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
Canonical name	ProofOfFatouLebesgueTheorem
Date of creation	2013-03-22 15:58:50
Last modified on	2013-03-22 15:58:50
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	13
Author	Wkbj79 (1863)
Entry type	Proof
Classification	msc 28A20