1.3 The Probability Set Function

Definition 1.3.1

\(\sigma-field\).

  • Let \(\beta\) be a collection of subset of \(\mathscr{C}\)1.

    We say \(\beta\) is a \(\sigma-field\),

  • if

    1. \(\emptyset \notin \beta\).
    2. \(C\in \beta \Rightarrow C^c\in \beta\). (cloed under complement)
    3. \(C_1, C_2\cdots\in\beta\Rightarrow\bigcup_{i=1}^{\infty}\). (closed under countable union)

Definition 1.3.2

\(Probability\).

  • \(\mathscr{C}\) : Sample Space.
  • \(\beta\) : Borel \(\sigma-field\).
  • P : Real-valued function defined on \(\beta\) Probability set function if it satisfied,
    1. \(P\left(c\right)\ge0, \quad \forall C\in\beta\quad\) (Non Negative)
    2. \(P\left(e\right)=1,\quad\) (Normality)
    3. \(C_1, C_2, \cdots\in\beta, \quad C_m\cap C_n = \emptyset,\quad \forall m\neq n\) \(\Rightarrow P\left(\bigcup_{i=1}^{\infty}C_i\right) = \sum_{i=1}^{\infty}P\left(i\right),\quad\)(Countable additivity)
Theorem 1.3.1
\[\begin{aligned} P\left(c\right) = 1-P\left(C^c\right), \quad \forall C\in\beta \end{aligned}\]
  • Proof.
    • \(\mathscr{C} = C\cup C^c\).
    • \(P\left(\mathscr{C}\right)= P\left(C\right) + P\left(C^c\right)\quad \left(\because\text{Countable Addictivity}\right)\).
    • \(P\left(\mathscr{C}\right) = 1\qquad\qquad\qquad\quad \left(\because Normality\right)\).
Theorem 1.3.2
\[\begin{aligned} P\left(\phi\right) = 0 \end{aligned}\]
  • Proof.
    • \(\mathscr{C}= \mathscr{C}\cup\emptyset\).
    • \(P\left(\mathscr{C}\right) = P\left(\mathscr{C}\right) + P\left(\emptyset\right)\).
Theorem 1.3.3
\[C_1\subset C_2 \Rightarrow P\left(C_1\right)\le P\left(C_2\right)\]
  • Proof.
    • \(C_2 = C_1\cup\left(C_2\cap C_1^c\right)\).
    • \(\phi\ \ = C_1\cap \left(C_1^c\cap C_2\right)\).
    • \(P\left(C_2\right)=P\left(C_1\right)+P\left(C_2\cap C_1^c\right)\).
Theorem 1.3.4
\[0\le P\left(C\right) \le 1, \quad \forall C \le \beta\]
  • Proof.
    • \(\phi \subset C \subset \mathscr{C}\).
    • \(0 = P\left(\phi\right) \le P\left(C\right) \le P\left(\mathscr{C}\right) = 1\).
Theorem 1.3.5
\[\begin{aligned} P\left(C_1\cup C_2\right) = & P\left(C_1\right) + P\left(C_2\right) - P\left(C_1\cap C_2\right) \end{aligned}\]
  • Proof.
\[\begin{aligned} \quad C_1 \cup C_2 = &\ C_1 \cup \left(C_2 \cap C_1^c\right) \\ \quad C_2 = &\ \left(C_1 \cap C_2\right) \cup \left(C_2 \cap C_1^c\right) \\ \quad P\left(C_1 \cup C_2\right) = &\ P\left(C_1\right) + P\left(C_2 \cap C_1^c\right) \\ \quad P\left(C_2\right) = & \ P\left(C_1 \cap C_2\right) + P\left(C_2\cap C_1^c\right) \end{aligned}\]

Remark 1.3.2

\[\left(\text{Inclusion Exclusion Formula}\right)\]

  • For 3 sets \(C_1, C_2, C_3\) it is not difficult to show. \(\begin{aligned} P\left(C_1\cup C_2 \cup C_3\right) = & \ p_1 - p_2 + p_3\\ \text{where}, \quad p_1 =&\ P\left(C_1\right) + P\left(C_2\right) + P\left(C_3\right)\\ p_2 =&\ P\left(C_1\cap C_2\right) + P\left(C_1\cap C_3\right) + P\left(C_2 \cap C_3\right)\\ p_3 =&\ P\left(C_1\cap C_2 \cap C_3\right)\\ \end{aligned}\)

  • In general inclusion exclusion formula, \(P\left(C_1\cup C_2 \cdots \cup C_k\right) = \ p_1 - p_2 + p+3 - \cdots + \left(-1\right)^{k+1}p_k,\)

  • where \(p_i\) is sum of probability of all possible intersection of sets. \(C_1, C_2, \cdots\) are called mutually exclusion.

  • If \(C_i \cap C_j = \phi,\quad \forall i \neq j\), mutually exclusive set \(c_1, c_2, \cdots\) are called exhaustive2. \(\left(\bigcup_{i=1}^{\infty}C_i\right)\)

Theorem 1.3.6 (Continuity of Probabilty set function)

Consider sequence of sets \(C_1, C_2,\cdots, C_n, \cdots\).

\[\begin{aligned} \lim_{n\rightarrow \infty}C_n =&\ \bigcup_{n=1}^{\infty}C_n, \quad \text{if } C_n \text{ is increasing set.} \\ \lim_{n\rightarrow \infty}C_n = &\ \bigcap_{n=1}^{\infty}C_n, \quad \text{if }C_n \text{ is decreasing set.} \end{aligned}\]

  • \(C_n\) : Increasing sequence of sets \(\Rightarrow \ \lim_{n\rightarrow \infty}P\left(C_n\right):=P\left(\lim_{n\rightarrow\infty}C_n\right)=P\left(\bigcup_{n=1}^{\infty}C_n\right)\) .

  • \(C_n\) : Decreasing sequence of sets \(\Rightarrow \ \lim_{n\rightarrow \infty}P\left(C_n\right):=P\left(\lim_{n\rightarrow\infty}C_n\right)=P\left(\bigcap_{n=1}^{\infty}C_n\right)\) .

  • proof

    . \(\begin{aligned} \text{Let } R_1 =&\ C_1 \\ R_n=&\ C_n\cap C_{n-1}, \quad n=2,3,\cdots\\ P\left(\lim_{n\rightarrow \infty}C_n\right) = &\ P\left(\cup_{n=1}^{\infty}C_n\right)\\ = &\ P\left(\cup_{n=1}^{\infty}R_n\right)\\ = &\ \sum_{n=1}^{\infty}P\left(R_n\right)\\ = &\ \lim_{n\rightarrow\infty}\sum_{j=1}^{n}P\left(R_j\right) \\ =&\ \lim_{n\rightarrow\infty}\left\{P\left(R_1\right)+\sum_{j=2}^{n}P\left(R_j\right)\right\}\\ = & \lim_{n\rightarrow\infty}\left\{P\left(R_1\right)+ \sum_{j=2}^{\infty}\left(P\left(C_j\right)-P\left(C_{j-1}\right)\right)\right\}\\ = &\ \lim_{n\rightarrow\infty}P\left(C_n\right) \end{aligned}\)

Theorem 1.3.7 (Bolle’s Inequality)

  • \(\left\{C_n\right\}\) : Arbitrary sequence of sets.

    • \(\Rightarrow P\left(\cup_{n=1}^{\infty}C_n\right)\le \sum_{n=1}^{\infty}P\left(C_n\right)\).

  • proof.

    • Let \(D_n=\ \bigcup_{i=1}^{\infty}C_i\)
    • Thus \(\left\{D_n\right\}\) is increasing Sequence of sets.
    • Since \(D_j = \ D_{j-1}\cup C_j,\quad j=1,2,\cdots\)
    \[\begin{aligned} P\left(D_j\right) =&\ P\left(D_{j-1}\right)P\left(C_j\right)-P\left(D_{j-1}\cap C_j\right)\le P\left(D_{j-1}\right)+P\left(C_j\right) \\ i.e. &\ P\left(D_j\right)-P\left(D_{j-1}\right)\le P\left(C_j\right) \end{aligned}\]
    • Now
    \[\begin{aligned} P\left(\cup_{n=1}^{\infty}C_n\right) = &\ P\left(\cup_{n=1}^{\infty}D_n\right)\\ = &\ \lim_{n\rightarrow\infty}P\left(D_n\right) \\ = &\ \lim_{n\rightarrow\infty}\left\{ P\left(D_1\right)+\sum_{j=2}^n\left(P\left(D_j\right)-P\left(D_{j-1}\right)\right) \right\} \\ \le & \lim_{n\rightarrow\infty}\left\{P\left(D_1\right)+\sum_{n=2}^{\infty}P\left(C_j\right)\right\}-\lim_{j=1}^{n}P\left(C_j\right) \\ =&\ \sum_{n=1}^{\infty}P\left(C_n\right) \end{aligned}\]

  1. Sample Space ↩

  2. partition ↩