On a measure space $(X,\mu)$, if $f_n \to f$ almost everywhere, and $\int_X |f_n|^p \to \int_X |f|^p$, does that imply $||f_n - f||_p \to 0$?
It is clear in the case $\mu(X) < \infty$ by Egorov theorem. But what about when $\mu(X)=\infty$? I am pretty confused here.
Yes. Assuming $0 < p < \infty$, we have $\lvert f_n - f\rvert^p \le M(p) (\lvert f_n\rvert^p + \lvert f\rvert^p)$ for all $n$, where $M(p)$ is the larger of the numbers $1, 2^{p-1}$. Let
$$g_n = M(p)(\lvert f_n\rvert^p + \lvert f\rvert^p) - \lvert f_n - f\rvert^p \quad(n = 1,2,3,\ldots)$$
Since $g_n$ is a sequence of nonnegative measurable functions converging to $2M(p)\lvert f\rvert^p$ pointwise a.e., we may apply Fatou's lemma to get $\liminf\int_X g_n\, d\mu \ge 2M(p)\|f\|_p^p$. Since $\|f_n\|_p \to \|f\|_p$, then $\liminf \int_X g_n\, d\mu = 2M(p)\|f\|_p^p - \limsup \|f_n - f\|_p^p$. Consequently $\limsup \|f_n - f\|_p^p \le 0$. Thus $\|f_n - f\|_p \to 0$.