Suppose $1 \le p < \infty$ and $f_n ,f \in L^p[0,1]$. If $f_n \to f$ almost everywhere, then prove that $||f_n-f||_p \to 0$ iff $||f_n||_p \to ||f||_p$.
The above problem does not come the real analysis text I am using as a reference (Real Analysis by Royden & Fitzpatrick). However, in that book is a proof of the above problem; in fact, a proof of more general case when the measure space finite or infinite. My question is, is there a reason the above problem uses a finite measure space? Is it possible that the proof of the above problem admits a "simpler" proof than the general case? (Okay: I realize that the proof of the general case is itself not very difficult).