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