Use the Monotone Convergence Thm, to show $\displaystyle\int f \le \liminf \int f_n$

Question

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I need to do the question at the top of this image.

I figured out that $g_n$ is an increasing sequence that is pointwise convergent to $f$. i.e. I know $\lim g_n(x) = \liminf f_n(x) = f(x)$.

Hence from MCT, I know $$\int f\ du = \int \lim\ g_n\ du = \lim \int g_n du$$

I can apply Fatou's Lemma since $f_n$ is a sequence of non-negative measurable functions. Hence, I know $$\int\lim inf\ f_n\ du\le \lim\inf \int f_n\ du$$

I need to prove that $$\int f\ du \leq \lim\inf \int f_n\ du$$

Answer 1

As the sequence $ \inf_{n>N} f_n $ is increasing, you get $$ \lim \int \inf_{n>N} f_n dx = \int \lim \inf_{n>N} f_n dx = \int \liminf f dx $$ and on the other hand, $$ \inf_{n>N} f_n \le f_{N+1} \implies \lim \int \inf_{n>N} f_n dx \le \liminf \int f_{N+1} $$