Good evening,
I have some problems with following task because I don't understand the context between the forms of convergence and $L$-integrability :
Let $f_n$ be an $L$-integrable sequence on $[1,\infty)$. Prove or disprove thefollowing statements:
a) If $f(_n)$ converges pointwise to $f$ on $[1,\infty)$, then f is $L$-integrable on $[1,\infty)$
b) If $f(_n)$ converges uniformly to $f$ on $[1,\infty)$, then $f$ is $L$-integrable on $[1,\infty)$
c) If $f(_n)$ converges uniformly to $f$ on $[1,\infty$) and $f$ is $L$-integrable on $[1,\infty)$, then: $\lim\limits_{n \rightarrow \infty}{} \int_{1}^{\infty} f_n(x) = \int_{1}^{\infty} f(x)\,dx$.
I guess a) and b) are not true but c) is true, but it not sure why I can't give a counterexample, but i guess if I take unbounded $f_n$?
Maybe you can help me (Sorry for the language I translated this exercise)
You are correct that $a)$ and $b)$ are false. For $a),$ simply check $f_n=1_{[1,n]}$ with limit $f=1$. For $b)$, check $f_n(x)=1_{[1,n]}(x) \frac{1}{x}$. Of course, that's also a counter example for $a)$.
As a matter of fact, $c)$ is also wrong. See $f(x)=0$ and $f_n=\frac{1}{n} 1_{[1,n+1]}$.
Note that none of these counter-examples involve any unbounded functions.