Let $(\rho_n)_n\subset L^1$ be a Dirac sequence. Study the equiintegrability of the following family:
(a) $f_n=\rho_n^2$,
(b) $g_n=\rho_1*\rho_n$ (convolution),
(c) $g_n=\rho_1*\rho_n^2$.
My attempt
Since for any $\epsilon>0$ we have $\int_{|x|<\epsilon}|\rho_n(x)|dx\rightarrow 1$, $(\rho_n)_n$ is not equiintegrable.
(b) Let $\epsilon>0$. Choose $\delta$ such that $m(A)\leq \delta$ implies $\int_A|\rho_1(x)|dx<\epsilon$. Then \begin{align*} \int_A|g_n(x)|dx&=\int_A|\rho_1*\rho_n|dx\\ &=\int_A\int_{\mathbb{R}}|\rho_1(x-y)| |\rho_n(y)|dydx\\ &=\int_{\mathbb{R}} |\rho_n(y)|\left(\int_A|\rho_1(x-y)|dx\right)dy\\ &\leq \epsilon \int_{\mathbb{R}} |\rho_n(y)|dy=\epsilon. \end{align*} Hence, the family $(g_n)_n$ is equiintegrable.
(a) I suspect its not equintegrable but I do not know what procee. \begin{align*} \int_A|f_n(x)|dx&=\int_A|\rho_n^2|dx. \end{align*} I hope to get this kind of inequality \begin{align*} \int_A|\rho_n^2|dx&\leq\int_A|\rho_n|dx \int_A|\rho_n|dx. \end{align*}