# Lebesgue vs Riemann: Trouble understanding a theorem on improper integrals

In the book "Measures, Integrals and Martingales" by R. Schilling, the Riemann integral is compared to the Lebesgue integral (Chapter 11).

I have trouble verifying the following statement for myself (it is the very first sentence of the proof to Corollary 11.9):

Riemann integrability of a measurable function $u \colon [0,\infty) \to \mathbb{R}$ that is Riemann integrable on every interval $[0,N]$ implies Riemann integrability of the positive and negative functions $u^+$ and $u^-$.

I tried to use the previous theorem (as suggested by the author), which goes as follows:

Theorem 11.8: Let $u \colon [a,b] \to \mathbb{R}$ be a measurable function.

(1) If $u$ is Riemann integrable then $u$ is Lebesgue integrable and the two integrals agree.

(2) A bounded function $f \colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if the points $(a,b)$ where $f$ is discontinuous have Lebesgue measure zero.

The problem is that I don't know whether $u$ is bounded so I cannot use part (2) of this theorem. How can I use this theorem otherwise?

• An unbounded function cannot be Riemann integrable. – njguliyev Sep 15 '13 at 17:25