I am trying to show these two statements:
Let $(\mathbb R^n,\mathcal M,m)$ where $m$ is the Lebesgue measure and $\mathcal M$ are the Lebesgue measurable sets, and let $(f_n)_{n \in \mathbb N}$ and $f$ in $L^p(\mathbb R^n)$ for $1 \leq p < \infty$.
(a) $||f_n-f||_{L^p} \to 0 \implies ||f_n||_{L^p(\mathbb R^n)} \to ||f||_{L^p(\mathbb R^n)}$
(b) If $f_n \to f$ a.e. and $||f_n||_{L^p(\mathbb R^n)} \to ||f||_{L^p(\mathbb R^n)}$, then $||f_n-f||_{L^p} \to 0 $
There is a suggestion for part (b) which is to apply Fatou's lemma to the sequence $g_n(X)=2^{p-1}(|f_n(x)|^p+|f(x)|^p)-|f_n(x)-f(x)|^p$.
I'll write what I could do:
For part (b), I could not show that the suggested sequence $(g_n)_{n \in \mathbb N}$ is non negative, so instead I've used the sequence $h_n(x)=2^p(|f_n(x)|^p+|f(x)|^p)-|f_n(x)-f(x)|^p$. To apply Fatou's lemma to this sequence let's show non-negativity of the terms:
$$|f_n(x)-f(x)|^p \leq (|f_n(x)|+|f(x)|)^p$$$$\leq 2^p\max\{|f_n(x)|,|f(x)|\}^p$$ $$\leq 2^p(|f_n(x)|^p+|f(x)|^p)$$
It follows that $h_n(x) \geq 0$. Notice that $\lim_{n \to \infty} h_n(x)=2^{p+1}|f(x)|^p$ a.e., applying Fatou's lemma we have $$\int_{\mathbb R^d} 2^{p+1}|f(x)|dx \leq \lim inf \int_{\mathbb R^d}2^p(|f_n(x)|^p+|f(x)|^p)-|f_n(x)-f(x)|^pdx$$$$=\int_{\mathbb R^d}2^p|f|^pdx+\int_{\mathbb R^d}2^p|f|^pdx-\lim sup \int_{\mathbb R^d}|f_n-f|^pdx$$
From here it follows $\lim sup \int_{\mathbb R^d}|f_n-f|^pdx=0$, so $||f_n-f||_{L^p(\mathbb R^d)}=0$.
I don't know what to do to prove (a), could someone help me with that part and tell me if my solution to (b) is correct? Thanks in advance.
If you would like to use the hint and show that $g_n(x)$ is positive, you can use the fact that $x^p$ is convex. \begin{align*} \big|f_n-f\big |^p &\leq \left( |f_n|+|f|\right)^p & \text{triangle inequality} \\ &=2^p\left( \frac{1}{2} |f_n| + \frac{1}{2}|f| \right)^p \\ &\leq 2^{p-1}\left( |f_n|^p+|f|^p \right) & \text{convexity of $x^p$.} \end{align*}
If you get lazy and not want to use Fatou, you can apply the generalized dominated convergence theorem.