Measurable functions such that $\int\underline{\text{lim}}f_n=0 $ and $\underline{\text{lim}}\int f_n=+\infty$.

Question

I need to find an example of a suquence of measurable functions $ f_n \geq0$ for $ n = 1,2, ... $ such that $\int\underline{\text{lim}}f_n=0 $ and $\underline{\text{lim}}\int f_n=+\infty$.

As I can define these measurable functions. Thanks

Answer 1

Hint: The integral of $n\chi_{[n,n+1]}$ is equal to $n$ (here $\chi_{[n,n+1]}$ is the indicator function on the interval $[n,n+1]$). What happens to this function pointwise as $n$ grows?

Answer 2

There are three things that can go wrong in integrals when you take limits of functions:

1. The support of the functions might become too large, and the values of the functions too small, so the functions converge to $0$ but the area under the graph to $\infty$.

2. The functions has some spike, so although they converge to $0$, the spike keeps the area under the graph large.

3. You might have some bump escaping to $\infty$, for example $f_n(x)=1$ on $[n+n+1]$ and $0$ otherwise.