# proof of Fatou-Lebesgue theorem

Since $\displaystyle\left|\int g\,d\mu\right|\leq\int|g|\,d\mu\leq\int\Phi\,d\mu<\infty$, we have that $\displaystyle\int g\,d\mu>-\infty$. Similarly, $\displaystyle\int h\,d\mu<\infty$.

The inequality  $\displaystyle\liminf_{n\to\infty}\int f_{n}\,d\mu\leq\limsup_{n\to\infty}\int f% _{n}\,d\mu$ 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 $-f_{n}\leq|f_{n}|\leq\Phi$) and integrable (since $k_{n}\leq|f_{n}|+\Phi\leq 2\Phi$), as is $\displaystyle k:=\liminf_{n\to\infty}k_{n}$. Fatou’s lemma yields that $\displaystyle\int k\,d\mu\leq\liminf_{n\to\infty}\int k_{n}\,d\mu$. Thus:

$\begin{array}[]{ll}\displaystyle\int g\,d\mu+\int\Phi\,d\mu&\displaystyle=\int% (g+\Phi)\,d\mu\\ \\ &\displaystyle=\int k\,d\mu\\ \\ &\displaystyle\leq\liminf_{n\to\infty}\int k_{n}\,d\mu\\ \\ &\displaystyle=\liminf_{n\to\infty}\int(f_{n}+\Phi)\,d\mu\\ \\ &\displaystyle=\liminf_{n\to\infty}\left(\int f_{n}\,d\mu+\int\Phi\,d\mu\right% )\\ \\ &\displaystyle=\liminf_{n\to\infty}\int f_{n}\,d\mu+\liminf_{n\to\infty}\int% \Phi\,d\mu\\ \\ &\displaystyle=\liminf_{n\to\infty}\int f_{n}\,d\mu+\int\Phi\,d\mu\end{array}$

Since $\displaystyle\int\Phi\,d\mu<\infty$, it follows that $\displaystyle\int g\,d\mu\leq\liminf_{n\to\infty}\int f_{n}\,d\mu$.

Note that $|-f_{n}|=|f_{n}|\leq\Phi$. Thus,

 $\displaystyle-\int h\,d\mu$ $\displaystyle=\int-h\,d\mu$ $\displaystyle=\int-\limsup_{n\to\infty}f_{n}\,d\mu$ $\displaystyle=\int\liminf_{n\to\infty}\left(-f_{n}\right)\,d\mu$ $\displaystyle\leq\liminf_{n\to\infty}\int-f_{n}\,d\mu$ by a previous , $\displaystyle=\liminf_{n\to\infty}\left(-\int f_{n}\,d\mu\right)$ $\displaystyle=-\limsup_{n\to\infty}\int f_{n}\,d\mu.$

Hence, $\displaystyle\limsup_{n\to\infty}\int f_{n}\,d\mu\leq\int h\,d\mu$. It follows that $\displaystyle-\infty<\int g\,d\mu\leq\liminf_{n\to\infty}\int f_{n}\,d\mu\leq% \limsup_{n\to\infty}\int f_{n}\,d\mu\leq\int h\,d\mu<\infty$. $\qed$

Title proof of Fatou-Lebesgue theorem ProofOfFatouLebesgueTheorem 2013-03-22 15:58:50 2013-03-22 15:58:50 Wkbj79 (1863) Wkbj79 (1863) 13 Wkbj79 (1863) Proof msc 28A20