ABSTRACT
Proof. Given s in J consider any sequence An —► 0 in R with λη φ 0 and s + A„ in J for all n. To prove (4) we need only show that |£ (s , t)dt is integrable on K and
(5)
Define gn on K by
(6)
for all t in K . By (6) and (ii)
(7)
By (3) and (6)
(8)
For each t where (1) holds the Theorem of the Mean applied to (6) gives gn(t) = §£(sn,t) for some sn between s and s + An. So (2) gives
(9)
The Dominated Convergence Theorem (Theorem 4(§2.8)) applied to (7), (8), and (9) gives (5) and (4). □
Theorem 1 provides a useful tool for evaluating certain in tegrals. The exercises yield the evaluation f R = π. The
Theorem 1.
