Prove Lp ball closed in L1

Question

Let $(X,\mathcal{M}, \mu)$ be a finite measure space.

Let $1\leq p\leq \infty$. Show that for fixed $M > 0$, the ball $||f||_{L^p}\leq M$ is closed in $L^1$

I'm going through some real analysis qual exams, and I'm a bit stuck on this question.

Suppose $\{f_n\}$ is a sequence of functions in the $L^p$ ball s.t. $f_n\to f$ in $L^1$. I'd like to show that $||f||_{L^p}\leq M$

I've tried the obvious $\int |f|^p\leq \int |f_n|^p + \int |f_n-f|^p$ but I don't think it's true that $\int |f_n-f|^p \to 0$ given that $\int |f_n-f|\to 0$

Any help would be appreciated.

Answer 1

I think I may have a solution actually.

Let $B_M$ denote the closed ball in $L^p$.
Let $\{f_n\}$ be a sequence of functions in $B_M$ such that $f_n\to f$ in $L^1$
Then, there is some subsequence $f_{n_k}\to f$ a.e.

Notice that $d|f|^p\chi_{|f|\leq \ell} << \mu$, so for any $\epsilon, \exists \delta $ s.t. if $\mu(E)<\delta, \int_E |f|^p\chi_{|f|\leq \ell}\ d\mu< \epsilon$.

By Egoroff's theorem, there is some measurable set $E$ s.t. $\mu(E)<\delta$ and $f_{n_k}\to f$ uniformly on $E^c$

Thus, for any $\ell\geq 1$, $\int |f|^p\chi_{|f|\leq \ell} = \int_E |f|^p\chi_{|f|\leq \ell} + \int_{E^c} |f|^p\chi_{|f|\leq \ell}\leq \epsilon + \int_{E^c}|f|^p$

However, $\int_{E^c}|f|^p\leq \int_{E^c}|f_{n_k}|^p + \int_{E^c}|f_{n_k} - f|^p \to M^p$ as $k\to \infty$ since $f_{n_k}\to f$ uniformly on $E^c$

Thus, for any $\ell$, $\int |f|^p\chi_{|f|\leq \ell} \leq M$, so by the monotone convergence theorem, $\int |f|^p\leq M^p$, so $f\in B_M$

Answer 2

Let $\Vert f-h_k \Vert_1 \to 0$, where $\Vert h_k \Vert_p \leq 1$. Then we can assume that there exists a subsequence of $(h_n)$ that converges to $f$ a.e whose p-norms are lesser than 1. Also note that by the reverse triangle inequality, we can replace $f$ with $|f|$ and $h_k$ with $|h_k|$.

Thus we may assume that $f,h_k \geq 0$ and $h_k \to f$ a.e. By Fatou's Lemma we have that $$ \int |f|^p \leq \lim \int |h_k|^p \leq 1 $$ Thus $\Vert f \Vert_p \leq 1$