Equivalences of Riemann Integral Based on p-Norm

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.


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.

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 Tag(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]: where ∆ P j = P(j) − P(j − 1) and ζ P j ∈ [P(j − 1), P(j)].Then, one defines the mesh of the partition P, then we say f is Riemann integrable [4] and the integral of f is defined as

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: For example, B = (1, 1.2, 1.5, 2.2, 2.6, 3) ∈ FPV (5)  [1,3], in which i.e., the set of all the finite partitions of [a, b].Observe that For example, if B = (1, 1.2, 1.5, 2.2, 2.6, 3) ∈ FPV (5) [1, 3], then • (Riemann norm) B := max{B j+1 − B j : 0 ≤ j ≤ | B| − 1}.This is exactly the usual definition of a mesh.
where H represents a sequence of tags, i.e., H ∈ ∏[ B].
FPV(δ) collects all the finite partition vectors whose Riemann norms are less than δ, while FPV p (δ) collects all the finite partition vectors whose p-norms are less than δ.One could easily verify that FPV p (δ) FPV(δ).The equality is obvious not true.For example, if B = (0, 0.2, 1, 1.5, 2) and where each C j ∈ FPV[a, b] and denotes the (j + 1)th partition vectors in C. and [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 .

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.Let us define the upper Darboux sum of f with respect to B as follows: } and the lower Darboux sum of f with respect to B as follows: Since f is a bounded function, these definitions are all well-defined.Definition 5. (Darboux Cauchy integrable [5,6]) If  Based on p-norm, we have the following new definitions.

Define
and Let ), one has 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.
for some s ∈ R. Furthermore, by the definitions of upper and lower Darboux sums of f with respect to By Equation (3), it follows directly that By Equations ( 2) and ( 4), one has Ā f ( i.e., by definition K < γ. By this, we know FPV p (δ) FPV(δ), since the equality is obviously false.
for some s ∈ R.
for some s ∈ R. Let us arbitrarily choose one B ∈ FPV p ( δ ).Let B ∈ REF( B ) be arbitrary.Then, Then, by the identical proof of Equation ( 4) in Lemma 1, the result follows.
for some s ∈ R. Now, take N¨ , C = N ¨ 4 , C .Let n ≥ N¨ , C be arbitrary.By the definitions of upper and lower Darboux integral, ∃ H, K ∈ Thus, 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.

Definition 4 .
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 FPV[a, b].For any P, Q ∈ FPV[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 REF( Q) = { P ∈ FPV[a, b] : P Q}.One could easily check that ∀δ > 0 and ∀ B ∈ FPV(δ)[REF( B) ⊆ FPV(δ)].If there exists s we say f is refinement-integrable, denoted by (REF) b a f = s.Furthermore, we use REF[a, b] to denote the set of all the refinement-integrable functions defined on [a, b].
we say f is Darboux Cauchy integrable and use DC[a, b] to denote the set of all the Darboux Cauchy integrable functions defined on [a, b].

Definition 6 .
(Darboux integrable [7], p. 120) If there exists s ∈ R such that s = b a f = ¯ b a f , we say f is Darboux integrable, denoted by (DI) b a f = s.Furthermore, we use DI[a, b] to denote the set of all the Darboux integrable functions defined on [a, b].

Example 1 .
we say f is p-integrable, denoted by (p) b a f = s.Furthermore, we use p[a, b] to denote the set of all the p-integrable functions defined on [a, b].In this example, we demonstrate that if f , g ∈ p[a, b], then b a f (x)x + b a g(x)dx = b a ( f + g)(x)dx.Suppose b a f (x)dx = c and b a g(x)dx = d.Let ˜ > 0 be arbitrary.By f , g ∈ p[a, b], one has there exist c, d and there exist δ f we say f is discrete Darboux Cauchy integrable, denoted by (DDC p ) b a f = s.Furthermore, we use DDC 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 denoted by (Dp) b a f = s.Furthermore, we use Dp[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 we say f is ranged Darboux Cauchy integrable, denoted by (RDC p ) b a f = s.Furthermore, we use RDC p [a, b] to denote the set of all the ranged Darboux Cauchy integrable functions defined on [a, b].
arbitrary.Then, the result f ∈ Dp[a, b] follows immediately from Equation (8) and Definition 9.