Setting

• For $i \in \mathbb{N}$, consider two sequences $f_i,g_i \in L^1(\mathbb{R})$ such that $$ f_i \rightarrow_{L^1} f \in L^1(\mathbb{R}) $$ and also $$ g_i \rightarrow_{L^1} g \in L^1(\mathbb{R}).$$

• Assume that all the functions involved are (1) striclty positive and that for any $i \in \mathbb{N}$, we have (2) $$ 0 < \int_\mathbb{R} f_i^2 g_i^{-1} < \infty$$ and also (3) $$ 0 < \int_\mathbb{R} f^2 g^{-1} < \infty $$

Question

Do we have the following convergence $$\int_\mathbb{R} f_i^2 g_i^{-1} \rightarrow \int_\mathbb{R} f^2 g^{-1} $$