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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
\[
\begin{gathered}
  \int {\mathop {\lim \inf }\limits_n } ({f_n} + g) \leqslant \lim \inf \int {({f_n} + g)} ; \hfill \\
  \int {\mathop {\lim \inf }\limits_n } (g - {f_n}) \leqslant \lim \inf \int {(g - {f_n})}  \hfill \\
\end{gathered} \;\;\; (*)
\]注意到上述積分…