Is Lebesgue integrable over (0,1)? What about over ? What kind of comparisons should I be trying?

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- April 24th 2009, 09:43 AMAmanda1990Is 1/(x^2 - 1) Lebesgue integrable on (0,1)?
Is Lebesgue integrable over (0,1)? What about over ? What kind of comparisons should I be trying?

- April 24th 2009, 01:52 PMredsoxfan325
I thought that everything that was Riemann integrable was also Lebesgue integrable.

- April 24th 2009, 10:41 PMAmanda1990
OK...so why is it Reimann integrable?Thanks.

- April 24th 2009, 11:02 PMredsoxfan325
Oh crap...sorry, I misread the question. I thought it said .

is definitely not Riemann integrable on . I don't know enough about Lebesgue integration to solve this problem. Sorry for wasting your time. - April 25th 2009, 12:10 AMOpalg
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).