Thread: Are the following Lebesgue integrable?

1. 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 = $[1,\infty)$ with $f(x) = (-1)^n/n$ if $n \leq x < n+1, n = 1,2,3...$

B) I = [0,1] with $f(x) = (-1)^{n(x)}/n(x)$ where n(x) is the largest integer n such that $nx \leq 1$

2. 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 = $[1,\infty)$ with $f(x) = (-1)^n/n$ if $n \leq x < n+1, n = 1,2,3...$

B) I = [0,1] with $f(x) = (-1)^{n(x)}/n(x)$ where n(x) is the largest integer n such that $nx \leq 1$
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.

3. 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?

4. 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.

5. Ouch. Good point...