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Mathematics 2018, 6(9), 168; doi:10.3390/math6090168

Article
Equivalences of Riemann Integral Based on p-Norm
School of Mathematics and Statistics, Baise University, No.21 Zhongshan Road, Baise 533000, Guangxi, China
Received: 11 August 2018 / Accepted: 6 September 2018 / Published: 13 September 2018

## Abstract

:
In the usual Riemann integral setting, the Riemann norm or a mesh is adopted for Riemann sums. In this article, we use the p-norm to define the p-integral and show the equivalences between the Riemann integral and the p-integral. The p-norm provides an alternative approach to define the Riemann integral. Based on this norm, we also derive some other equivalences of the Riemann integral and the p-integral.
Keywords:
p-integral; Riemann integral; Darboux integral
MSC:
28A10

## 1. Introduction

In this article, only the bounded functions defined on a closed interval $[ a , b ]$ are considered. A function f is used to denote a bound function defined on $[ a , b ]$. To begin with, the usual settings of a Riemann integral are listed for comparison. Then, some notations for the preparation of our settings of the p-integral are introduced. Although the p-norm and the Riemann norm turn out to be equivalent, the p-norm has many merits in connecting functional analysis and integrals. Since the p-norm is massively used in functional analysis, by defining an alternative Riemann integral via this norm, one could further look at the typical the Riemann integral from a new aspect. This might further extend the Riemann integral to other territories. Defining the p-norm from the beginning and deriving all the equivalences directly gives us some insightful knowledge between all these equivalences.

#### 1.1. Background

Let $P = { P ( 0 ) , P ( 1 ) , … , P ( n ) }$ denote a partition of $[ a , b ]$, in which $P ( 0 ) = a < P ( 1 ) < P ( 2 ) < ⋯ < P ( n ) = b$. Let $| P |$ denote the length of the partition, in this case n. Let $T a g ( P )$ denote the set of all the tags of P. The usual setting of the Riemann integral is based on the concept of meshes [1]. Firstly, one defines a Riemann sum corresponding to a partition P of $[ a , b ]$ and its tags $ζ P$ as follows [2,3]:
$σ ( f , P , ζ P ) = ∑ j = 1 | P | f ( ζ j P ) · Δ j P ,$
where $Δ j P = P ( j ) − P ( j − 1 )$ and $ζ j P ∈ [ P ( j − 1 ) , P ( j ) ]$. Then, one defines the mesh of the partition P,
$λ ( P ) = m a x { Δ j P : j ∈ { 1 , 2 , … , | P | } .$
Moreover, if there exists $I ∈ R$ such that $∀ ϵ > 0$ $∃ δ ϵ > 0$ such that for all partition P of $[ a , b ]$
$λ ( P ) < δ ϵ → ∀ ζ P ∈ T a g ( P ) | I − σ ( f , P , ζ P ) | < ϵ ,$
then we say f is Riemann integrable [4] and the integral of f is defined as $∫ a b f ( x ) d x = I$ or
$∫ a b f ( x ) d x = lim λ ( P ) → 0 f ( ζ j P ) · Δ j P .$

#### 1.2. Notations

In this article, we use the $p$-norm (where $p > 1$) instead of a mesh (or the Riemann norm) to derive some equivalences of the Riemann integral. Unlike the usual setting that combines partitions and tags, they are reintroduced via two individual parts: partition vectors and their corresponding product (tag) spaces. For a (partition) vector $v → = ( v 0 , v 1 , … , v n ) ∈ R n + 1$, we use $| v → |$ to denote the length of $v →$, i.e., n in this case and $v → j$ to denote the jth element of $v →$, i.e., $v j$ in this case. To unify the presentation of the Riemann integral and the p-integral, some notations are defined beforehand:
• $F P V ( n ) [ a , b ]$ denotes the set of all the finite partition vectors of $[ a , b ]$ whose length is n, i.e., $F P V ( n ) [ a , b ] = { B → ∈ R n + 1 : B 0 = a < B 1 < … < B j < B j + 1 < … < B n = b }$. For example, $B → = ( 1 , 1.2 , 1.5 , 2.2 , 2.6 , 3 ) ∈ F P V ( 5 ) [ 1 , 3 ]$, in which $B 0 = 1 , B 1 = 1.2 , … , B 5 = 3$ and $| B → | = 5$.
• $F P V [ a , b ] = ⋃ n = 1 ∞ F P V ( n ) [ a , b ]$, i.e., the set of all the finite partitions of $[ a , b ]$. Observe that $F P V [ a , b ] ⊆ ⋃ n = 1 ∞ R n + 1$.
• $∏ [ B → ] = ∏ j = 0 | B → | − 1 [ B j , B j + 1 ]$. For example, if $B → = ( 1 , 1.2 , 1.5 , 2.2 , 2.6 , 3 ) ∈ F P V ( 5 ) [ 1 , 3 ]$, then $( 1.1 , 1.4 , 1.5 , 2.4 , 2.9 ) ∈ ∏ [ B → ]$. $∏ [ B → ]$ represents the space where the tags are located, given a partition $B →$ of $[ a , b ]$.
• $∏ F P V [ a , b ] = ∏ j = 1 ∞ F P V [ a , b ]$. $∏ F P V [ a , b ]$ represents an ordered sequence of finite partition vectors.
• (Riemann norm) $〈 〈 B → 〉 〉 : = m a x { B j + 1 − B j : 0 ≤ j ≤ | B → | − 1 }$. This is exactly the usual definition of a mesh.
• (Riemann sum) $A f ( B → , H → ) : = ∑ j = 0 | B → | − 1 ( B j + 1 − B j ) · f ( H j )$, where $H → j ∈ [ B j , B j + 1 ]$, where $H →$ represents a sequence of tags, i.e., $H → ∈ ∏ [ B → ]$.
• (p-norm) $∥ B → ∥ p = ∑ j = 0 | B → | − 1 | B j + 1 − B j | p p$, where $p > 1$. For example, if $B → = ( 1 , 2 , 4 , 7 , 10 )$, then $∥ B → ∥ 3 = ( 1 + 2 3 + 3 3 + 3 3 ) 3 = 63 3$. By exploiting Minkowski inequality, one could easily verify $∥ · ∥ p$ is a norm.
• For any $δ > 0$, define $F P V ( δ ) = { B → ∈ F P V [ a , b ] : 〈 〈 B → 〉 〉 < δ }$.
• For any $δ > 0$, define $F P V p ( δ ) = { B → ∈ F P V [ a , b ] : ∥ B → ∥ p < δ }$.
$F P V ( δ )$ collects all the finite partition vectors whose Riemann norms are less than $δ$, while $F P V p ( δ )$ collects all the finite partition vectors whose p-norms are less than $δ$. One could easily verify that $F P V p ( δ ) ⊊ F P V ( δ )$. The equality is obvious not true. For example, if $B → = ( 0 , 0.2 , 1 , 1.5 , 2 )$ and $δ = 1$, then $B → ∈ F P V ( 1 )$, but $B → ∉ F P V p ( 1 )$. We use the notation $C → = ( C → 0 , C → 1 , C → 2 , … , C → n , … ) ∈ ∏ F P V [ a , b ]$ to denote a sequence of partition vectors in $F P V [ a , b ]$, where each $C → j ∈ F P V [ a , b ]$ and denotes the $( j + 1 )$th partition vectors in $C →$.
Definition 1.
$[ 0 ] = { C → ∈ ∏ F P V [ a , b ] : lim j → ∞ 〈 〈 C → j 〉 〉 = 0 }$.
Definition 2.
$[ 0 ] p = { C → ∈ ∏ F P V [ a , b ] : lim j → ∞ ∥ C → j ∥ p = 0 }$.
For example, if $C → = ( C → 0 , C → 1 , C → 2 , … , C → n , … ) =$ $( ( a , b ) , ( a , a + 1 2 ( b − a ) , b ) ,$ $( a , a + 1 3 ( b − a )$, $a + 2 3 ( b − a ) , b ) , … , ( a , a + 1 n ( b − a ) ,$ $a + 2 n ( b − a ) , … , a + n − 1 n ( b − a ) , b ) , … ) ,$ then one has
$lim j → ∞ 〈 〈 C → j 〉 〉 = lim j → ∞ b − a j + 1 = 0 ,$
and
$lim j → ∞ ∥ C → j ∥ p = lim j → ∞ j · b − a j p p = 0 ,$
i.e., $C → ∈ [ 0 ]$ and $C → ∈ [ 0 ] p$. $[ 0 ]$ is the set of all the sequences of partitioned vectors whose Riemann norms converge to 0, while $[ 0 ] p$ is the set of all the sequences of partition vectors whose p-norms converge to 0. Since Riemann norm and p-norm is equivalent, $[ 0 ] = [ 0 ] p$.

## 2. Definitions

To compare the differences between our settings and the usual Riemann integral and to facilitate our introduction of the p-integral, we rephrase some terminologies regarding the Riemann integral and have the following definitions.
Definition 3.
(Riemann integrable) If there exists $s ∈ R$ such that $∀ ϵ > 0$ $∃ δ ϵ > 0$ such that $∀ B → ∈ F P V ( δ ϵ )$ and $∀ H → ∈ ∏ [ B → ]$, $| A f ( B → , H → ) − s | < ϵ$, we say f is Riemann-integrable, denoted by $( R ) ∫ a b f = s$. Furthermore, we use $R [ a , b ]$ to denote the set of all the Riemann-integrable functions defined on $[ a , b ]$.
Let us define a partial ordering ⪯ on $F P V [ a , b ]$. For any $P → , Q → ∈ F P V [ a , b ]$, $P → ⪯ Q →$ if and only if $∀ j ∈ { 1 , 2 , … , | Q → | }$ $∃ k ∈ { 1 , 2 , … , | P → | }$ such that $Q → ( k ) = P → ( j )$. If $P → ⪯ Q →$, we say P is a refinement of $Q →$. Then, we define the set of all the refinements of $Q →$ by $R E F ( Q → ) = { P → ∈ F P V [ a , b ] : P → ⪯ Q → }$. One could easily check that $∀ δ > 0$ and $∀ B → ∈ F P V ( δ ) [ R E F ( B → ) ⊆ F P V ( δ ) ]$.
Definition 4.
If there exists $s ∈ R$ such that $∀ ϵ ′ > 0$ $∃ B → ϵ ′ ∈ F P V [ a , b ]$ $∀ B → ∈ R E F ( B → ϵ ′ )$ $∀ H → ∈ ∏ [ B → ] | A f ( B → , H → ) − s | < ϵ ′$, we say f is refinement-integrable, denoted by $( R E F ) ∫ a b f = s$. Furthermore, we use $R E F [ a , b ]$ to denote the set of all the refinement-integrable functions defined on $[ a , b ]$.
Let us define the upper Darboux sum of f with respect to $B →$ as follows:
$A ¯ f ( B → ) = ∑ j = 0 | B → | − 1 [ B j − B j − 1 ] · f ∗ ( [ B j − 1 , B j ] ) ,$
where $f ∗ ( [ B j − 1 , B j ] ) = s u p { f ( x ) : x ∈ [ B j − 1 , B j ] }$ and the lower Darboux sum of f with respect to $B →$ as follows:
$A ̲ f ( B → ) = ∑ j = 0 | B → | − 1 [ B j − B j − 1 ] · f ∗ ( [ B j − 1 , B j ] ) ,$
where $f ∗ ( [ B j − 1 , B j ] ) = i n f { f ( x ) : x ∈ [ B j − 1 , B j ] }$. Since f is a bounded function, these definitions are all well-defined.
Definition 5.
(Darboux Cauchy integrable [5,6]) If
we say f is Darboux Cauchy integrable and use $D C [ a , b ]$ to denote the set of all the Darboux Cauchy integrable functions defined on $[ a , b ]$.
Define $∫ ̲ a b f = s u p { A ¯ f ( B → ) : B → ∈ F P V [ a , b ] }$ and $∫ ¯ a b f = i n f { A ¯ f ( B → ) : B → ∈ F P V [ a , b ] }$. Since $∀ B → ∈ F P V [ a , b ] [ A ̲ f ( B → ) ≤ A ¯ f ( B → ) ]$, one has $∫ ̲ a b f ≤ ∫ ¯ a b f$.
Definition 6.
(Darboux integrable [7], p. 120) If there exists $s ∈ R$ such that $s = ∫ ̲ a b f = ∫ ¯ a b f$, we say f is Darboux integrable, denoted by $( D I ) ∫ a b f = s$. Furthermore, we use $D I [ a , b ]$ to denote the set of all the Darboux integrable functions defined on $[ a , b ]$.
Based on p-norm, we have the following new definitions.
Definition 7.
(p-integrable) If there exists $s ∈ R$ such that $∀ ϵ ˜ > 0$ , we say f is $p$-integrable, denoted by $( p ) ∫ a b f = s$. Furthermore, we use $p [ a , b ]$ to denote the set of all the $p$-integrable functions defined on $[ a , b ]$.
Example 1.
In this example, we demonstrate that if $f , g ∈ p [ a , b ]$, then $∫ a b f ( x ) x + ∫ a b g ( x ) d x = ∫ a b ( f + g ) ( x ) d x$. Suppose $∫ a b f ( x ) d x = c$ and $∫ a b g ( x ) d x = d$. Let $ϵ ˜ > 0$ be arbitrary. By $f , g ∈ p [ a , b ]$, one has there exist $c , d$ and there exist $δ ˜ ϵ ˜ 2 f$, $δ ˜ ϵ ˜ 2 g$ such that
$∀ B → ∈ F P V p ( δ ˜ ϵ ˜ 2 f ) ∀ H → ∈ ∏ [ B → ] | A f ( B → , H → ) − c | < ϵ ˜ 2 ,$
and
$∀ B → ∈ F P V p ( δ ˜ ϵ ˜ 2 g ) ∀ H → ∈ ∏ [ B → ] | A f ( B → , H → ) − c | < ϵ ˜ 2 .$
Let $δ ˜ ϵ ˜ = m i n { δ ˜ ϵ ˜ 2 f , δ ˜ ϵ ˜ 2 g } .$ Let $B → ∈ F P V p ( δ ˜ ϵ ˜ )$ and let $H → ∈ ∏ [ B → ]$ be arbitrary. By the property that $F P V p ( δ ˜ ϵ ˜ ) ⊆ F P V p ( δ ˜ ϵ ˜ 2 f ) , F P V p ( δ ˜ ϵ ˜ 2 g ) ,$ one has
$∑ j = 0 | B → | − 1 ( B j + 1 − B j ) · f ( H j ) − c < ϵ ˜ ˜ 2 ,$
$∑ j = 0 | B → | − 1 ( B j + 1 − B j ) · g ( H j ) − d < ϵ ˜ ˜ 2 ,$
i.e.,
$∑ j = 0 | B → | − 1 ( B j + 1 − B j ) ( f + g ) ( H j ) − ( c + d ) ≤ ∑ j = 0 | B → | − 1 ( B j + 1 − B j ) · f ( H j ) − c + ∑ j = 0 | B → | − 1 ( B j + 1 − B j ) · g ( H j ) − d < ϵ ˜ ,$
i.e., $f + g ∈ p [ a , b ]$ and
$∫ a b ( f + g ) ( x ) d x = c + d = ∫ a b f ( x ) x + ∫ a b g ( x ) d x .$
Since the main purpose of this article is to show all sorts of equivalences of integrals and since the Riemann integral and the p-norm integral are equivalent, we do not focus too much on how to reinvent the wheel. The interested readers could simply use the p-norm integral to construct a whole parallel results with respect to the Riemann integral.
Definition 8.
(Discrete Darboux Cauchy integrable) If there exists $s ∈ R$ such that $∀ ϵ ¨ > 0$ $∀ C → ∈ [ 0 ] p$ $∃ N ¨ ϵ ¨ , C →$ , we say f is discrete Darboux Cauchy integrable, denoted by $( D D C p ) ∫ a b f = s$. Furthermore, we use $D D C p [ a , b ]$ to denote the set of all the discrete Darboux Cauchy integrable functions defined on $[ a , b ]$.
Definition 9.
(Discrete $p$-integrable) If there exists $s ∈ R$ such that $∀ ϵ ˇ > 0$ $∀ C → ∈ [ 0 ] p$ $∀ m ≥ N ˇ ϵ ˇ , C →$ $∀ H → ∈ ∏ [ C → m ] | A f ( C → m , H → ) − s | < ϵ ˇ$, we say f is discrete p-integrable, denoted by $( D p ) ∫ a b f = s$. Furthermore, we use $D p [ a , b ]$ to denote the set of all the discrete p-integrable functions defined on $[ a , b ]$.
Definition 10.
(Ranged Darboux Cauchy integrable) If there exists $s ∈ R$ such that $∀ ϵ ¯ > 0$ $∀ B → ∈ F P V p ( δ ¯ ϵ ¯ ) [ A ¯ f ( B → ) − A ̲ f ( B → ) < ϵ ¯ ]$, we say f is ranged Darboux Cauchy integrable, denoted by $( R D C p ) ∫ a b f = s$. Furthermore, we use $R D C p [ a , b ]$ to denote the set of all the ranged Darboux Cauchy integrable functions defined on $[ a , b ]$.

## 3. Theorems

Claim 1.
$R [ a , b ] ⊆ R E F [ a , b ]$.
Proof.
Let $f ∈ R [ a , b ]$ be arbitrary. We show $f ∈ R E F [ a , b ]$. Let $ϵ ′ > 0$ be arbitrary. By Definition 3, it follows
for some $s ∈ R$. Choose one $B → ϵ ′ ∈ F P V ( δ ϵ ′ )$. Then, $R E F ( B → ϵ ′ ) ⊆ F P V ( δ ϵ ′ )$ and thus, by (1),
$∀ B → ∈ R E F ( B → ϵ ′ ) ∀ H → ∈ ∏ [ B → ] | A f ( B → , H → ) − s | < ϵ ′ .$
By Definition 4, this completes our proof. ☐
Claim 2.
$R [ a , b ] = D C [ a , b ] = D I [ a , b ]$.
Proof.
The proofs could be found in ([4], Theorem 6.6) and ([7], Theorems 3.2.6 and 3.2.7). ☐
Lemma 1.
$R [ a , b ] = R E F [ a , b ]$.
Proof.
Due to Claims 1 and 2, it suffices to show $R E F [ a , b ] ⊆ D C [ a , b ]$. Let $f ∈ R E F [ a , b ]$ be arbitrary. We show $f ∈ D C [ a , b ]$. Let $ϵ ^ > 0$ be arbitrary. Then, by Definition 4
$∃ B → ϵ ^ 4 ∈ F P V [ a , b ] ∀ B → ∈ R E F ( B → ϵ ^ 4 ) ∀ H → ∈ ∏ [ B → ] | A f ( B → , H → ) − s | < ϵ ^ 4$
for some $s ∈ R$. Furthermore, by the definitions of upper and lower Darboux sums of f with respect to $B → ϵ ^ 4$, one has $∃ H → ϵ ^ 4 , K → ϵ ^ 4 ∈ ∏ [ B → ϵ ^ 4 ]$ such that
$A ¯ f ( B → ϵ ^ 4 ) − ϵ ^ 4 < A f ( B → ϵ ^ 4 , H → ϵ ^ 4 ) A f ( B → ϵ ^ 4 , K → ϵ ^ 4 ) < A ̲ f ( B → ϵ ^ 4 ) + ϵ ^ 4 .$
By Equation (3), it follows directly that
$A ¯ f ( B → ϵ ^ 4 ) − A ̲ f ( B → ϵ ^ 4 ) − ϵ ^ 2 < A f ( B → ϵ ^ 4 , H → ϵ ^ 4 ) − A f ( B → ϵ ^ 4 , K → ϵ ^ 4 ) ≤ | A f ( B → ϵ ^ 4 , H → ϵ ^ 4 ) − A f ( B → ϵ ^ 4 , K → ϵ ^ 4 ) | = | ( A f ( B → ϵ ^ 4 , H → ϵ ^ 4 ) − s ) + ( − A f ( B → ϵ ^ 4 , K → ϵ ^ 4 ) + s ) | ≤ | ( A f ( B → ϵ ^ 4 , H → ϵ ^ 4 ) − s ) | + | A f ( B → ϵ ^ 4 , K → ϵ ^ 4 ) − s | .$
By Equations (2) and (4), one has $A ¯ f ( B → ϵ ^ 4 ) − A ̲ f ( B → ϵ ^ 4 ) < ϵ ^ .$ Hence, by Definition 5, $f ∈ D C [ a , b ]$. ☐
Claim 3.
$∥ K → ∥ p < γ ⇒ 〈 〈 K → 〉 〉 < γ , ∀ K → ∈ F P V [ a , b ]$.
Proof.
By definition, $∥ K → ∥ p = ∑ i = 0 | K → | − 1 | K → i + 1 − K → i | p p < γ$, i.e.,
$∀ i ∈ { 0 , 1 , 2 , … , | K → − 1 | } [ | K → i + 1 − K → i | < γ ] ,$
i.e., by definition $〈 〈 K → 〉 〉 < γ$. ☐
By this, we know $F P V p ( δ ) ⊊ F P V ( δ )$, since the equality is obviously false.
Lemma 2.
$R [ a , b ] ⊆ p [ a , b ]$.
Proof.
Let $f ∈ R [ a , b ]$ be arbitrary. We show $f ∈ p [ a , b ]$. Let $ϵ ˜ > 0$ be arbitrary. Then, by Definition 3
for some $s ∈ R$. Let us take $δ ˜ ϵ ˜ = δ ϵ ˜$. Let $B → ∈ F P V p ( δ ˜ ϵ ˜ ) = F P V p ( δ ϵ ˜ )$ be arbitrary. Then, by definition, one has $∥ B → ∥ p = ∑ j = 0 | B → | − 1 | B → j + 1 − B → j | p p < δ ˜ ϵ ˜$, i.e., by Claim 3, $〈 〈 B → 〉 〉 < δ ˜ ϵ ˜$, i.e., $F P V p ( δ ˜ ϵ ˜ ) ⊊ F P V ( δ ˜ ϵ ˜ ) = F P V ( δ ϵ ˜ )$, then the result $f ∈ p [ a , b ]$ follows immediately from Equation (5) and Definition 7. ☐
Since each $F P V p ( δ )$ is a proper subset of $F P V ( δ )$, it appears that $R [ a , b ] = p [ a , b ]$ is doubtful. However, the following lemma proves otherwise.
Lemma 3.
$p [ a , b ] ⊆ R E F [ a , b ]$.
Proof.
Let $f ∈ p [ a , b ]$ be arbitrary. We show $f ∈ R E F [ a , b ]$. Let $ϵ ′ > 0$ be arbitrary. Then, by Definition 7
for some $s ∈ R$. Let us arbitrarily choose one $B → ϵ ′ ∈ F P V p ( δ ˜ ϵ ′ )$. Let $B → ∈ R E F ( B → ϵ ′ )$ be arbitrary. Then,
$∥ B → ∥ p = ∑ i = 0 | B → | − 1 | B → i + 1 − B → i | p p ≤ ∥ B → ϵ ′ ∥ p = ∑ j = 0 | B → ϵ ′ | − 1 | B → j + 1 ϵ ′ − B → j ϵ ′ | p p < δ ˜ ϵ ′ ,$
i.e., $R E F ( B → ϵ ′ ) ⊆ F P V p ( δ ˜ ϵ ′ )$. Then, the result $f ∈ R E F [ a , b ]$ follows immediately from Equation (6). ☐
Theorem 1.
$p [ a , b ] = R [ a , b ]$.
Proof.
From Lemmas 1–3, one has
$R [ a , b ] ⊆ p [ a , b ] ⊆ R E F [ a , b ] = R [ a , b ]$
and thus the result follows immediately. ☐
Example 2.
Take $[ a , b ] = [ 0 , 1 ]$ and $f ( x ) = x$ for example. Then, $f ∈ R [ a , b ]$. Here, based on the definition of p-norm integral, we give a detailed proof showing $f ∈ p [ a , b ]$. Let $ϵ ˜ > 0$ be arbitrary. Let $N ϵ ˜ ∈ N$ be the natural number satisfying $1 N ϵ ˜ ≤ ϵ ˜ < 1 N ϵ ˜ − 1$ for all $N ϵ ˜ ≥ 2$. Observe that when $ϵ ˜$ decreases, $N ϵ ˜$ increases. Now, take $δ ˜ ϵ ˜ = 1 N ϵ ˜$. Let $B → ∈ F P V p ( 1 N ϵ ˜ )$ and $H → ∈ ∏ [ B → ]$ be arbitrary. Then,
$∑ j = 0 | B → | − 1 ( B → j + 1 − B → j ) · B → j ≤ A f ( B → , H → ) = ∑ j = 0 | B → | − 1 ( B → j + 1 − B → j ) · f ( H → j ) ≤ ∑ j = 0 | B → | − 1 ( B → j + 1 − B → j ) · B → j + 1 .$
Since $B → ∈ F P V p ( 1 N ϵ ˜ )$, one has
$( ∑ j = 0 | B → | − 1 B → j p ) 1 p < 1 N ϵ ˜ ,$
i.e., $( ∑ j = 0 | B → | − 1 B → j p ) < ( 1 N ϵ ˜ ) p ,$ i.e., $B → j < 1 N ϵ ˜$ for all $0 ≤ j ≤ | B → | − 1$. Hence, one has
$1 N ϵ ˜ · [ 1 N ϵ ˜ + 2 1 N ϵ ˜ + … + ( N ϵ ˜ − 1 ) · 1 N ϵ ˜ ] ≤ ∑ j = 0 | B → | − 1 ( B → j + 1 − B → j ) · B → j ,$
$∑ j = 0 | B → | − 1 ( B → j + 1 − B → j ) · B → j + 1 ≤ 1 N ϵ ˜ · [ 1 N ϵ ˜ + 2 1 N ϵ ˜ + … + N ϵ ˜ · 1 N ϵ ˜ ] .$
Hence, one has
$( 1 N ϵ ˜ ) 2 · [ ( N ϵ ˜ − 1 ) N ϵ ˜ 2 ] ≤ A f ( B → , H → ) ≤ ( 1 N ϵ ˜ ) 2 · [ ( ( N ϵ ˜ ) ( N ϵ ˜ + 1 ) 2 ] ,$
i.e.,
$1 2 − 1 N ϵ ˜ ≤ A f ( B → , H → ) ≤ 1 2 + 1 N ϵ ˜ .$
Since $lim ϵ ˜ → 0 1 N ϵ ˜ = 0$, one has $f ∈ p [ a , b ]$.
Lemma 4.
$D p [ a , b ] ⊆ D C [ a , b ]$.
Proof.
Let $f ∈ D p [ a , b ]$ be arbitrary, we show $f ∈ D C [ a , b ]$. Let $C → ∈ [ 0 ] p$ be arbitrary. Let $ϵ ^ > 0$ be arbitrary. Then, by Definition 9,
$∀ m ≥ N ˇ ϵ ^ 4 , C → ∀ H → ∈ ∏ [ C → m ] | A f ( C → m , H → ) − s | < ϵ ^ 4 ,$
for some $s ∈ R$. Take $B → ϵ ^ = C → N ˇ ϵ ^ 4 , C →$. Then, by the definitions of upper and lower Darboux integral, one has $∃ H → ϵ ^ 4 , K → ϵ ^ 4 ∈ ∏ [ B → ϵ ^ 4 ]$ such that
$A ¯ f ( B → ϵ ^ 4 ) − ϵ ^ 4 < A f ( B → ϵ ^ 4 , H → ϵ ^ 4 ) A f ( B → ϵ ^ 4 , K → ϵ ^ 4 ) < A ̲ f ( B → ϵ ^ 4 ) + ϵ ^ 4 .$
Then, by the identical proof of Equation (4) in Lemma 1, the result follows. ☐
Lemma 5.
$R [ a , b ] ⊆ D p [ a , b ]$.
Proof.
Let $f ∈ R [ a , b ]$ be arbitrary. We show $f ∈ D p [ a , b ]$. Let $ϵ ˇ > 0 , C → ∈ [ 0 ] p$ be arbitrary. By Definition 3
$∃ δ ϵ ˇ > 0 ∀ B → ∈ F P V ( δ ϵ ˇ ) ∀ H → ∈ ∏ [ B → ] | A f ( B → , H → ) − s | < ϵ ˇ ,$
for some $s ∈ R$. By $C → ∈ [ 0 ] p$, it follows
Now, take $γ = δ ϵ ˇ$, one has i.e., by Claim 3, $〈 〈 C k → 〉 〉 < δ ϵ ˇ$, i.e., $F P V p ( δ ϵ ˇ ) ⊆ F P V ( δ ϵ ˇ )$. Take $N ˇ ϵ ˇ , C → = N δ ϵ ˇ$. Let $m ≥ N ˇ ϵ ˇ , C → = N δ ϵ ˇ$, $H → ∈ ∏ [ C m → ]$ be arbitrary. Then, the result $f ∈ D p [ a , b ]$ follows immediately from Equation (8) and Definition 9. ☐
Theorem 2.
$R [ a , b ] = D p [ a , b ]$.
Proof.
From Claim 2, and Lemmas 4 and 5, one has
$R [ a , b ] ⊆ D p [ a , b ] ⊆ D C [ a , b ] = R [ a , b ]$
and thus the result follows immediately. ☐
Lemma 6.
$D D C p [ a , b ] ⊆ D C [ a , b ]$.
Proof.
Let $f ∈ D D C p [ a , b ]$ be arbitrary. We show $f ∈ D C [ a , b ]$. Let $ϵ ^ > 0$ be arbitrary. Let $C → ∈ [ 0 ] p$ be arbitrary. By Definition 8,
Now, take $B → ϵ ^ = C → N ¨ ϵ ^ , C →$. Then, $A ¯ f ( B → ϵ ^ ) − A ̲ f ( B → ϵ ^ ) < ϵ ^$ and thus the result $f ∈ D C [ a , b ]$ follows immediately from Definition 5. ☐
Lemma 7.
$D p [ a , b ] ⊆ D D C p [ a , b ]$.
Proof.
Let $f ∈ D p [ a , b ]$ be arbitrary. We show $f ∈ D D C p [ a , b ]$. Let $ϵ ¨ > 0$, $C → ∈ [ 0 ] p$ be arbitrary. By $f ∈ D p [ a , b ]$, it follows
for some $s ∈ R$. Now, take $N ¨ ϵ ¨ , C → = N ϵ ¨ 4 , C →$. Let $n ≥ N ¨ ϵ ¨ , C →$ be arbitrary. By the definitions of upper and lower Darboux integral,
$A ¯ f ( C → n ) − ϵ ¨ 4 < A f ( C → n , K → ) , A f ( C → n , H → ) < A ̲ f ( C → n ) + ϵ ¨ 4 .$
Thus,
$A ¯ f ( C → n ) − A ̲ f ( C → n ) < A f ( C → n , K → ) − A f ( C → n , H → ) + ϵ ¨ 2 < | A f ( C → n , K → ) − s | + | A f ( C → n , H → ) − s | + ϵ ¨ 2 < ϵ ¨ 4 + ϵ ¨ 4 + ϵ ¨ 2 = ϵ ¨ .$
Thus, the result $f ∈ D D C p [ a , b ]$ follows immediately from Definition 8. ☐
Theorem 3.
$D D C p [ a , b ] = D C [ a , b ]$.
Proof.
By Claim 2, Theorem 2, and Lemmas 6 and 7,
$D C [ a , b ] = R [ a , b ] = D p [ a , b ] ⊆ D D C p [ a , b ] ⊆ D C [ a , b ]$
and thus the result follows immediately. ☐
Lemma 8.
$R D C p [ a , b ] ⊆ D C [ a , b ]$.
Proof.
Let $f ∈ R D C p [ a , b ]$ be arbitrary. We show $f ∈ D C [ a , b ]$. Let $ϵ ^ > 0$ be arbitrary. By Definition 10,
Now, choose one $B → ϵ ^ ∈ F P V p ( δ ¯ ϵ ^ ) ⊆ F P V [ a , b ]$, then the result $f ∈ D C [ a , b ]$ follows immediately via Definition 5. ☐
Lemma 9.
$R [ a , b ] ⊆ R D C p [ a , b ]$.
Proof.
Let $f ∈ R [ a , b ]$ be arbitrary. We show $f ∈ R D C p [ a , b ]$. Let $ϵ ¯ > 0$ be arbitrary. By Definition 3,
for some $s ∈ R$. Now, take $δ ¯ ϵ ¯ = δ ϵ ¯ 4$. Let $B → ∈ F P V p ( δ ¯ ϵ ¯ )$ be arbitrary. By Claim 3, $B → ∈ F P V p ( δ ¯ ϵ ¯ ) ⊊ F P V ( δ ¯ ϵ ¯ ) = F P V ( δ ϵ ¯ 4 ) .$ By the definitions of upper and lower Darboux integral, one has $∃ H → ,$
$A f ( B → , H → ) < A ̲ f ( B → ) + ϵ ¯ 4 A ¯ f ( B → ) − ϵ ¯ 4 < A f ( B → , K → ) .$
Thus, by Equation (10) and the identical proof of Equation (4) in Lemma 1, one has
$A ¯ f ( B → ) − A ̲ f ( B → ) < ϵ ¯ .$
Thus the result $f ∈ D D C p [ a , b ]$ follows immediately from Definition 10. ☐
Theorem 4.
$R D C p [ a , b ] = D C [ a , b ]$.
Proof.
From Claim 2, and Lemmas 8 and 9, $D C [ a , b ] = R [ a , b ] ⊆ R D C p [ a , b ] ⊆ D C [ a , b ]$, the result follows immediately. ☐
Theorem 5.
$R [ a , b ] = R E F [ a , b ] = D C [ a , b ] = D I [ a , b ] = p [ a , b ] = D p [ a , b ] = D D C p [ a , b ] = R D C p [ a , b ]$.
Proof.
By Lemma 1, Claim 2, and Theorems 1–4, the result follows immediately. ☐

## 4. Conclusions

In this article, we have shown how to give a new equivalent definition of the Riemann integral based on the p-norm. This norm could easily link functional analysis and the Riemann integral. By directly defining an alternative definition based on this norm, one gains more insightful knowledge about all sorts of equivalences of the Riemann integral. Furthermore, we also derive some related equivalences based on this norm. The p-norm provides an alternative to the usual definition of the Riemann integral.

## Funding

This research was funded by the Natural Science Foundation of Fujian Province, China (Grant No. 2017J01566).

## Conflicts of Interest

The author declares no conflict of interest.

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