I used following state when I solved some problems.
But I can't prove that state. How can I prove that?
Let $1
$f_k\in L^p$ for $k\ge 1$ such that
There is an $a$ with $\|f_k\|_p \leq a$ for $k=1,2,...$
$\lim f_k(x)=:f(x)$ exists a.$\,$e.
Then $f\in L^p$ (i.$\,$e. $\|f\|_p < \infty$).