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I was required to prove Monotone Convergent Theorem as a corollary of Fatou lemma,i.e using Fatou lemma to prove the MCT.

Monotone enter image description here

The hint I was given is let $f_n$ be a sequence of increasing function, $lim f_n = f$ . let $b_n=f-f_n$ to prove. Due to my knowledge of limsup and liminf is rusty and superficial. I tried to prove as follows, however I am not sure the way I do is right or not, so please point out the mistake.

Apply Fatou Lemma for $f_n$ :

$ \int \liminf f_n\le\liminf\int f_n \\$

since $\lim f_n = f$ , then $\liminf f_n = f$

$\int f\le\liminf\int f_n \\$ (1)

Apply Fatou Lemma for $b_n$ :

$\int (\liminf (f-f_n))\le\liminf\int(f-f_n)\\$

$\int(f+\liminf (-f_n))\le\liminf(\int f-\int f_n)\\$

$\int f+ \int \liminf (-f_n)\le\int f+\liminf(-\int f_n) \\$

$-\int\limsup(f_n)\le-\limsup(\int f_n)$

since $\lim f_n = f$ , then $\limsup f_n = f$

$\int f \ge\limsup(\int f_n)\ge\liminf\int f_n $ (2)

(1),(2) : $\int f = \liminf\int f_n = \lim \int f_n$, due to $f_n\to f$.

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One problem is that $\int f$ could be infinite, so you can't cancel it from both sides of the inequality $$\int f+ \int \liminf (-f_n)\le\int f+\liminf(-\int f_n).$$

Since $f_n \le f_{n+1}$ you have that $\lim f_n$ exists, and also that $\displaystyle \int f_n \le \int f_{n+1}$ so $\displaystyle \lim \int f_n$ exists too. Thus you don't need to use $\liminf$ and $\limsup$.

In the present context (I'm guessing $L^+$ denotes nonnegative measurable functions) Fatou's lemma states that $$\int \lim f_n \le \lim \int f_n.$$ The other inequality follows from integrating $f_n \le \lim f_n$.

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