Page 77 - Vector Analysis

§3.1 The Double Integrals 73

“ð” Let ε ą 0 be given. By assumption there exists a partition P of A such that U (f, P) ´
L(f, P) ă ε. Then

żż

0 ď f (x) dx ´ f (x) dx ď U (f, P) ´ L(f, P) ă ε .

AA

żż

Since ε ą 0 is given arbitrary, we must have f (x)dx = f (x)dx; thus f is Riemann

integrable over A. AA

˝

Definition 3.10. Let P = t∆1, ∆2, ¨ ¨ ¨ , ∆N u be a partition of a bounded set A Ď Rn. A
collection of N points tξ1, ¨ ¨ ¨ , ξN u is called a sample set for the partition P if ξk P ∆k for
all k = 1, ¨ ¨ ¨ , N . Points in a sample set are called sample points for the partition P.

Let A Ď Rn be a bounded set, and f : A Ñ R be a bounded function. A Riemann
sum of f for the the partition P = t∆1, ∆2, ¨ ¨ ¨ , ∆N u of A is a sum which takes the form

N

ÿ f A(ξi)νn(∆k) ,

k=1

where the set Ξ = tξ1, ξ2, ¨ ¨ ¨ , ξN u is a sample set for the partition P.

Theorem 3.11 (Darboux). Let A Ď Rn be a bounded set, and f : A Ñ R be a bounded
function with extension f A given by (3.1). Then f is Riemann integrable over A if and only
if there exists I P R such that for every given ε ą 0, there exists δ ą 0 such that if P is a
partition of A satisfying }P} ă δ, then any Riemann sums for the partition P belongs to the
interval (I ´ ε, I + ε). In other words, f is Riemann integrable over A if and only if there
exists I P R such that for every given ε ą 0, there exists δ ą 0 such that

ˇ N ˇ
ˇ ˇ
ˇ ÿ A (ξk )ν (∆k ) I ˇ (3.2)

f ´ ă ε

k=1

whenever P = t∆1, ¨ ¨ ¨ , ∆N ( is a partition of A satisfying }P} ă δ and tξ1, ξ2, ¨ ¨ ¨ , ξN u is a
sample set for P.

Proof. The boundedness of A guarantees that A Ď [ r , r ]n for some r ą 0. Let R =
´
[ ]n. 22
´ r r
,
22

“ð” Suppose the right-hand side statement is true. Let ε ą 0 be given. Then there exists
δ ą 0 such that if P = t∆1, ¨ ¨ ¨ , ∆N ( is a partition of A satisfying }P} ă δ, then for