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,

Hence, $\displaystyle \limsup_{n \to \infty} \int f_n \, d\mu \le \int h \, d\mu$ It follows that $\displaystyle -\infty < \int g \, d\mu \le \liminf_{n \to \infty} \int f_n \, d\mu \le \limsup_{n \to \infty} \int f_n \, d\mu \le \int h \, d\mu < \infty$ $\qedsymbol$