Uniformly bounded sequence of $L^{2}$ functions and a limit Let $f_{n}: \mathbb{R}^{d} \rightarrow \mathbb{R}$ such that $\sup_{n}\|f_{n}\|_{L^{2}} < \infty$. Furthermore suppose $f_{n} \rightarrow f$ pointwise almost everywhere for some $f$. The problem I am working on is to show that $$\int_{\mathbb{R}^{d}}||f_{n}|^{2} - |f_{n} - f|^{2} - |f|^{2}|\, dx \rightarrow 0$$ as $n \rightarrow \infty$.
First I noticed that $||f_{n}|^{2} - |f_{n} - f|^{2} - |f|^{2}| \rightarrow 0$ pointwise almost everywhere. However, I can't think of any function which dominates this expression.
 A: As pointed out by Giuseppe Negro in a comment above, this is a special case of a result in the book by Lieb and Loss. In this special case, the following argument seems simpler to me:
Put $C=\sup_n \|f_n\|$, and observe the Fatou's Lemma implies $\|f\|\leq C$, i.e. $f\in L^2$. Now we compute
$$
||f_n|^2-|f_n-f|^2-|f|^2|=|2\mbox{Re }f_n\bar f+2|f|^2|\leq 2|f|(|f_n|+|f|).
$$
Invoking the Cauchy-Schwarz inequality followed by Minkowski's inquality, we obtain
$$
\int_E ||f_n|^2-|f_n-f|^2-|f|^2|dx\leq 2\|1_Ef\|(\|f_n\|+\|f\|)\leq 4C\|1_E f\|,
$$
for any measurable set $E$. Assume now that we are given $\epsilon>0$. Using the fact that $f\in L^2$, we can pick a measurable set $A$ such that $4C\|1_A f\|<\epsilon/3$, and $K=A^c$ is compact.
At this point, further application of the above inequality allows us to reduce the the case of a pointwise convergent, uniformly integrable sequence on a finite measure space, which finishes the argument if one is familiar with the theory of probability spaces (actually, convergence in probability is sufficient). If not, the argument can be finished by envoking Egorov's theorem as follows:
There are disjoint sets $K_1,K_2$ such that $K_1\cup K_2=K$, $f_n\rightarrow f$ uniformly on $K_1$ and $m(K_2)<(\epsilon/(12C\|f\|))^2$. Pick $N$ such that $||f_n|^2-|f_n-f|^2-|f|^2|<\epsilon/(3m(K_1))$ on $K_1$ for all $n\geq N$. Then
\begin{align*}
\int ||f_n|^2-|f_n-f|^2-|f|^2|dx&\leq 4C\|1_A f\|+\frac{\epsilon}{3m(K_1)}\cdot m(K_1)+4C\|1_{K_2}f\|\\
&\leq\frac{\epsilon}{3}+\frac{\epsilon}{3}+4C\|f\|m(K_2)^{1/2}<\epsilon
\end{align*}
for all $n\geq N$.
