By C. Andreian Cazacu, N. Boboc, M. Jurchescu, I. Suciu
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Additional info for Complex Analysis. 5th Romanian-Finnish Seminar
Evaluate JQ xe~xdx. 3. Let / : J —>• R. Suppose 3A 6 R such that for every e > 0 there are integrable functions g and h with g < f < h and A — e < fTg < fjh < A+e. Show that / is integrable with Jj f = A. 4. Let / : / —» R be integrable over / and suppose that g : I —> R is equal to / except possibly at countably many points in I. Show that g is integrable with fIg = JI f. 5. Let f(t) = sini if t e [0,1]\Q and f(t) = t if t G Q n [0,1]. Show that / is integrable over [0,1] and calculate JQ f. 6.
17. Let / , g : [a, b] —> R be continuous on (a, b] and g' absolutely integrable over [a, b]. Assume F(t) = Jt / i s bounded. a+ F(c)g(c) exists. Chapter 5 Convergence Theorems The principal reason that the Lebesgue integral is favored over the Riemann integral is the fact that convergence theorems of the form lim ft fk = J 7 (lim fk) hold for the Lebesgue integral under very general conditions. The major convergence theorems of this type are the Monotone Convergence Theorem (MCT) and the Dominated Convergence Theorem (DCT).
Then by the first inequality f(ti)£(Ji)- 0 < £ (mew - / / ) = £ /(*0'(-*) - / /< £ j+ ^ Jji ' j+ Jji and 0 < - E (s(uWi) - fj /) = E |/eo w - /<7 £ / so the second inequality follows. Henstock's Lemma asserts that if 7 is a gauge on I such that 7-fine tagged partitions of I induce Riemann sums which give good approximations to the value of the integral over J, then likewise any 7-flne partial tagged partition Henstock's Lemma and Improper Integrals 25 induces Riemann sums which give good approximations to the value of the integral over the union of the intervals in the partial tagged partition.