I'm trying prove of Vitali's theorem to myself from Theory of Statistical Hypotheses by Lehmann and Romano:
Suppose $f_n$ and $f$ are real-valued measurable functions with $f_n(x) \rightarrow f(x)$, except possibly on a set having $\mu$ measure $0$. Assume $$ \limsup_{n\rightarrow \infty} \int f_n^2(x) d\mu (x) \leq \int f^2(x) d\mu (x) < \infty .$$ Then, $$\int \left | f_n(x) - f(x) \right |^2 d\mu (x) \rightarrow 0$$
I'm wondering what the difference is between proving that $\int \left | f_n(x) - f(x) \right |^2 d\mu (x) \rightarrow 0$ and proving $f_n(x) \rightarrow f(x)$.
I understand that we're not directly finding that $f_n(x) \rightarrow f(x)$, but normally if the area under the squared error is $0$ then the functions are equal. What is different when this happens asyptotically?
Or does this theorem actually say that $f_n(x) \rightarrow f(x)$? If that is the case then what is the reason for writing it this way? Is there something significant about the notation that indicates something about the result?
Thanks.