Let $\{f_n\}$ and $\{g_n^\delta\}$ be sequences of non-negative functions in $L^1(\mathbb{R}^N)$, converging weakly respectively to $f$ and $g^\delta$. Assume that $g_n^\delta$ converges strongly to $f_n$ as $\delta\rightarrow 0$, uniformly with respect to $n$, i.e., $$\lim_{\delta\rightarrow 0}\sup_{n>0}\|f_n-g_n^\delta\|_{L^1}=0.$$ 1-What can we say about the convergence from $g^\delta$ to $f$ as $\delta\rightarrow 0$?
2-I can see that $g^\delta\rightharpoonup f$ weakly in $L^1$. Do we get any extra properties on this convergence if we assume that $g_n^\delta$ also converges a.e. to $f_n$ as $\delta\rightarrow 0$?
It seems that the assumption of uniform strong convergence gives that $g^\delta\to f$ as $\delta$ goes to zero. Indeed, we know that if $h_n\to h$ weakly, then $\left\lVert h\right\rVert\leqslant \liminf_{n\to + \infty} \left\lVert h_n\right\rVert$. Apply this fact to $h_n=f_n-g_n^{\delta}$ for a fixed $\delta$, one gets $$\left\lVert f-g^{\delta} \right\rVert_1 \leqslant \liminf_{n\to + \infty} \left\lVert f_n-g_n^{\delta}\right\rVert_1\leqslant \sup_{n\geqslant 1} \left\lVert f_n-g_n^{\delta}\right\rVert_1 $$
and the latter quantity goes to zero as $\delta$ goes to zero.