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proof of Fatou-Lebesgue theorem
Since $\displaystyle \left| \int g \, d\mu \right| \le \int |g| \, d\mu \le \int \Phi \, d\mu < \infty$ , we have that $\displaystyle \int g \, d\mu > - \infty$ . Similarly, $\displaystyle \int h \, d\mu < \infty$ .
The inequality $\displaystyle \liminf_{n \to \infty} \int f_n \, d\mu \le \limsup_{n \to \infty} \int f_n \, d\mu$ is obvious by definition of $\liminf$ and $\limsup$ .
Define a sequence of functions $k_n \colon X \to \mathbb{R}$ by $k_n(x)=f_n(x)+\Phi (x)$ . Then each $k_n$ is nonnegative (since $-f_n \le |f_n| \le \Phi$ ) and integrable (since $k_n \le |f_n|+\Phi \le 2\Phi$ ), as is $\displaystyle k := \liminf_{n \to \infty} k_n$ . Fatou's lemma yields that $\displaystyle \int k \, d\mu \le \liminf_{n \to \infty} \int k_n \, d\mu$ . Thus:
Since $\displaystyle \int \Phi \, d\mu < \infty$ , it follows that $\displaystyle \int g \, d\mu \le \liminf_{n \to \infty} \int f_n \, d\mu$ .
Note that $|-f_n|=|f_n| \le \Phi$ . Thus,
| $\displaystyle -\int h \, d\mu$ | $\displaystyle = \int -h \, d\mu$ |
| $\displaystyle = \int -\limsup_{n \to \infty} f_n \, d\mu$ | |
| $\displaystyle = \int \liminf_{n \to \infty} \left( -f_n \right) \, d\mu$ | |
| $\displaystyle \le \liminf_{n \to \infty} \int -f_n \, d\mu$ by a previous argument, | |
| $\displaystyle = \liminf_{n \to \infty} \left( -\int f_n \, d\mu \right)$ | |
| $\displaystyle = -\limsup_{n \to \infty} \int f_n \, d\mu.$ |
Hence, $\displaystyle \limsup_{n \to \infty} \int f_n \, d\mu \le \int h \, d\mu$ . It follows that $\displaystyle -\infty < \int g \, d\mu \le \liminf_{n \to \infty} \int f_n \, d\mu \le \limsup_{n \to \infty} \int f_n \, d\mu \le \int h \, d\mu < \infty$ . $\qedsymbol$