Integral

Supremum and Infimum

Definition

Def Upper Bound and Lower Bound For a set , if exists such that for all , , then is an upper bound for . Similarly, if for all , , then is called a lower bound.

Definition

Def Supremum and Infimum is called supremum of a set if is an upper bound and for all upper bound , . Similarly, is infimum of if it is the greatest upper bound.

Lemma

Lemma

• iff for all there exists such that .
• iff for all there exists such that .

Definition

Def Maximum and Minimum Suppose . If there exists such that is supremum of , then is called the maximum of . Similarly, is called the minimum if is the infimum lying in .

Partition and Refinement

Definition

Def Partition of Interval Given interval , a set is a partition of if

• .
• is finite.
• and .

By convention, label the elements in as:

Definition

Def Refinement A partition is called refinement of if .

Proposition

Prop If is a partition of then is a refinement of .

Proposition

Prop Suppose and are partitions of , then is a refinement of and .

Riemann-Darboux Integrability

Definition

Def Lower Sum and Upper Sum Suppose is a bounded function, and elements of partition on be . For each , , we defineThen the lower sum and upper sum are defined by

Proposition

Prop For all , there is some partition such that

Riemann-Darboux Integrability

Suppose is a bounded function, is integrable if where and . And we define the integral of be

Lemma

Lemma Suppose be bounded function, and are partitions of . Then,

• .
• If is a refinement of then and .
• .

Proof The first proposition arises from . And implies the third proposition.