# Question on Fatou theorem

Can we replace inequality:

$$\int_X f(x)\text{d}\mu \leq \underline\lim_{n\to\infty} \int_X f_n(x) \text{d}\mu$$

in Fatou's lemma replace by equality. If not, how can it be proved?

No, in general both expressions are not equal. An easy counterexample: Define $$f_n(x) := \begin{cases} n & x \leq \frac{1}{n} \\ 0 & \text{otherwise} \end{cases}, \qquad x \in [0,1]$$ Then $f_n \to f :=0$ almost surely (with respect to the Lebesgue measure on $[0,1]$) and
$$0 = \int_0^1 f(x) \, dx< \liminf_{n \to \infty} \int_0^1 f_n(x) \, dx = 1$$