$(4)($a$)$ Assume $(x)<mi\; id="EditorElement\_20743"></mi>$ and, for some $p>1,f_n{?}^{L_p}f$ as $n.$ Is it true that ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{f}_{n}d{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}fd$ as

$n?$ Justify your answer.

$($b$)$ Assume $(x)=$ and, for some $p>1,f_n{?}^{L_p}f$ as $n.$ Moreover, assume $f_n,fin{L}_1.$ Is it true

that ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{f}_{n}d{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}fd$ as $n?$ Justify your answer.