Let $f : [0,1] \rightarrow \mathbb{R}$, $f_k : [0,1] \rightarrow \mathbb{R}$ such that $\sup_{k \in \mathbb{N}} \int_{0}^{1} |f_k(x)|^4 \: dx < \infty $ and $$ \lim_{k\rightarrow \infty} \int_{0}^{1} |f_k(x) - f(x)| \: dx = 0.$$ Show that $ \int_{0}^{1} |f(x)|^4 \: dx < \infty$.
I found that $$ \int_{0}^{1} |f(x)|^4 \: dx \leq 2^4 \int_{0}^{1} |f_k(x) - f(x)|^4 \: dx + 2^4 \int_{0}^{1} |f_k(x)|^4 \: dx,$$ and I am not sure how to proceed after this.
Any help would be appreciated.
Can you show there is a subsequence $\{f_{k_j}\}$ of $\{f_k\}$ that converges pointwise almost everywhere to $f$?
If so, this is a quick application of Fatou's lemma: $$\int_0^1 |f(x)|^4 \, dx = \int_0^1 \lim_j |f_{k_j}(x)|^4 \, dx \le \liminf_j \int_0^1 |f_{k_j}(x)|^4 \, dx < \infty.$$