Is Lebesgue integrable over (0,1)? What about over ? What kind of comparisons should I be trying?
The integral of diverges at x=1. That is to say, it diverges as an improper Riemann integral. But that implies that it does not exist as a Lebesgue integral, on either of the intervals (0,1) or (1,∞).
To prove that formally, on the interval (1,∞) you could define a sequence of functions
Then increases to as , but . It follows from the monotone convergence theorem that is not (Lebesgue) integrable on (1,∞). A similar argument shows that it is not integrable on (0,1).