Fields of Sets

Definition(Field)

A class \(\mathcal{F}\) of subsets of \(\Omega\) is a over \(\Omega\) if

  1. \(\Omega \in \mathcal{F}.\)
  2. \(\mathcal{F}\) is closed under complementation: \(A \in \mathcal{F}\) implies \(A^c\in \mathcal{F}\).
  3. \(\mathcal{F}\) is closed under finite unions: \(A,B \in \mathcal{F}\) implies \(A \cup B\in \mathcal{F}\).

  

Remark

Note that 1 could be replaced by

  1. \(\phi \in {F}\),

and 3 replaced by

  1. \(\mathcal{F}\) is closed under finite intersections: \(A,B \in \mathcal{F}\) implies \(A \cap B\in \mathcal{F}\).

  

Definition(\(\sigma\)-Field)

A class \(\mathcal{F}\) of subsets of \(\Omega\) is a if

  1. \(\Omega \in \mathcal{F}.\)
  2. \(\mathcal{F}\) is closed under complementation: \(A \in \mathcal{F}\) implies \(A^c\in \mathcal{F}\).
  3. \(\mathcal{F}\) is closed under countable unions: \(if A_1,A_2, \ldots \in \mathcal{F}\), then \(\cup_n A_n \mathcal{F}\).

  

Remark

If \(\mathcal{F}\) is \(\sigma\)-field, then it is a field. But, the reverse does not hold.

  

Lemma

If \(\mathcal{F}\) is a field and \(\mathcal{F}\) is closed under countable disjoint unions, then \(\mathcal{F}\) is a \(\sigma\)-field.

(You can construct disjoint sets \(B_n, n\ge 1\) which is increasing sequence of sets, \(\cup_n A_n = \uplus_n B_n\). Because \(\mathcal{F}\) is closed under countable disjoint union, \(\cup_n A_n = \uplus_n B_n \in \mathcal{F}\).)

  

Proposition

\(\sigma(\mathcal{C})\) is the smallest \(\sigma\)-field containing \(\mathcal{C}\).

  

Proposition

Note that

  1. If \(\mathcal{F}\) is a \(\sigma\)-field, then \(\sigma(\mathcal{F})= \mathcal{F}\).

  2. If \(\mathcal{C} \subset \mathcal{D}\), then \(\sigma(\mathcal{C})\subset \sigma(\mathcal{D})\).

  3. If \(\mathcal{C} \subset \mathcal{D} \subset \sigma (\mathcal{C})\), then \(\sigma(\mathcal{C})= \sigma(\mathcal{D})\).

  

Definition(Borel \(\sigma\)-field on \(\mathbb{R}\))

The on \(\mathbb{R}=(-\infty, \infty)\), denoted by \(\mathcal{R}\) is defined to be the smallest \(\sigma\)-field on \(\mathbb{R}\) containing

  1. \(\mathcal{I}_0 = \{(a,b]: -\infty<a\le<b<\infty\}\)
  2. \(\mathcal{I}_1 =\{(-\infty,b]:-\infty<b<\infty\}\)
  3. \(\mathcal{I}_2 =\{(a, \infty):-\infty<a<\infty\}\)
  4. \(\mathcal{I}_3 =\{(-\infty,b):-\infty<b<\infty\}\).
  5. \(\mathcal{I}_4 =\{[a,\infty):-\infty<a<\infty\}\).
  6. \(\mathcal{I}_5 =\{(a,b)-\infty<a\le<b<\infty\}\).
  7. \(\mathcal{T}= \{\mbox{open sets in }\mathbb{R}\}\)
  8. \(\mathcal{S}= \{\mbox{closed sets in }\mathbb{R}\}\)

e.g. \(\sigma(\mathcal{I}_0)= \mathcal{R}.\)

*실수에서의 Entire set \(\Omega\)는 \(\mathbb{R}\)이라고 생각하고 \(\Omega\)의 Subset들을 any interval in the real line \(\mathbb{R}\) 이라고 생각하자. 그렇다면 그 subset들의 class(1번부터 8번까지의 class 중 하나)의 the smallest \(\sigma\)-field를 우리는 Borel \(\sigma\)-field 라고 생각하자.