Real Analysis/Riemann integration

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DefinitionEdit

Riemann integration is the formulation of integration most people think of if they ever think about integration. It is the only type of integration considered in most calculus classes; many other forms of integration, notably Lebesgue integrals, are extensions of Riemann integrals to larger classes of functions. The Riemann integral was developed by Bernhard Riemann in 1854 and was, when invented, the first rigorous definition of integration applicable to not necessarily continuous functions.

We will first define some preliminary ideas.

PartitionsEdit

DefinitionEdit

Let

A Partition is defined as the ordered n-tuple of real numbers such that

Norm of a PartitionEdit

Let be a partition given by

Then, the Norm (or the "mesh") of is defined as

Tagged PartitionEdit

Let be a partition

A Tagged Partition is defined as the set of ordered pairs such that . The points are called Tags.

RiemannEdit

Riemann sum of a function

Riemann SumsEdit

Let

Let be a tagged partition of

The Riemann Sum of over with respect to is given by

Riemann IntegralEdit

Let

Let

We say that is Integrable on if and only if for every , there exists such that for every partition satisfying , we have that

is said to be the integral of over , and is written as

or as

PropertiesEdit

Theorem (Uniqueness)Edit

Let be integrable on

Then the integral of is unique

ProofEdit

Assume, if possible that are both integrals of over . Consider

As are integrals, there exist such that for all that satisfy and for all that satisfy

Let . Hence, if is a partition satisfying , then we have and that

That is, , which is an obvious contradiction. Hence the integral of is unique.


We now state (without proof) two seemingly obvious properties of the integral.

TheoremEdit

Let be integrable and let

Then:

(i)

(ii)

Theorem (Boundedness Theorem)Edit

Let be Riemann integrable. Then, is bounded over

ProofEdit

Assume if possible that is unbounded. For every divide the interval into parts. Hence, for every , is unbounded on at least one of these parts. Call it .

Now, let be given. Consider an arbitrary . Let be a tagged partition such that and , where is taken so as to satisfy .

Thus we have that . But as is arbitrary, we have a contradiction to the fact that is Riemann integrable.

Hence, is bounded.

IntegrabilityEdit

We now study classes of Riemann integrable functions. The first "constraint" on Riemann integrable functions is provided by the Cauchy Integrability Criterion.

Theorem (Cauchy Criterion)Edit

Let

Then,

(i) is Riemann integrable on if and only if

(ii) For every , there exists such that if are two partitions satisfying then

ProofEdit

()Let and let be given.

Then, there exists such that for every partition satisfying ,we have

Now, let partitions be such that .

Thus we have that , that is

() For every , consider such that for all partitions satisfying , we have .

Without loss of generality, we can assume that when . For every , let be a partition such that

The sequence is a Cauchy sequence, and hence it has a limit .

Now, for every , we have a such that implies .

Thus

Theorem (Squeeze Theorem)Edit

Let

Then,

(i) is Riemann integrable on if and only if

(ii) For every , there exist Riemann integrable functions such that

for all and

ProofEdit

()Take . It is easy to see that

()Let . Then, there exist functions such that . Further, if and , then there exist such that if a partition satisfies then and then

Now let be an interval satisfying .

Now, we can easily see that . Hence, is a Cauchy sequence, with a limit , and as in the previous proof, we can show that