Let $(X, \mathcal{M}, \mu)$ be a measure space, and let $\{f_n\}_{n=1}^{\infty}$ be a sequence of nonnegative extended real-valued measurable functions defined on $X$. Suppose also that $\lim_{n\to\infty}f_n=f$. Are there any conditions which can be imposed on $\{f_n\}_{n=1}^{\infty}$ other than the hypotheses for LDCT or LMCT so that equality holds in Fatou's Lemma; that is
$$\int_X \liminf_{n\to\infty} f_n d\mu\ =\ \liminf_{n\to\infty}\int_{X} f_n\ d\mu\hspace{.1in}\ ?$$
