The function $f_n$: $\mathbb{R}\rightarrow \mathbb{R}$ is defined by $f_n(x):=\chi_{[0,\infty)}(x)\frac{1}{n}\exp(-\frac{x}{n})$, where $n\in \mathbb{N}$.
Now: How can I show that $f(x):=\lim\limits_{n \rightarrow \infty}{f_n(x)}$ is lebesgue integrable?
I know that f(x) has to be nonnegative and Riemann integrable (How can I show that?). Are there any more criteria?
My problem is mostly that I dont know how to handle the $\chi(x)$.
EDIT: (Im new here, so please tell me in case I sould open a new question for this)
G. Sassatelli helped me in the comments to solve the first part of my problem.
Now I have to show $\int_\mathbb{R} f(x) d\lambda(x) \neq \lim\limits_{n \rightarrow \infty}{\int_\mathbb{R} f_n(x) d\lambda(x)}$.
Can someone give me a hint how to do that?