Definition(measure)

A measure \(\mu\) is a set function defined on a field \(\mathcal{F}\) if

  1. \(\mu\) is nonnegative: \(0\le \mu(A) \le \infty\) for all \(A\in \mathcal{F}\)

  2. \(\mu(\phi)=0\)

  3. \(\mu\) is countably additive: if \(A_1, A_2,\ldots \in \mathcal{F}\) are disjoint and \(\uplus_n A_n \in \mathcal{F}\), then \(\mu (\uplus_n A_{n})= \sum_n \mu (A_{n})\)

  

위 조건을 충족하는 (\(\Omega, \mathcal{F}, \mu\))를 Measurable space라고 부른다.

  

Remark

If \(\mu(\Omega)<\infty\), then \(\mu\) is called a ￩finite measure.

For \(A_1, A_2, \ldots \in \mathcal{F}\), if \(\cup_n A_n = \Omega\) and \(\mu(A_n)<\infty\) for all \(n\), then \(\mu\) is called a \(\sigma\)-finite measure.

  

Definition(counting measure)

Given \(\Omega\) and \(\mathcal{F}\)=\(2^\Omega\), Let \(\mu(A)=\#(A)\), where \(\#(A)\) is the number of points in \(A\) if \(A\) is finite, or \(\infty\) otherwise. Then, \(\mu(A)=\#(A)\) is called Counting measure on \(\Omega\).

  ￣

Definition(discrete measure)

Suppose that \(\Omega\) is countable and \(p:\Omega \rightarrow [0,\infty].\). For any set \(A\in 2^\Omega\), let \(\mu(A)=\sum_{\omega\in A} P(\omega)\), where an empty sum is taken to be zero. Then, \(\mu\) is called a discrete measure.

  

Theorem(fundamental properties of measure)

For any measure \(\mu\) on a field \(\mathcal{F}\) and sets \(A,B,A_1,A_2,\ldots \in \mathcal{F}\).

  1. Monotonicity: If \(A\subset B\), then \(\mu(A) \le \mu(B)\).

  2. Subtractivity: If \(A\subset B\) and \(\mu(A)<\infty\), then \(\mu(B-A)=\mu(B)-\mu(A)\).

  3. Countable subadditivity: If \(\cup_n A_n \in \mathcal{F}\), then \(\mu(\cup_n A_n) \le \sum_n \mu(A_n)\).

  4. Continuity from below: If \(A_n\uparrow A\), then \(\mu(A_n)\uparrow \mu(A)\).

  5. Continuity from above: If \(A_n\downarrow A\) and \(\mu(A_1)<\infty\), then \(\mu(A_n)\downarrow \mu(A)\).

  

  

Theorem

Definition(\(\pi\)-system) A class \(\mathcal{P}\) of subsets of \(\Omega\) is called a \(\pi\)-system if it is closed under finite intersection.

Definition(\(\lambda\)-system) A class \(\mathcal{L}\) of subsets of \(\Omega\) is called a \(\lambda\)-system if

  1. \(\Omega\in \mathcal{L}\);
  2. \(A\in \mathcal{L}\) implies \(A^c \in \mathcal{L}\) (closed under complementation);
  3. For \(A_1,A_2,\ldots \in \mathcal{L}\) and \(A_m\cap A_n=\phi\) \(m\ne n\), \(\uplus_n A_n\in \mathcal{L}\) (closed under countable disjoint union).

This can be replaced by

  1. \(\Omega\in \mathcal{L}\);
  2. \(A, B \in \mathcal{L}\), \(A\subset B\) implies \(B-A \in \mathcal{L}\) (closed under subtraction);
  3. For \(A_1,A_2,\ldots \in \mathcal{L}\), \(A_n\uparrow A\) implies \(A \in \mathcal{L}\).

  

Lemma

A class that is both a \(\pi\)-system and \(\lambda\)-system is a \(\sigma\)-field.

  

Theorem(Dynkin’s \(\pi-\lambda\) theorem)

If \(\mathcal{P}\) is a \(\pi\)-system and \(\mathcal{L}\) is a \(\lambda\)-system, then \(\mathcal{P}\subset \mathcal{L}\) implies \(\sigma(\mathcal{P}) \subset \mathcal{L}\).