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Math 554 – Riemann Integration
Handout #9a (Nov. 29)
Defn. A collection of n + 1 distinct points of the interval [a, b]
P := {x 0 = a < x 1 < · · · < xi− 1 < xi < · · · < b =: xn}
is called a partition of the interval. In this case, we define the norm of the partition by
‖P ‖ := max 1 ≤ i≤n ∆xi.
where ∆xi := xi − xi− 1 is the length of the i-th subinterval [xi− 1 , xi].
Defn. For a given partition P , we define the Riemann upper sum of a function f by
U (f, P ) :=
∑^ n
i=
Mi ∆xi
where Mi denotes the supremum of f over each of the subintervals [xi− 1 , xi]. Similarly, we define the Riemann lower sum of a function f by
L(f, P ) :=
∑^ n i=
mi ∆xi
where mi denotes the infimum of f over each of the subintervals [xi− 1 , xi]. Since mi ≤ Mi, we note that L(f, P ) ≤ U (f, P ).
for any partition P.
Defn. Suppose P 1 , P 2 are both partitions of [a, b], then P 2 is called a refinement of P 1 , denoted by
P 1 ≺ P 2 ,
if as sets P 1 ⊆ P 2.
Note. If P 1 ≺ P 2 , it follows that ‖P 2 ‖ ≤ ‖P 1 ‖ since each of the subintervals formed by P 2 is contained in a subinterval arising from P 1.
Lemma. If P 1 ≺ P 2 , then L(f, P 1 ) ≤ L(f, P 2 ).
and U (f, P 2 ) ≤ U (f, P 1 ).
Proof. Suppose first that P 1 is a partition of [a, b] and that P 2 is the partition obtained from P 1 by adding an additional point z. The general case follows by induction, adding one point at at time. In particular, we let
P 1 := {x 0 = a < x 1 < · · · < xi− 1 < xi < · · · < b =: xn}
and P 2 := {x 0 = a < x 1 < · · · < xi− 1 < z < xi < · · · < b =: xn}
for some fixed i. We focus on the upper Riemann sum for these two partitions, noting that the inequality for the lower sums follows similarly. Observe that
U (f, P 1 ) :=
∑^ n
j=
Mj ∆xj
and
U (f, P 2 ) :=
i∑− 1
j=
Mj ∆xj + M (z − xi− 1 ) + M˜ (xi − z) +
∑^ n
j=i+
Mj ∆xj
where M := sup[xi− 1 ,z] f (x) and M˜ := sup[z,xi] f (x). It then follows that U (f, P 2 ) ≤ U (f, P 1 ) since
M, M˜ ≤ Mi. 2
Defn. If P 1 and P 2 are arbitrary partitions of [a, b], then the common refinement of P 1 and P 2 is defined as the formal union of the two.
Corollary. Suppose P 1 and P 2 are arbitrary partitions of [a, b], then
L(f, P 1 ) ≤ U (f, P 2 ).
Proof. Let P be the common refinement of P 1 and P 2 , then
L(f, P 1 ) ≤ L(f, P ) ≤ U (f, P ) ≤ U (f, P 2 ). 2
Defn. The lower Riemann integral of f over [a, b] is defined to be
∫ (^) b
a
f (x)dx := sup all partitions P of [a,b]
L(f, P ).
Similarly, the upper Riemann integral of f over [a, b] is defined to be
∫ (^) b
a
f (x)dx := inf all partitions P of [a,b]
U (f, P ).
By the definitions of least upper bound and greatest lower bound, it is evident that for any function f there holds ∫ (^) b
a
f (x)dx ≤
∫ (^) b
a
f (x)dx.
Defn. A function f is Riemann integrable over [a, b] if the upper and lower Riemann integrals coincide. We denote this common value by
∫ (^) b a f^ (x)^ dx.
Theorem. A necessary and sufficient condition for f to be Riemann integrable is given è > 0, there exists a partition P of [a, b] such that
(∗) U (f, P ) − L(f, P ) < è.
