Lebesgue Monotonous Convergence Theorem for $f_n = \chi_{[0,n]}$

Question

Lebesgue Monotonous Convergence Theorem for $f_n = \chi_{[0,n]}$.

Let $(\mathbb{R},\mathbb{B},\mu)$ the measure space. If $f_n = \chi_{[0,n]}$, then the sequence is monotone converging to $f=\chi_{[0,\infty]}$ (Already verified). Although the $f_n$ are uniformly bounded by $1$ and the integrals of all $f_n$ are finite, we have $\int f d\mu = \infty$

The LMCT says $\lim \int f_n d\mu = \lim \int \chi_{[0,n]} d\mu= \lim n = \infty$ should be equal to $\int \lim f_n d\mu = \int f d\mu = \infty$. Well, it seems alright, but my book says to me to wonder if I can apply LMCT here, and I want to understand if there is any problem with what I wrote.

Thanks.

Answer 1

Yes, you can apply it. You don't even need to worry about uniformly bounded.