Let $\{f_n\}$ be a sequence of measurable functions on a domain $E$ and $p$ be a positive finite real number such that
(a) $\{f_n\}$ converges to a measurable function $f$ almost everywhere, and;
(b) $\lim _{n \rightarrow \infty} (||f_n||_p – ||f||_p) = 0.$
Prove that $||f_n – f||_p \rightarrow 0.$
I believe what we have to show is that $$\lim_{n \rightarrow \infty} \int_E |f_n – f|^p = 0.$$
My approach is to hope to invoke the Dominated Convergence theorem, i.e., prove that there exists some $g \in L^1(E)$ such that $|f_n| \leq g$ for almost everywhere for all $n$, then we can have
$$\int_E |f_n – f| \rightarrow 0.$$
Then by (a), we can find an integer $K$ such that $$|f_k – f| < 1$$ for all $
k \geq K$. Then
$$0 \leq \int _E |f_k – f|^p \leq \int _E |f_k – f|.$$
Since the integral on the right approaches $0$, we conclude that the same is true for the middle term and we will be done.
However, I have no idea how to obtain the $g$ as mentioned before, as well as use condition (b) given in the question. Some help would be appreciated.
Best Answer
This question has been answered many many times for $p \geq 1$. For $0<p<1$ this follows from Fatou's Lemma: $|f-f_n|^{p} \leq |f|^{p}+|f_n|^{p}$ in this case so $|f|^{p}+|f_n|^{p} -|f-f_n|^{p} $ is non-negative Fatou's Lemma applied to this sequence gives the result immediately.