7.1: Upper and Lower Integrals
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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.
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):xi−1≤x≤xi}
and
Mi=sup{f(x):xi−1≤x≤xi}.
We call
L(f,P)=n∑i=1mi(xi−xi−1)
the lower sum of f determined by P and
U(f,P)=n∑i=1Mi(xi−xi−1)
the upper sum of f determined by P.
If P1 and P2 are both partitions of [a,b] and P1⊂P2, then we call P2 a refinement of P1.
If P1 and P2 are both partitions of [a,b], then we call the partition P=P1∪P2 the common refinement of P1 and P2.
Suppose P1={x0,x1,…,xn} is a partition of [a,b],s∈(a,b), s∉P1, 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 xi−1<s<xi and let
w1=inf{f(x):xi−1≤x≤s},W1=sup{f(x):xi−1≤x≤s},w2=inf{f(x):s≤x≤xi},W2=sup{f(x):s≤x≤xi},mi=inf{f(x):xi−1≤x≤xi},
and
Mi=sup{f(x):xi−1≤x≤xi}.
Then w1≥mi,w2≥mi,W1≤Mi, and W2≤Mi. Hence
L(f,P2)−L(f,P1)=w1(s−xi−1)+w2(xi−s)−mi(xi−xi−1)=w1(s−xi−1)+w2(xi−s)−mi(s−xi−1)−mi(xi−s)=(w1−mi)(s−xi−1)+(w2−mi)(xi−s)≥0
and
U(f,P1)−U(f,P2)=Mi(xi−xi−1)−W1(s−xi−1)−W2(xi−s)=Mi(s−xi−1)+Mi(xi−s)−W1(s−xi−1)−W2(xi−s)=(Mi−W1)(s−xi−1)+(Mi−W2)(xi−s)≥0.
Thus L(f,P1)≤L(f,P2) and U(f,P2)≤U(f,P1). Q.E.D.
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.
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.
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.
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_f≤U(f,P).
But this shows that the lower integral is a lower bound for any upper sum. Hence
∫baf≤∫baf.
Q.E.D.