the Lebesgue integral can be evaluated as an improper Riemann integral if the integrand is nonnegative

Using the gamma function as an example, my textbook states that since the integrand of the Lebesgue integral is nonnegative, the Lebesgue integral can be evaluated as an improper Riemann integral (Yeh, 2014). However, the condition for the equivalence between improper Riemman integral and Lebesgue integral in Apostol (1974) only requires that the Riemman integration of |f| on a finite interval is bounded.

My question is how to deduce the equivalence relationship between improper Riemman integral and Lebesgue integral from the first condition (i.e., the integrand of the Lebesgue integral is nonnegative).

• Can you state the actual conditions? I'm guessing that "Yeh, 2014" refers to this book: amazon.com/Real-Analysis-Theory-Measure-Integration-ebook/dp/… which most people probably don't have handy... – Math1000 Jun 7 '15 at 14:21
• @Math1000 Yes, I do refer to the book "Real analysis: Theory of measure and integration" by J. Yeh. In the book, the condition is not stated in a theorem or proposition but in an example, without any further explanation. – Song Wang Jun 7 '15 at 14:40