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Mathematics LibreTexts

7.1: Upper and Lower Integrals

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Definition

Given a closed interval [a,b]R with a<b, we call any finite subset of [a,b] which includes both a and b a partition of [a,b].

For convenience, whenever we consider a partition P of an interval [a,b] we will index the elements in increasing order, starting with 0. That is, if |P|=n+1 and P={x0,x1,,xn}, then

a=x0<x1<x2<<xn=b.

Definition

Suppose P={x0,x1,,xn} is a partition of [a,b] and f:[a,b]R is bounded. For i=1,2,,n, let

mi=inf{f(x):xi1xxi}

and

Mi=sup{f(x):xi1xxi}.

We call

L(f,P)=ni=1mi(xixi1)

the lower sum of f determined by P and

U(f,P)=ni=1Mi(xixi1)

the upper sum of f determined by P.

Definition

If P1 and P2 are both partitions of [a,b] and P1P2, then we call P2 a refinement of P1.

Definition

If P1 and P2 are both partitions of [a,b], then we call the partition P=P1P2 the common refinement of P1 and P2.

lemma 7.1.1

Suppose P1={x0,x1,,xn} is a partition of [a,b],s(a,b), sP1, and f:[a,b]R is bounded. If P2=P1{s}, then L(f,P1)L(f,P2) and U(f,P2)U(f,P1).

Proof

Suppose xi1<s<xi and let

w1=inf{f(x):xi1xs},W1=sup{f(x):xi1xs},w2=inf{f(x):sxxi},W2=sup{f(x):sxxi},mi=inf{f(x):xi1xxi},

and

Mi=sup{f(x):xi1xxi}.

Then w1mi,w2mi,W1Mi, and W2Mi. Hence

L(f,P2)L(f,P1)=w1(sxi1)+w2(xis)mi(xixi1)=w1(sxi1)+w2(xis)mi(sxi1)mi(xis)=(w1mi)(sxi1)+(w2mi)(xis)0

and

U(f,P1)U(f,P2)=Mi(xixi1)W1(sxi1)W2(xis)=Mi(sxi1)+Mi(xis)W1(sxi1)W2(xis)=(MiW1)(sxi1)+(MiW2)(xis)0.

Thus L(f,P1)L(f,P2) and U(f,P2)U(f,P1). Q.E.D.

Proposition 7.1.2

Suppose P1 and P2 are partitions of [a,b], with P2 a refinement of P1. If f:[a,b]R is bounded, then L(f,P1)L(f,P2) and U(f,P2)U(f,P1).

Proof

The proposition follows immediately from repeated use of the previous lemma. Q.E.D.

Proposition 7.1.3

Suppose P1 and P2 are partitions of [a,b]. If f:[a,b]R is bounded, then L(f,P1)U(f,P2).

Proof

The result follows immediately from the definitions if P1=P2. Otherwise, let P be the common refinement of P1 and P2. Then

L(f,P1)L(f,P)U(f,P)U(f,P2).

Q.E.D.

Definition

Suppose a<b and f:[a,b]R is bounded. We call

ba_f=sup{L(f,P):P is a partition of [a,b]}

the lower integral of f over [a,b] and

¯baf=inf{U(f,P):P is a partition of [a,b]}

the upper integral of f over [a,b].

Note that both the lower integral and the upper integral are finite real numbers since the lower sums are all bounded above by any upper sum and the upper sums are all bounded below by any lower sum.

Proposition 7.1.4

Suppose a<b and f:[a,b]R is bounded. Then

ba_f¯baf.

Proof

Let P be a partition of [a,b]. Then for any partition Q of [a,b], we have L(f,Q)U(f,P). Hence U(f,P) is an upper bound for any lower sum, and so

ba_fU(f,P).

But this shows that the lower integral is a lower bound for any upper sum. Hence

bafbaf.

Q.E.D.


This page titled 7.1: Upper and Lower Integrals is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform.

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