←Fundamental Theorem of Calculus | Real AnalysisDarboux Integral |
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Another popular definition of "integration" was provided by Jean Gaston Darboux and is often used in more advanced texts. In this chapter, we will define the Darboux integral, and demonstrate the equivalence of Riemann and Darboux integrals.

## Upper and Lower SumsEdit

### DefinitionEdit

Let

Let be a (finite) partition of

For every define:

and

The **Upper Sum** of with respect to is defined as

The **Lower Sum** of with respect to is defined as

### DefinitionEdit

A partition is said to be a **Refinement** for a given partition iff

### TheoremEdit

Let

Let be a partition and let be a refinement of . Then,

(i)

(ii)

#### ProofEdit

Let and let be such that . Also, let , and

Obviously, , but as is arbitrary, we have that

Similarly, we can prove

## Darboux IntegrationEdit

Let

We say that is **Darboux Integrable** on if and only if

, where the supremum is taken over the __Set of all partitions on that interval__

is also written as

### TheoremEdit

Let

is Darboux integrable over if and only if for every , there exists a partition on such that

#### ProofEdit

()Let and let be given. Thus, by Gap Lemma, there exists a partition such that both , and hence

()Let be any partition on . Observe that is a lower bound of the set is any partition and that is an upper bound of the set is any partition

Thus, let and . As , we have that cannot be true. Also, as are a supremum and infimum respectively, is also not possible. Hence, (say).

As , we have that

## Equivalence of Riemann and Darboux IntegralsEdit

At first sight, it may appear that the Darboux integral is a special case of the Riemann integral. However, this is illusionary, and indeed the two are equivalent.

### LemmaEdit

(1) Let be Darboux Integrable, with integral

Define function

(2) Then

#### ProofEdit

Let . Consider set of tagged partitions such that

Let be the set of where and

note that and that the set indeed contains all partitions with

Now, for , we can construct such that

Hence,

i.e.

### TheoremEdit

Let

(1) is Riemann integrable on iff

(2) is Darboux Integrable on

#### ProofEdit

() Let be given.

(1) tagged partition such that .

Let partitions and be the same refinement of but with different tags.

Therefore, and

i.e., by the triangle inequality,

Gap Lemma ,

being arbitrary, using Theorem 2.1, we have that is Darboux Integrable.

()Let be given.

(2), Theorem 2.1 partition such that

Hence, as

By Lemma 3.1, if

Thus, if we put , we have (1)

We note here that the crucial element in this proof is Lemma 3.1, as it essentially is giving an order relation between and , which is not directly present in either the Riemann or Darboux definition.