $\overline{B}$. If f is non-negative on $\overline{B}$, then ${\int }_{B}f(x)\mathrm{d}x\ge 0.$

Proof.

Let $({\phi }_m^{\prime })\subseteq E$ be a sequence that converge uniformly to f on $\overline{B}$. For each $m\in \mathbb{N}$ and $x\in \overline{B}$, define ${\phi }_m(x)=max\{0,{\phi }_m^{\prime }(x)\}$. It is easy to verify that $({\phi }_m)\subseteq E$ converges uniformly to f. Fix $m\in \mathbb{N}$, and let $\sum _{i=1}^k{a}_i{\chi }_{S_i}$be the standard representation of ${\chi }_B{\phi }_m$. Clearly $a_i\ge 0$, so ${\int }_B{\phi }_m(x)\mathrm{d}x=\sum _{i=1}^k{a}_i|S_i|\ge 0.$This last inequality holds for all $m\in \mathbb{N}$. Making $m\to \mathrm{\infty }$ now, we obtain ${\int }_{B}f(x)\mathrm{d}x=\underset{m\to \mathrm{\infty }}{lim}{\int }_B{\phi }_m(x)\mathrm{d}x\ge 0$which is what we wanted to prove.

Theorem 4.3

Assume that $B\in \mathcal{F}$, and suppose that f and g are regulated functions on $\overline{B}$. If $f\ge g$ on $\overline{B}$, then ${\int }_{B}f(x)\mathrm{d}x\ge {\int }_{B}g(x)\mathrm{d}x.$

Proof.

By hypothesis $f-g\ge 0$ on $\overline{B}$. The conclusion readily follows now from the linearity of the integral and Theorem 4.2.

5. Illustrative examples

The purpose of the present section is to present some concrete examples to illustrate the applicability of the theorems above. In the following, we will observe the nomenclature employed in those results. We just need to point out that B will represent a box in ${\mathbb{R}}^n$, as defined at the beginning of Section 3.

1. To start with, observe that if f is a constant function equal to $c\in \mathbb{R}$ in all B, then f is integrable on B and, moreover, ${\int }_{B}f(x)\mathrm{d}x=c|B|.$In particular, it follows from this fact that the integral of the constant function 1 over B is equal to the volume of B.

2. If f and g are continuous functions, then they are regulated and, in that case, the properties in Theorem 4.1 are satisfied. It is worth pointing out that the class of generalized regulated functions obviously includes the continuous functions. Moreover, regulated functions are obviously bounded.

3. An easy application of the monotonicity property results if we consider the function $f:\overline{B}\to \mathbb{R}$ given by $f(x)=x_1^2+\dots +x_n^2$, where $x=(x_1,\dots ,x_n)$. In this case, it is obvious (both from the definition and the monotonicity of the integral) that ${\int }_{B}f(x)\mathrm{d}x\ge 0.$Moreover, if we let $g:\overline{B}\to \mathbb{R}$ be given as $g(x)=x_1^2$, then evidently ${\int }_{B}f(x)\mathrm{d}x\ge {\int }_{B}g(x)\mathrm{d}x.$These properties were known for functions defined on an interval of $\mathbb{R}$. The advantage, in this case, is that they have been extended theoretically to functions defined in ${\mathbb{R}}^n$. These and other trivial examples confirm the validity of the theoretical results demonstrated in the present manuscript.

The previous were some easy consequences from the analytical results derived in the present work. However, the following are more elaborate applications of the extended Cauchy integral for generalized regulated real-valued functions on ${\mathbb{R}}^n$. For the sake of illustration, the functions considered will be defined on ${\mathbb{R}}^2$. Our examples will be numerically oriented rather than theoretical.

In the first of our examples, we provide a means to estimate the value of the generalized Cauchy integral of a function, and the approximate error associated with that estimation.

Example 5.1

Define the function $f:{\mathbb{R}}^2\to \mathbb{R}$ by $f(x,y)=5-\frac{1}{5}(x^2+y^2).$For the sake of illustration, let us fix the box $B=[0,4]\times [0,3]\subseteq {\mathbb{R}}^2$. Observe that the function f is continuous, so regulated. Thus, f is regulated in the generalized sense of this work and so, integrable in the sense of Cauchy. Let us consider uniform partitions of the intervals $[0,4]$ and $[0,3]$ of the forms $P_1:x_0<x_1<x_2<x_3<x_4$ and $P_2:y_0<y_1<y_2<y_3$, respectively. Notice then that $P_1=\{0,1,2,3,4\}$ and $P_2=\{0,1,2,3\}$, and these partitions induce a partition of B into twelve squares of equal area. These squares will be represented by $R_i$ and, evidently, $|R_i|=1$, for each $i=1,2,\dots ,12$. For convenience, Figure 1 shows the graph of the function f on the rectangle B. It is worthwhile pointing out that the rectangles on the xy-plane represent the squares $R_i$ used in this example.

Figure 1. Graph of the function $z=5-\frac{1}{5}(x^2+y^2)$ as a function of x and y on the domain $(x,y)\in [0,4]\times [0,3]$. The rectangles on the xy-plane represent a partition of B into equal squares of area equal to 1.

Let $\varphi ,\psi :B\to \mathbb{R}$ be step functions defined as follows: for each $i\in \{1,2,\dots ,12\}$ and $(x,y)\in R_i$, agree that $\begin{array}{rl}\varphi (x,y)& =min\{f(x,y):(x,y)\in R_i\},\\ \psi (x,y)& =max\{f(x,y):(x,y)\in R_i\}.\end{array}$For simplification, let us convey that the values for ϕ and ψ on the squares $R_i$ are given by $a_i$ and $b_i$, respectively, for each $i=1,2,\dots ,12$. Obviously, the inequalities $\varphi \le f\le \psi$ are satisfied. Using the definition of the integral for step functions, we obtain that $\begin{array}{rl}{\int }_B\varphi (t)\mathrm{d}t& =\sum _{i=1}^{12}{a}_i|R_i|\\ & =f(1,1)+f(2,1)+f(3,1)+f(4,1)\\ & +f(1,2)+f(2,2)\\ & +f(3,2)+f(4,2)+f(1,3)+f(2,3)\\ & +f(3,3)+f(4,3)\\ & =\frac{23}{5}+\frac{20}{5}+\frac{15}{5}+\frac{8}{5}+\frac{20}{5}+\frac{17}{5}\\ & +\frac{12}{5}+\frac{5}{5}+\frac{15}{5}+\frac{12}{5}+\frac{7}{5}+0\\ & =\mathrm{30.8.}\end{array}$In a similar fashion, observe that $\begin{array}{rl}{\int }_B\psi (t)\mathrm{d}t& =\sum _{i=1}^{12}{b}_i|R_i|\\ & =f(0,0)+f(1,0)+f(2,0)\\ & +f(3,0)+f(0,1)+f(1,1)\\ & +f(2,1)+f(3,1)+f(0,2)+f(1,2)\\ & +f(2,2)+f(3,2)\\ & =\mathrm{47.6.}\end{array}$Integrability of the functions ϕ, f and ψ together with Theorem 4.3 assure then that $30.8={\int }_B\varphi (t)\mathrm{d}(t)\le {\int }_{B}f(t)\mathrm{d}t\le {\int }_B\psi (t)\mathrm{d}t=\mathrm{47.6.}$Moreover, elementary algebra establishes that $\begin{array}{rl}& |{\int }_{B}f(t)\mathrm{d}t-39.2|\\ & =|{\int }_{B}f(t)\mathrm{d}t-\frac{1}{2}[{\int }_B\varphi (t)\mathrm{d}t+{\int }_B\psi (t)\mathrm{d}t]|\\ & \le \frac{1}{2}[{\int }_B\psi (t)\mathrm{d}t-{\int }_B\varphi (t)\mathrm{d}t]=\mathrm{8.4.}\end{array}$As a conclusion, the numerical value of the Cauchy integral of the function f on B is estimated as 39.2, with a numerical error equal to 8.4. As expected by the theory developed in this work, better estimates can be obtained by choosing step functions over finer mesh grids on the x and y directions.

In the following example, we obtain the value of the extended Cauchy integral for the function in the previous example. The approach provided in the solution may be extended to calculate the integral for any polynomial function.

Example 5.2

Consider the function f and the set B from Example 5.1. For each $n\in \mathbb{N}$, let $P_1^{(n)}$ and $P_2^{(n)}$ be the partitions of $[0,4]$ and $[0,3]$, respectively, consisting of sub-intervals of lengths equal to $\frac{1}{n}$. Let $P^{(n)}$ be the partition of B induced by $P_1^{(n)}$ and $P_2^{(n)}$, for each $n\in \mathbb{N}$. It is obvious that P consists of $12n^2$ identical squares of equal area. Denote those squares by $R_i^{(n)}$, for each $i=1,2,\dots ,12n^2$, and note that $|R_i|=\frac{1}{n^2}$. Define now the function ${\psi }_n:B\to \mathbb{R}$ by ${\psi }_n(x,y)=max\{f(x,y):(x,y)\in R_i^{(n)}\},$for each $i\in \{1,2,\dots ,12n^2\}$ and $(x,y)\in R_i$. Clearly, ${\psi }_n$ is a step function, for each $n\in \mathbb{N}$. We will check now that ${\psi }_n$ converges uniformly on B to the function f.

Fix $n\in \mathbb{N}$, and observe that each square $R_i$ has the form $[x_{\ast },x_{\ast }+\frac{1}{n}]\times [y_{\ast },y_{\ast }+\frac{1}{n}]$, for suitable $x_{\ast }\in P_1^{(n)}$ and $y_{\ast }\in P_2^{(n)}$. The definition of the function ${\psi }_n$ shows that ${\psi }_n(x,y)=f(x_{\ast },y_{\ast })$, for each $(x,y)\in R_i$. Moreover, $\begin{array}{rl}& |{\psi }_n(x,y)-f(x,y)|\\ & \le |f(x_{\ast },y_{\ast })-f(x_{\ast }+\frac{1}{n},y_{\ast }+\frac{1}{n})|\\ & =\frac{2}{5}[\frac{1}{n^2}+\frac{1}{n}x+\frac{1}{n}y]\le \frac{2}{5}[\frac{1}{n^2}+\frac{7}{n}].\end{array}$From this fact, we can readily establish that the sequence of step functions $({\psi }_n)$ converges uniformly to f on B. It follows then that (8) ${\int }_{B}f(t)\mathrm{d}t=\underset{n\to \mathrm{\infty }}{lim}{\int }_B{\psi }_n(t)\mathrm{d}t.$(8) To calculate each of the integrals on the right-hand side, observe that the nodes of the partitions $P_1^{(n)}:x_0<x_1<\dots <x_{4n}$ and $P_2^{(n)}:y_0<y_1<\dots y_{3n}$ are defined, respectively, by $x_i=i/n$ and $y_j=j/n$, for each $i=0,1,\dots ,4n$ and $j=0,1,\dots ,3n$. Let $R_{\mathrm{ij}}=[i/n,(i+1)/n]\times [j/n,(j+1)/n]$, and observe that ${\psi }_n(x,y)=f(i/n,j/n)$, for each $(x,y)\in R_{\mathrm{ij}}$, $i=0,1,\dots ,4n$ and $j=0,1,\dots ,3n$.

For convenience, we allow $a_{\mathrm{ij}}=f(i/n,j/n)$ in what follows. In such way, $\begin{array}{rl}& {\int }_B{\psi }_n(t)\mathrm{d}t\\ & =\sum _{i=0}^{4n-1}\sum _{j=0}^{3n-1}{a}_{\mathrm{ij}}|R_{\mathrm{ij}}|=\frac{1}{n^2}\sum _{i=0}^{4n-1}\sum _{j=0}^{3n-1}[5-\frac{i^2+j^2}{5n^2}]\\ & =60-\frac{1}{5n^4}\sum _{i=0}^{4n-1}\sum _{j=0}^{3n-1}(i^2+j^2)\\ & =60-\frac{1}{5n^4}[3n\sum _{i=0}^{4n-1}{i}^2+4n\sum _{j=1}^{3n-1}{j}^2]\\ & =60-\frac{2n(4n-1)(8n-1)}{5n^3}-\frac{2n(3n-1)(6n-1)}{5n^n}\end{array}$Here, we employed the well-known formula for the sum of squares in the last step. Substituting this last identity into Equation (8(8) ${\int }_{B}f(t)\mathrm{d}t=\underset{n\to \mathrm{\infty }}{lim}{\int }_B{\psi }_n(t)\mathrm{d}t.$(8) ) and taking then the limit when n tends to infinity, we easily obtain that ${\int }_{B}f(t)\mathrm{d}t=40.$Alternatively, we could have applied the linearity of the generalized Cauchy integral to calculate separately the integrals of each summand in the expression of f. Adding the values of those integrals would lead us to the same result for the integral of f.

It is important to notice here that the estimate derived in Example 5.1 (which is equal to 39.2) is in qualitative agreement with the value of the Cauchy integral of f obtained in the last example (which is equal to 40). Moreover, observe that the error obtained in the former example constitutes an upper-bound estimate.

The last example establishes the existence of non-integrable functions.

Example 5.3

Let $B=[0,1]\times [0,1]$, and consider the function $f:{\mathbb{R}}^2\to \mathbb{R}$ given by $f(x,y)=\{\begin{array}{ll}0,& x,y\in \mathbb{Q},\\ 1,& \mathrm{otherwise}.\end{array}$Using contradiction, it is easy to observe that there is no sequence of simple functions defined on B which converges uniformly to f. It follows that the function is not integrable in the extended sense of Cauchy.

6. Conclusions

Using appropriate families of functions and generalized regulated functions, the present manuscript extended Cauchy's integral to real-valued functions defined in ${\mathbb{R}}^n$. As it has been done for other theories [13], the present work provides a self-contained and detailed generalized theory for the generalized Cauchy integral. This extension was reached by defining the integral as the limit of a sequence of integrals of appropriate step functions that converge to a generalized regulated function. In addition, some basic properties of this integral were proved, like the linearity, the preservation of non-negativity and the monotonicity. We consider that one of the advantages of this integral is that it is not necessary to study measure theory to characterize the integrable functions, unlike the Riemann integral. This could be more advantageous from a pedagogical point of view, especially as a topic for elementary analysis courses. On the other hand, the authors of this manuscript are convinced that there are many more properties associated with this integral. For instance, one could study its relationship with the classical Riemann–Darboux integral. Here, it is worth pointing out that the author of [14] showed that every regulated function is Riemann–Darboux integrable, but the converse is not true in general. A natural direction of investigation is whether a similar situation prevails with generalized regulated functions. Obviously, this is an interesting topic for research that the authors intend to tackle in the near future.

As one of the anonymous reviewers of this manuscript pointed out, the examples provided in this work are still simple. This is due to the fact that the theory formulated herein can still be developed tremendously. With more theory, better examples can be provided to illustrate the applications of this integration theory. To that effect, counterparts of strong theorems (like the Lebesgue's dominated convergence theorem, the monotone convergence theorem, Fatou's lemma, Fubini's theorem, the fundamental theorem of calculus, etc.) are required. However, the derivation of those results (if possible) will entail much more efforts and time and would require additional manuscripts. In its present form, our work has tackled the difficulties to extend Cauchy's integral to $R^n$, and the arguments have been nontrivial (as it can be checked above). Knowing that there is still a lot more to be done in this topic, we have included the adjective ‘foundational’ in the title of this manuscript to emphasize the foundational aspect of this theory. As part of future works, we will strive to develop this theory even more towards stronger properties of the Cauchy integral in $R^n$ which will lead to more complicated and interesting applications.

Acknowledgments

The authors would like to thank the anonymous reviewers and the associate editor in charge of handling this manuscript for all their suggestions. Their comments and criticisms helped us to improve the quality of this work. The authors confirm that all the research meets ethical guidelines and adheres to the legal requirements of the study country.

Data availability statement

Data sharing is not applicable to this article as no new data were created or analysed in this study.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

One of the authors (J.E.M.-D.) was funded by the National Council of Science and Technology of Mexico through grant A1-S-45928.