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[測度論] Dominated Convergence Theorem

Theorem: Dominated Convergence Theorem DCT  (real-valued functions) 令 sequence $\{f_n\} \subset L^1$ 滿足 (a) $f_n \to f$ almost everywhere (b) 存在非負函數 $g \in L^1$ 使得 $|f_n| \leq g$ almost everywhere for all $n$ 則\[ f \in L^1 \]且 \[ \lim_n \int f_n =\int f \] Proof:  先證 $f \in L^1$: 由於  $f_n \to f$ almost everywhere  且 $f_n \in L^1$,可知 $f $ measurable (see Proposition 2.11/2.12 )。由於 $|f_n| \leq g$ almost everywhere 故 $|f| \leq g$ almost everywhere,故 $f \in L^1$。 接著我們證 $\lim_n \int f_n =\int f$:注意到 $|f_n| \leq g$ almost everywhere,故我們有 \[ -g \leq f_n \leq g \text{ almost everywhere} \]換言之,我們有 \[ f_n + g \geq 0\text{ almost everywhere} \]與 \[ g-f_n \geq 0 \text{almost everywhere} \]也就是說 $\{f_n + g\}, \{g-f_n\} \in L^+$。由 Fatou Lemma 我們有 \[ \begin{gathered}   \int {\mathop {\lim \inf }\limits_n } ({f_n} + g) \leqslant \lim \inf \int {({f_n} + g)} ; \hfill \\   \int {\mathop {\lim \inf }\limits_n } (g - {f_n}) \leqslant \lim \inf \int {(g - {f_n})}  \hfill \\ \end{ga