Part of a HW question reads:
Let $X=\mathbb{N}$ and let $\mathscr{F}=\mathscr{B}$ (borel sets), and let $\mu$ be the Lebesgue measure on the borel sets. If $f_n=\chi_{[0,n]}$, then the sequence converges to $f=\chi_{[0,\infty]}$. Although the functions are bounded by $1$ and the integrals of each of the $f_n$ are all finite, we have $\int f d\mu = + \infty$.
I am so confused, how is it that $\int f d\mu = + \infty$?
Our set is $\mathbb N$. Our measurable sets are the borel sets (so singleton sets belonging to $\mathbb{N}$, which are closed, unions and intersections of these singleton sets belonging to $\mathbb{N}$ would be examples of measurable sets). But we have the Lebesgue measure as the measure of our measurable space $(\mathbb{N},\mathscr{B})$ and I thought this measure assigns $0$ to countable sets. So our space should have measure $0$. How is it that $\int f d\mu = + \infty$?
There is definitely something I haven't learned or understood correctly.