2.5. Limits and Integration

이 챕터는 전부 다 매우 중요하다.

  

Theorem(Monotone Convergence Theorem)

If \(0 \le f_n \uparrow f\) a.e., then \(\int f_n d\mu \uparrow \int f\mbox{ }d\mu\).

  

매우 중요하다.

Corollary

If \(f_n \ge 0\) for all \(n\ge 1\), then \[ \int \sum_{n=1}^\infty f_n\mbox{ } d\mu = \sum_{n=1}^\infty\int f_n\mbox{ }d\mu. \]

• \(f_n\)이 nonnegative일 때, summation이 밖으로 빠져나올 수 있다.

• \(g_N(\omega):=\sum_{n=1}^Nf_n(\omega)\uparrow \sum_{n=1}^\infty f_n(\omega)=:g(\omega)\)를 이용해 증명 가능하다.

  

Fatou’s Lemma를 위해

Theorem(Extended Monotone Convergence Theorem)

If \(g\le f_n\uparrow f\) a.e., where \(\int g\mbox{ }d\mu>-\infty\), then \(-\infty<\int f_n \mbox d\mu \uparrow \int f \mbox d\mu\).

• 만약 \(\int g\mbox{ }d\mu=\infty\)라면 trivial하다. 때문에, let \(\int |g|\mbox{ }d\mu < \infty\). This implies \(|g|< \infty\) a.e.

• \(0\le f_n-g \uparrow f-g\) a.e., \(\stackrel{\text{monotone}}{\implies} 0\le \int(f_n-g)d\mu\uparrow \int(f-g)d\mu\stackrel{\text{linearity}}{\implies}-\infty< \int g\mbox{ }d\mu \le \int f_n \mbox{ }d\mu \uparrow \int f\mbox{ }d\mu\).

  

역시 매우 중요

Theorem((Extended) Fatou’s Lemma)

If \(f_n\ge g\) a.e., where \(\int g\mbox{ }d\mu>-\infty\), then \[ \int \liminf_nf_nd\mu\le \liminf_n\int f_nd\mu. \]

• Note that \(\liminf\) is nondecreasing function.

• \(g\le \inf_{k\ge n}f_k \uparrow \liminf_n f_n\) a.e. \(\stackrel{\text{extended M.C.T}}{\implies} \lim_n \int \inf_{k \ge n}f_k \mbox{ }d\mu=\int \liminf_n f_n \mbox{ }d\mu\).

• \(g\le \inf_{k\ge n}f_k \le f_n\) \(\forall n\), \(\int g\mbox{ }d\mu>-\infty\)

  \(\stackrel{\text{ext. monotone}}{\implies}\int \inf_{k\ge n}f_k\mbox{ }d\mu \le \int f_n \mbox{ }d\mu \implies \liminf_n\int \inf_{k\ge n}f_k\mbox{ }d\mu \le \liminf_n\int f_n \mbox{ }d\mu\)(End).

  

매우 중요

Theorem(Lebesgue Dominated Convergence Theorem)

If \(|f_n|\le g\) a.e. for all \(n\ge 1\), where \(g\) is integrable, and if \(f_n\rightarrow f\) a.e., then \(f\) and the \(f_n\) are integrable and \(\int f_n\mbox{ }d\mu \rightarrow \int f\mbox{ }d\mu\).

• 조건 3가지를 잘 기억해야 한다.
  • \(|f_n|\le g\) a.e. for all \(n\ge 1\);
  • \(g\) is integrable;
  • \(f_n\rightarrow f\) a.e.

  

D.C.T의 Variation

Corollary(Bounded Convergence Theorem)

Suppose that \(\mu\) is a finite measure. If \(f_n\rightarrow f\) a.e. and the \(f_n, n\ge 1\) are a.e. uniformly bounded(by some finite constant), then \(\int f_n\mbox{ }d\mu\rightarrow \int f\mbox{ }d\mu\).

• We can take \(g(\omega)=c\) for all \(\omega\), where \(c>0\), \(g\) is the function in D.C.T(결국 단순한 D.C.T의 variation).

• Then, \(0\le \int g\mbox{ }d\mu= \int c\mbox{ }d\mu=c\int I_\Omega\mbox{ }d\mu=c\mu(\Omega)<\infty\).

• Uniformly bounded: \(|f_n|\le c\) a.e \(\iff \mu(\{\omega: |f_n(\omega)>c \})=0\) where \(c\) is not dependent on \(n\).

  

Corollary

If \(\sum_{n=1}^\infty f_n\) converges a.e. and \(|\sum_{k=1}^n f_k|\le g\) a.e. for all \(n\ge 1\), where g is integrable, then both \(\sum_{n=1}^\infty f_n\) and the individual \(f_n\)s are integrable and \(\int\sum_{n=1}^\infty f_n \mbox{ }d\mu=\sum_{n=1}^\infty\int f_n\mbox{ }d\mu\).

• Define \(h_n:=\sum_{k=1}^n f_k\), \(h:=\sum_{n=1}^\infty f_n\).

1. \(|h_n|\le g\) a.e. for all \(n\ge 1\);

2. \(g\) is integrable;

3. \(h_n\rightarrow h\)(\(h\) exists from the given assumption).

• Thus, D.C.T can be applied s.t. \[ \int h_n \mbox{ }d\mu\rightarrow \int h\mbox{ }d\mu \\ \implies \lim_n \int \sum_{k=1}^n f_k\mbox{ }d\mu=\int \sum_{n=1}^\infty f_k\mbox{ }d\mu\\ \implies \lim_n \sum_{k=1}^n \int f_k\mbox{ }d\mu=\int \sum_{n=1}^\infty f_k\mbox{ }d\mu\\ \implies\sum_{n=1}^\infty \int f_n\mbox{ }d\mu=\int \sum_{n=1}^\infty f_k\mbox{ }d\mu. \]

  

  

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