For a sequence of $\mathbb{R}$-valued (Borel-)measurable functions $\{h_n\}_{n=1,2,\cdots}$ let us consider $$ \int_0^1\frac{h_n(x)}{1-x}dx:=\lim_{\epsilon\to0}\int_0^{1-\epsilon}\frac{h_n(x)}{1-x}dx. $$ Here we regard the integral in the right-hand side as Lebesgue integral. Assume that the limit exists for each $n=1,2,\cdots$. Then is the assumptions of Fatou's lemma enough to establish $$ \liminf_{n\to\infty}\int_0^1\frac{h_n(x)}{1-x}dx\ge\int_0^1\liminf_{n\to\infty}\frac{h_n(x)}{1-x}dx? $$
I'm confused to answer this question because I have to change the order $\lim_{\epsilon\to0}$ and $\liminf_{n\to\infty}$. However, in this case, it's not always that $\lim_{\epsilon\to0}$ or $\liminf_{n\to\infty}$ have any uniformity (such as Moore-Osgood Theorem). I don't know whether well-known convergence theorems are valid for improper integrals or not.
Please tell me ideas, answers or references if you have. Thank you in advance.