Lemma 1 (Barbălat)   Let be Riemann integrable and uniformly continuous then

Note that if is non-negative, then Riemann integrability is the same as being in the sense of Lebesgue, but if oscillates then the Lebesgue integral may not exist.

Further note that the uniform continuity is required to prevent sharp “spikes” that might prevent the limit from existing. For example suppose we add a spike of height 1 and area at every integer. Then the function is continuous and (and thus Riemann integrable), but would not have a limit at infinity.

Bibliography

1

Hartmut Logemann, Eugene P. Ryan. Asymptotic behaviour of nonlinear systems. The American Mathematical Monthly, 111(10):864-889, 2004.