Thread: Are the following Lebesgue integrable?

(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

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...