I want to find a sequence of measurable non negative functions $(f_n)$ with point wise limit a.e. such that the integral of the limit is different to the limit of the integrals and both limits are finite.
So I thought I could just let $f_n = \frac{1}{n}\chi_{(0,n)}$. On one hand, $$\lim_n f_n = 0$$ and so, $\int \lim_n f_n = \int 0 = 0$. But for each $n$, $$\int f_n = \frac{1}{n}\mu((0,n)) = \frac{n}{n} = 1$$ and therefore, $\lim_n \int f_n = \lim_n 1 = 1$.
Is this correct? Thank you in advance!
Your reasoning is correct. This example, which I like to call the "melting ice cube," is a good example of a sequence of functions that converge pointwise almost everywhere (indeed, the sequence converges everywhere, and even uniformly so!), but which fails to converge in $L^1$ (in particular, you cannot pass the limit through the integral).
Another nice example is the "traveling box" example. Let $f_n = \chi_{[n,n+1]}$. Observe that $f_n \to 0$ pointwise, but $$ \int f_n = \int \chi_{[n,n+1]} = m([n,n+1]) = 1,$$ where $m$ is the Lebesgue measure. The traveling box is maybe slightly more interesting as a counter-example, as the convergence is not uniform (though it is still pointwise).