(You can use standard results in any proofs). Are the following 2 functions Lebesgue integrable over the interval I? A) I = with if B) I = [0,1] with where n(x) is the largest integer n such that
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Originally Posted by Amanda1990 (You can use standard results in any proofs). Are the following 2 functions Lebesgue integrable over the interval I? A) I = with if B) I = [0,1] with where n(x) is the largest integer n such that For A), remember that the Lebesgue integral is an absolute integral. In other words, if f is integrable then so is |f|. For B), show that f is measurable. Since it is bounded, on a bounded interval, it must then be integrable.
Thanks for you help. However for B) I've got that f is not integrable. Surely f is unbounded, as 1/n(x) can become arbitrarily large for small x?
Originally Posted by Amanda1990 Thanks for you help. However for B) I've got that f is not integrable. Surely f is unbounded, as 1/n(x) can become arbitrarily large for small x? If x is small then n(x) will be large, so 1/n(x) will be small.
Ouch. Good point...
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