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,

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

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