Fatou's Lemma

If a Sequence $\{f_n\}$ of Nonnegative measurable functions is defined on a measurable set $E$, then

$$\int\limits_E \mathop{\rm lim\ inf}_{n\to\infty} f_n\,d\mu \leq \mathop{\rm lim\ inf}_{n\to\infty} \int\limits_E f_n\,d\mu.$$

References

Zeidler, E. Applied Functional Analysis: Applications to Mathematical Physics. New York: Springer-Verlag, 1995.