I am thinking I can use the function $f_{n}= n1_{[0,1/n]}$ This will work beacuse $f_{n}\rightarrow 0 $ in measure because $1/n \rightarrow 0$ as $n$ gets bigger, but when we take the integral of $f_{n}$ over $[0,1]$ then we will clearly get 1 for all n, no matter larger or smaller, which will essentially mean the limit is not 0.
I want to see if my understanding is correct here?
What I always had in my mind is since the integral of $f_{n}$ is finite over $[0,1]$, I would simply think the integral is convergent. But in the problem above I simply do not have the limit 0. But the function still converge in $L^{1}$. What is wrong with my understanding? Or my example is completely out of track?
Can somebody pull me out of this confusion. Thank you in advance.