Let $g_n =e^{x} f_n \in L_1$ and $g=e^{x} f \in L_1$ with $f_n, f\in L_1$. If $f_n \to f$ in $L_1$ does $g_n \to g$ in $L_1$

Question

Let $g_n =e^{x} f_n \in L_1$ and $g=e^{x} f \in L_1$ with $f_n, f\in L_1$.

Suppose that $f_n \to f$ in $L_1$. That is, \begin{align} \int |f-f_n| \to 0 \end{align}

Does this imply that $g_n \to g$ in $L_1$?

Answer 1

No. Let $f=g=0$ and let $f_n(x)=n e^{-x} 1_{[n,n+1]}(x)$. Then, $$ \|f_n\|_{L^1}\leq ne^{-n}\to 0, $$ but $$ \|g_n\|_{L^1}=n\not\to 0 $$