Hello all, I'm going through Foundations of Mathematical Analysis by Johnsonbaugh and Pfaffenberger, and I read a curious line that I was hoping someone here could clear me up on (perhaps I'm thinking too much into it). The begininning of the section on the Lebesgue integral introduces a sequence of Riemann integrable functions which converge (somehow) to a function, ƒ,which is not Riemann integrable. The authors go on to write.... "Why is ƒ not Riemann integrable? The fault lies not with the average used, namely α(x)=x, but rather with the sets averaged over, i.e., intervals." where α(x) is the integrator. Generally, when I think of α(x), I think of it as a weighting function. So, what do the authors here mean that α(x)=x is the "average used"?