Show that $\int_{\mathbb R^d}(|f_n|^p-|f_n-f|^p)\,dx\to\int_{\mathbb R^d}|f|^p\,dx.$

Question

Let $1\leq p<\infty$. Suppose $\{f_n\}\subset L^p(\mathbb R^d)$ satisfies $\int_{\mathbb R^d} |f_n|^p\,dx\leq M$ and $f_n\to f$ for almost every $x\in\mathbb R^d$. Show that $$\int_{\mathbb R^d}(|f_n|^p-|f_n-f|^p)\,dx\to\int_{\mathbb R^d}|f|^p\,dx.$$

Applying Fatou's lemma I get that $\int_{\mathbb R^d}|f|^p\,dx\leq \liminf_{n\to\infty} \int_{\mathbb R^d} |f_n|^p\,dx\leq M$, so $f\in L^p(\mathbb R^d)$. But I can't move forward. If I had $\int_{\mathbb R^d} |f_n|^p\,dx\to \int_{\mathbb R^d}|f|^p\,dx$, I could prove that $\int_{\mathbb R^d}|f_n-f|^p\,dx\to 0$ so we are done. But it is not always true. Any help please?

Answer 1

First, notice that as a consequence of Fatou's lemma, the function $f$ belongs to $L^p$.

Define $$ g_n=\left\lvert f_n\right\rvert^p-\left\lvert f_n-f\right\rvert^p-\left\lvert f\right\rvert^p. $$ For two non-negative numbers $a$, $b$, the following inequality holds $$ \left\lvert a^p-b^p\right\rvert\leqslant \max\{a,b\}^{p-1}p\left\lvert b-a\right\rvert. $$ Consequently, $$\tag{*} \left\lvert g_n\right\rvert\leqslant c_p \left\lvert f\right\rvert^p+c_p' \left\lvert f_n\right\rvert^{p-1}\left\lvert f\right\rvert, $$ where $c_p$ and $c_p'$ depend only on $p$. The sequence $(g_n)$ converges to $0$ almost everywhere. We will see that $\int \left\lvert g_n\right\rvert\to 0$. For a fixed $R$, let $B_R$ be the ball of $\mathbb R^d$ of radius $R$ centered at $0$ and $\mathbf 1_{B_R}$ the associated indicator function.

Fix a positive $\varepsilon$. Fix $R$ such that $\int_{B_R^c} \left\lvert f\right\rvert^pdx\lt \varepsilon$. There exists a $\delta$ such that if $A$ has measure smaller than $\delta$, then for all $n$, $\int_A \left\lvert g_n\right\rvert\leqslant\varepsilon$: this follows from (*) and an application of Hölder's inequality for $p>1$.

Applying Egoroff's theorem to $(g_n\mathbf 1_{B_R})$, we can find a set $A\subset B_R$ such that $\sup_{x\in B_R\setminus A}\left\lvert g_n\right\rvert \to 0$ and $\lambda(A)\lt \delta$. Therefore, $$ \int \left\lvert g_n\right\rvert= \int_A \left\lvert g_n\right\rvert +\int_{B_R\setminus A} \left\lvert g_n\right\rvert +\int_{B_R^c}\left\lvert g_n\right\rvert \leqslant C_p\varepsilon+R^d \sup_{x\in B_R\setminus A}\left\lvert g_n\right\rvert. $$