I am asked to show that for p > 0. I have a lot of it so far, and am basically on the cusp.
. Now the obvious calculus thing to do is integrate by parts and I'm done. But to do that, assuming I'm treating this like a Riemann integral, I have to evaluate from 0 to 1. Well okay, 1 is easy. Log of 0 isn't defined, though, and I'm not sure how to do a proofy demonstration that this is whole thing is zero. So that leads me to think that maybe I shouldn't be treating this like a Riemann integral and instead should find some sequence of step functions which approaches what I'm trying to integrate, i.e. to argue it as a Lebesgue integral. But finding a sequence of step functions which approaches seems daunting. Ideas?
... However, a similar idea might be to build the step-functions which approximate this integral on intervals [s, 1] where s is ever closer to 0 and the step functions are constant on smaller partitions as the sequence approaches the function. I'm toying with this idea right now.
How does this sound? I create the sequence of functions defined by . Obviously this sequence is increasing and approaching , and each individual function is Riemann integrable and equal to... Hm. I guess I'm stuck here, I don't know how to integrate log yet. So this looks like the wrong way to go.