Let $X = \mathbb R$, $\textbf{X} = \textbf{B}$ and $\lambda$ the Lebesgue measure on $\textbf{X}$.
I have the following: $f_n = (\frac{1}{n})\chi_{[n, +\infty)}$.
I need to find the following: $\displaystyle \lim \int f_n d\lambda$.
I honestly have no idea how to find this, since $(f_n)$ is not a monotone increasing sequence of functions I cannot say that $$\int f d\lambda = \lim \int f_n d\lambda$$ where $f=0$, the uniform limit of $f_n$.
For a set $A$ of finite measure, we have $\int a\chi_{A} \text{d} \lambda = a\lambda(A)$ for all $a \in \mathbb{R}$.
We note that for $f_n$, we have $g_{n,m} = \frac{1}{n} \chi_{[n,m]}$ for $m>n$ converges pointwise to $f_n$ and is increasing to $f_n$, so in this case we may interchange limit and integration. So we have $$ \int f_n \text{d} \lambda = \lim_{m \to \infty} \int g_{n,m} \text{d} \lambda = \lim_{m \to \infty} \frac{m-n}{n} = \infty.$$ So for all $n \in \mathbb{N}$, we have $\int f_n \text{d} \lambda = \infty$, and so is their limit as $n \to \infty$.