Integral problem:

Let $f_n(x) =\begin{cases} \dfrac{1}{x}~~~\text{if}~~~ x \in [1/n, 1]\ 0, ~~~\text{otherwise} \end{cases}$, for every $n$ in $\mathbb{N}$.
Is $f_n$ Lebesgue integrable?

Question

Integral problem:

Let $f_n(x) =\begin{cases} \dfrac{1}{x}~~~\text{if}~~~ x \in [1/n, 1]\ 0, ~~~\text{otherwise} \end{cases}$, for every $n$ in $\mathbb{N}$.
Is $f_n$ Lebesgue integrable?

anonymous · Answer

I'm pretty sure it is not, since at some point i'd have $n \rightarrow \infty$   $\Rightarrow [1/n, 1] \rightarrow [0, 1]$ and this is definately not Riemann or Lebesgue integrable (if you look at the improper Riemann integral) Even if only had $(0,1]$ I'd still have the same problem. (or am I wrong)?

thomas5267 · Answer

I don't know anything about Lebesgue integral but I think it might even be Riemann integrable for every $n\in\mathbb{N}\setminus \{0\}$.  For every $n$, the $f_n$ has finite support and the integral is finite as well, and $\infty\notin \mathbb{N}$.

anonymous · Answer

Ooh yeah, so infinity isn't in the Naturals. It seems like that completely went over my head, then everything makes a lot more sense :)

anonymous · Answer

Thanks a lot for that.