Chapter 1
Introduction
Definition 1.6 (Probability measure)
Let $ \Omega$ be a nonempty set and $ {\mathcal F}$ be a $ \sigma$-field of subsets of $ \Omega$. A map $ P: {\mathcal F} \rightarrow [0, \ 1]$ is said to be a probability measure if P satisfies
(i) $ P(\Omega) \ = \ 1,$
(ii) if $ A_1, \, A_2, \cdots \in {\mathcal F}$ are pairwise disjoint, then
$\displaystyle P\Big(\cup^{\infty}_{n=1} A_n \Big) \ = \ \sum^{\infty}_{n=1} P(A_n) \, . $
 
Definition 1.7 (Probability space).
The triplet $ ({\Omega}, {\mathcal F}, P) $; where $ \Omega$, a nonempty set (sample space), $ \mathcal F$, a $ \sigma$-field and $ P$, a probability measure; is called a probability space.